SECTION 6: TIME-DOMAIN ANALYSIS - College of...

ESE 330 ndash Modeling amp Analysis of Dynamic Systems

SECTION 6 TIME-DOMAIN ANALYSIS

K Webb ESE 330

This first sub-section of notes continues where the previous section left off and will explore the difference between the forced and natural responses of a dynamic system

Natural and Forced Responses2

K Webb ESE 330

3

Natural and Forced Responses

In the previous section we used Laplace transforms to determine the response of a system to a step input given zero initial conditions The driven response

Now consider the response of the same system to non-zero initial conditions only The natural response

K Webb ESE 330

4

Natural Response

119910 + 119887119887119898119898119910 + 119896119896

119898119898119910119910 = 0 (1)

Use the derivative property to Laplace transform (1) Allow for non-zero initial-conditions

1199041199042119884119884 119904119904 minus 119904119904119910119910 0 minus 119910 0 + 119887119887119898119898119904119904119884119884 119904119904 minus 119887119887

119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 0 (2)

Same springmassdamper system Set the input to zero Second-order ODE for displacement

of the mass

K Webb ESE 330

5

Natural Response

Solving (2) for 119884119884 119904119904 gives the Laplace transform of the output due solely to initial conditions

Laplace transform of the natural response

119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(3)

Consider the under-damped system with the following initial conditions

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01119898119898119904119904

K Webb ESE 330

6

Natural Response

Substituting component parameters and initial conditions into (3)

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)

Remember itrsquos the roots of the denominator polynomial that dictate the form of the response Real roots ndash decaying exponentials Complex roots ndash decaying sinusoids

For the under-damped case roots are complex Complete the square Partial fraction expansion has the form

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

Multiply both sides of (5) by the denominator of the left-hand side

015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032

Equating coefficients and solving for 1199031199031 and 1199031199032 gives

1199031199031 = 015 1199031199032 = 0115

The Laplace transform of the natural response

119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response

Inverse Laplace transform is the natural response119910119910 119905119905 = 015119890119890minus2119905119905 cos 3464 119905119905 + 0115119890119890minus2119905119905 sin 3464 119905119905 (7)

Under-damped response is the sum of decaying sine and cosine terms

K Webb ESE 330

9

Driven Response with Non-Zero ICrsquos

Now Laplace transform allowing for both non-zero input and initial conditions


119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904

Solving for 119884119884 119904119904 gives the Laplace transform of the response to both the input and the initial conditions

119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10


Laplace transform of the response has two components

119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

Total response is a superposition of the initial condition response and the driven response

Both have the same denominator polynomial Same roots same type of response

Over- under- critically-damped

Natural response - initial conditions Driven response - input

K Webb ESE 330

11


Laplace transform of the total response

119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898

Transform back to time domain via partial fraction expansion

119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2

Solving for the residues gives

1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12


Total response119910119910 119905119905 = 00625 + 00875119890119890minus2119905119905 cos 3464 119905119905 + 00794119890119890minus2119905119905 sin 3464 119905119905

Superposition of two components Natural response

due to initial conditions

Driven response due to the input

K Webb ESE 330

Next wersquoll apply the Laplace transform to the entire state-space model in matrix form just as we did for single differential equations

Solving the State-Space Model13

K Webb ESE 330

14

Solving the State-Space Model

Wersquove seen how to use the Laplace transform to solve individual differential equations

Now wersquoll apply the Laplace transform to the full state-space system model

First wersquoll look at the same simple example Later wersquoll take a more generalized approach

State-space model is

119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909

Note that because this model was derived from a bond graph model the state variables are now momentum and displacement

K Webb ESE 330

15

Laplace Transform of the State-Space Model

For now focus on the state equation Output is a linear combination of states and inputs Determining the state trajectory is the important thing

Use the derivative property to Laplace transform the state equation

119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904

Rearranging to put all transformed state vectors on the left-hand side

119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

Can factor out the transformed state vector from the left-hand side Must multiply 119904119904 by a 2 times 2 identity matrix

119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

Note the form of (3) The LHS is 119904119904119868119868 minus 119860119860 119831119831 119904119904 where 119860119860 is the system matrix Everything on the RHS reduces to a 2 times 1 vector A known matrix times a vector of unknowns equals a

known vector If we can solve for 119875119875 119904119904 andor 119883119883 119904119904 we can inverse

transform to get 119901119901 119905119905 andor 119909119909 119905119905 Use Cramerrsquos Rule

K Webb ESE 330

18

Cramerrsquos Rule

Given a matrix equation

119808119808119808119808 = 119858119858

We can solve for elements of 119808119808 as follows

119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808

The matrix 119808119808119894119894 is formed by replacing the 119894119894119905119905119905 column of 119808119808 with the vector 119858119858

K Webb ESE 330

19


According to Cramerrsquos Rule

119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887

119898119898 119909119909 0 minus minus1119898119898119901119901 0 minus 1

119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)

According to the output equation from (1)119884119884 119904119904 = 119883119883 119904119904

Equation (4) is identical to (8) from the previous subsection of notes which we arrived at differently

K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)

Next sub in parameter values ICrsquos and an input Use PFE to inverse transform to 119910119910(119905119905) Again consider the under-damped system

Let the input be a 1119873119873 step 119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21


The Laplace transform of the output becomes

119884119884 119904119904 =015119904119904 +06 minus minus01minus1119904119904

1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)

Inverse transform via partial fraction expansion

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)

Multiply both sides by left-hand-side denominator0151199041199042 + 07119904119904 + 1 = 11990311990311199041199042 + 41199031199031119904119904 + 161199031199031 + 11990311990321199041199042 + 21199031199032119904119904 + 34641199031199033119904119904

Equating coefficients and solving yields1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

K Webb ESE 330

22


The Laplace transform of the system response is

119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)

The time-domain response is119910119910 119905119905 = 00625 + 00875119890119890minus2119905119905 cos 3464119905119905 + 00794119890119890minus2119905119905 sin 3464119905119905 (9)

Transient portion Due to initial conditions

and input step Decays to zero

Steady-State portion Due to constant input Does not decay

K Webb ESE 330

23


Now wersquoll apply the Laplace transform to the solution of the state-space model in general form

119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809

For now focus on the state equation only Output is derived from states and inputs

Laplace transform of the state equation

119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0

Factoring out the transformed state from the left-hand side

119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

Remember the dimensions of each term in (10) 119904119904119816119816 minus 119808119808 119899119899 times 119899119899 119831119831 119904119904 119899119899 times 1 119904119904119816119816 minus 119808119808 119831119831 119904119904 119899119899 times 1

119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1

Apply Cramerrsquos rule to solve for the Laplace transform of the 119894119894119905119905119905 state variable

119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)

The matrix 119904119904119816119816 minus 119808119808 119894119894 is formed by replacing the 119894119894119905119905119905 column of 119904119904119816119816 minus 119808119808 with 119809119809119809119809 119904119904 + 119808119808 0 An 119899119899 times 1 vector of known values

K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)

Denominator of (11) is the determinant of 119904119904119816119816 minus 119808119808 119904119904119816119816 minus 119808119808 is an 119899119899 times 119899119899 matrix

Each diagonal term is a first-order polynomial in 119904119904 One term in the determinant is the trace of the matrix the product terms along

the diagonal 119904119904119816119816 minus 119808119808 is an 119951119951119957119957119957119957-order polynomial in 119956119956

The characteristic polynomial

Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)

Roots of Δ 119904119904 are values of 119904119904 that satisfy the characteristic equation

Δ 119904119904 = 0 (13) Poles of (11) Eigenvalues of system matrix 119808119808

K Webb ESE 330

26


Denominator of every state variablersquos Laplace transform contains the characteristic polynomial

119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)

A characteristic of the system Remember denominator roots (ie poles) determine the nature of

the response Real roots ndash decaying exponentials Complex roots ndash decaying sinusoids

Responses of all state variables have same components Numerators of transforms determine the differences

Output transform has the same denominator Δ 119904119904 Linear combination of states and input Response includes the same sinusoidal andor exponential components

K Webb ESE 330

27


Assume zero initial conditions 119808119808 0 = 120782120782 SISO system single input ndash 119880119880 119904119904 is a scalar transform

Can factor out the input from the numerator of (14)

119904119904119816119816 minus 119808119808 119894119894 = 119880119880 119904119904 119904119904119816119816 minus 119808119808 119894119894lowast

where 119904119904119816119816 minus 119808119808 119894119894lowast is the 119899119899 times 119899119899 matrix formed by replacing the 119894119894119905119905119905 column of 119904119904119816119816 minus 119808119808 with the 119899119899 times 1vector 119809119809 119880119880 119904119904 appears in every term of one column of 119904119904119816119816 minus 119808119808 119894119894 119880119880 119904119904 appears in every term of the determinant

K Webb ESE 330

28


Can now write the Laplace transform of the state variable response as

119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)

119873119873119873119873119898119898119894119894 119904119904 is in general different for each state variable At most an 119899119899 minus 1 119904119904119905119905 order polynomial in 119904119904

Components of every state variable (and output) response determined by The characteristic polynomial Δ 119904119904 The input 119880119880 119904119904

Numerator 119873119873119873119873119898119898119894119894 119904119904 determines exact response Weighting of each sinusoidal andor exponential component

K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)

Laplace transform of each state variable response 119883119883119894119894 119904119904 is the Laplace transform of the input scaled by

119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904

In the next sub-section wersquoll explore a related concept ndashtransfer functions

119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330

Transfer Functions30

K Webb ESE 330

31

Transfer Functions

Now come back to the full state-space model including the output equation ndash (SISO case assumed here - 119873119873 and 119910119910 are scalars)

119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873

Assume zero initial conditions and Laplace transform the whole model

119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)

Simplify the state equation as before

119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904

Solving for the state vector

119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions

Substituting (3) into (2) gives the Laplace transform of the output

119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904

Factoring out the input119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D 119880119880 119904119904 (4)

Transform of the output is the input scaled by the stuff in the square brackets

Dividing through by the input gives the transfer function

119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)

Ratio of systemrsquos output to input in the Laplace domain assuming zero initial conditions

An alternative to the state-space (time-domain) model for mathematically representing a system

K Webb ESE 330

33

Transfer Matrix ndash MIMO Systems

For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809

119898119898 inputs 119809119809 is 119898119898 times 1 119809119809 is 119899119899 times 119898119898 119901119901 outputs 119858119858 is 119901119901 times 1 119810119810 is 119901119901 times 119899119899

Transfer function becomes a 119953119953 times 119950119950 matrix

119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811

Transfer function 119866119866119894119894119894119894 119904119904 relates the 119894119894119905119905119905 output to the 119895119895119905119905119905input

119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904

Wersquoll continue to assume SISO systems in this course

K Webb ESE 330

34

Transfer Functions

System output in the Laplace domain is the input multiplied by the transfer function

119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904

We saw earlier that state variables are given by

119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

where Δ 119904119904 = 119904119904119816119816 minus 119808119808 is the characteristic polynomial

Output is linear combination of states and input so wersquod expect the denominator of 119866119866 119904119904 to be Δ 119904119904 as well Is it What is the denominator of 119866119866 119904119904

K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)

The matrix inverse term in (5) is given by

119904119904119816119816 minus 119808119808 minus1 =119886119886119886119886119895119895 119904119904119816119816 minus 119808119808119904119904119816119816 minus 119808119808

where the numerator is the adjoint of 119904119904119816119816 minus 119808119808 Equation (5) can be rewritten as

119866119866 119904119904 = 119810119810 119886119886119886119886119894119894 119904119904119816119816minus119808119808 119809119809+D 119904119904119816119816minus119808119808119904119904119816119816minus119808119808

(6)

Transfer function denominator is the characteristic polynomial Poles of the transfer function are roots of Δ 119904119904

System poles or eigenvalues Eigenvalues of the system matrix 119808119808 Along with the input system poles determine the nature of the time-domain

response

K Webb ESE 330

This sub-section of notes takes a bit of a tangent to explain the use of the term eigenvalues when referring to system poles

Eigenvalues36

K Webb ESE 330

37

Eigenvalues Wersquove been using the term eigenvalue when referring to system poles ndash why Recall from linear algebra the eigenvalue problem

119808119808119808119808 = 120582120582119808119808 (1)

where 119808119808 is an 119899119899 times 119899119899 matrix119808119808 is an 119899119899 times 1 vector ndash an eigenvector120582120582 is a scalar ndash an eigenvalue

Eigenvalue problem involves finding both the eigenvalues and the eigenvectorsthat satisfy (1)

Eigenvalues and eigenvectors are specific to (characteristics of) the matrix 119808119808 An 119899119899 times 119899119899 matrix will have at most 119899119899 eigenvalues and 119899119899 corresponding eigenvectors

Equation (1) says An 119899119899 times 1 eigenvector 119808119808 left-multiplied by an 119899119899 times 119899119899 matrix 119808119808 results in an 119899119899 times 1 vector The resulting vector is the eigenvector scaled by the eigenvalue 120582120582 Result is in the same direction as 119808119808 ndash ie not rotated

K Webb ESE 330

38

Eigenvalues and Eigenvectors

Geometrically multiplication of a vector by a matrix results in two things Scaling and rotation

Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1

0 Compute the product

119858119858 = 119808119808119808119808

In both cases results have different magnitudes and different directions

K Webb ESE 330

39


Multiplication of a matrix and one of its eigenvectors results in scaling only No rotation

The 2 times 2 matrix

119808119808 = 1 34 2

has two eigenvectors (normalized)

1198081198081 = minus07070707 and 1198081198082 = minus06

minus08and two corresponding eigenvalues

λ1 = minus2 and λ2 = 5

such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40


A full-rank 119899119899 times 119899119899 matrix will have 119899119899 pairs of eigenvalues and eigenvectors

To find all eigenvalues and eigenvectors that satisfy (1)

119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782

and factor out the eigenvector term

120582120582119816119816 minus 119808119808 119808119808 = 0 (2)

If 120582120582119816119816 minus 119808119808 minus1 exists then 119808119808 = 120782120782 which is the trivial solution and of no interest

Wersquore interested in values of 120582120582 and 119808119808 that satisfy (2) when 120582120582119816119816 minus 119808119808 is not invertible ndash when it is singular

K Webb ESE 330

41


Want to find values of 120582120582 for which (120582120582119816119816 minus 119808119808) is singular A matrix is singular if its determinant is zero

120582120582119816119816 minus 119808119808 = 0 (3)

Equation (3) is the characteristic equation for 119808119808 120582120582119816119816 minus 119808119808 is the characteristic polynomial Δ 120582120582 An 119899119899119905119905119905-order polynomial in 120582120582

Eigenvalues of matrix 119808119808 are all 119899119899 values of 120582120582 that satisfy (3) Roots of the characteristic polynomial Find the corresponding eigenvectors by substituting 120582120582 into (2) and

solving for 119808119808

Letting 120582120582 = 119904119904 (3) becomes the denominator of the system transfer function 119866119866 119904119904

K Webb ESE 330

Using the Transfer Function to Determine System Response42

K Webb ESE 330

43

Using 119866119866 119904119904 to determine System Response

System output in the Laplace domain can be expressed in terms of the transfer function as

119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)

Laplace-domain output is the product of the Laplace-domain input and the transfer function

Response to two specific types of inputs often used to characterize dynamic systems Impulse response Step response

Wersquoll use the approach of (1) to determine these responses

K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0

intminusinfininfin 120575120575 119905119905 119886119886119905119905 = 1

Laplace transform of the impulse function is

ℒ 120575120575 119905119905 = 1

Impulse response in the Laplace domain is

119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904

The transfer function is the Laplace transform of the impulse response Impulse response in the time domain is the inverse transform of the

transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response

Step function 1 119905119905 = 0 119905119905 lt 01 119905119905 ge 0

Laplace transform of the step function

ℒ 1 119905119905 = 1119904119904

Laplace-domain step response

119884119884 119904119904 = 1119904119904 119866119866 119904119904

Time-domain step response

119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904

Recall the integral property of the Laplace transform

ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591

The step response is the integral of the impulse response

K Webb ESE 330

46

First- and Second-Order Systems

All transfer functions can be decomposed into 1st- and 2nd-order terms by factoring Δ 119904119904 Real poles ndash 1st-order terms Complex-conjugate poles ndash 2nd-order terms

These terms and therefore the poles determine the nature of the time-domain response Real poles ndash decaying exponentials Complex-conjugate poles - decaying sinusoids

All time-domain responses will be a superposition of decaying exponentials and decaying sinusoids These are the natural modes or eigenmodes of the system

Instructive to examine the responses of 1st- and 2nd-order systems Gain insight into relationships between pole location and response

K Webb ESE 330

Response of First-Order Systems47

K Webb ESE 330

48

First-Order System ndash Impulse Response

First-order transfer function

119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591

where 120591120591 is the system time constant Impulse response

119896119896 119905119905 = ℒminus1 119866119866 119904119904 = 119860119860119890119890minus120590120590119905119905 = 119860119860119890119890minus119905119905120591120591

119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49


Initial slope is inversely proportional to time constant

Response completes 63 of transition after one time constant

Decays to zero as long as the pole is negative

K Webb ESE 330

50

Impulse Response vs Pole Location

Increasing 120590120590 corresponds to decreasing 120591120591 and a faster response

K Webb ESE 330

51

First-Order System ndash Step Response Step response in the Laplace domain

119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590

Inverse transform back to time domain via partial fraction expansion

119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590


119910119910 119905119905 =119860119860120590120590minus119860119860120590120590119890119890minus120590120590119905119905 = 119861119861 minus 119861119861119890119890minus

119905119905120591120591

K Webb ESE 330

52

First-Order System ndash Step Response



Almost completely settled after 7120591120591

K Webb ESE 330

53

Step Response vs Pole Location


K Webb ESE 330

54

Pole Location and Stability


119866119866 119904119904 =119860119860

119904119904 minus 119901119901

where 119901119901 is the system pole Impulse response is

119896119896 119905119905 = 119860119860119890119890119901119901119905119905

If 119901119901 lt 0 119896119896 119905119905 decays to zero Pole in the left half-plane System is stable

If 119901119901 gt 0 119896119896 119905119905 grows without bound Pole in the right half-plane System is unstable

K Webb ESE 330

Response of Second-Order Systems55

K Webb ESE 330

56

Second-Order Systems

Second-order transfer function

119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)

where 120596120596119886119886 is the damped natural frequency

Can also express the 2nd-order transfer function as

119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)

where 120596120596119894119894 is the un-damped natural frequency and 120577120577 is the damping ratio

120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894

Two poles at 11990411990412 = minus120590120590 plusmn 1205901205902 minus 1205961205961198941198942 = minus120577120577120596120596119894119894 plusmn 120596120596119894119894 1205771205772 minus 1

K Webb ESE 330

57

Categories of Second-Order Systems The 2nd-order system poles are

11990411990412 = minus120577120577120596120596119894119894 plusmn 120596120596119894119894 1205771205772 minus 1

Value of 120577120577 determines the nature of the poles and therefore the response

120635120635 gt 120783120783 Over-damped Two distinct real poles ndash sum of decaying exponentials ndash treat as two first-order terms 1199041199041 = minus1205901205901 1199041199042 = minus1205901205902

120635120635 = 120783120783 Critically-damped Two identical real poles ndash time-scaled decaying exponentials 11990411990412 = minus120590120590 = minus120577120577120596120596119894119894 = minus120596120596119894119894

120782120782 lt 120635120635 lt 120783120783 Under-damped Complex-conjugate pair of poles ndash sum of decaying sinusoids 11990411990412 = minus120590120590 plusmn 119895119895120596120596119886119886 = minus120577120577120596120596119894119894 plusmn 119895119895120596120596119894119894 1 minus 1205771205772

120635120635 = 120782120782 Un-damped Purely-imaginary conjugate pair of poles ndash sum of non-decaying sinusoids 11990411990412 = plusmn119895119895120596120596119894119894

K Webb ESE 330

58

2nd-Order Pole Locations and Damping

K Webb ESE 330

59

Second-Order Poles - 0 le 120577120577 le 1

Can relate 120590120590 120596120596119886119886 120596120596119894119894 and 120577120577to pole location geometry

120596120596119894119894 is the magnitude of the poles

120577120577 is a measure of system damping

120577120577 = 120590120590120596120596119894119894

= sin 120579120579

120577120577 = 0 Two purely imaginary poles

120577120577 = 1 Two identical real poles

K Webb ESE 330

60

Impulse Response ndash Critically-Damped

For 120577120577 = 1 the transfer function reduces to

119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2

Impulse response119896119896 119905119905 = ℒminus1 119866119866 119904119904

119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61


Speed of response is proportional to 120590120590

K Webb ESE 330

62

Impulse Response ndash Under-Damped For 0 lt 120577120577 lt 1 the transfer function is

119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942

Complete the square on the denominator

119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2

Rewrite in the form of a damped sinusoid

119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862

Inverse Laplace transform for the time-domain impulse response

119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63

Under-Damped Impulse Response vs 120596120596119894119894

119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905 = 119861119861119890119890minus120577120577120596120596119894119894119905119905 sin 120596120596119894119894 1 minus 1205771205772119905119905

K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65

Impulse Response ndash Un-Damped


119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942

Putting into the form of a sinusoid

119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942

Inverse transform to get the time-domain impulse response

119896119896 119905119905 = ℒminus1 119866119866 119904119904

An un-damped sinusoid

119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66

Un-Damped Impulse Response vs 120596120596119894119894

119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67

Second-Order Step Response

The Laplace transform of the step response is

119884119884 119904119904 =1119904119904119866119866 119904119904

The time-domain step response for each damping case can be derived as the the inverse transform of 119884119884 119904119904

119910119910 119905119905 = ℒminus1 119884119884 119904119904

or as the integral of the corresponding impulse response

119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68

Critically-Damped Step Response vs 120590120590

119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902

1 minus 119890119890minus120590120590119905119905 minus 120590120590119905119905119890119890minus120590120590119905119905

K Webb ESE 330

69

Under-Damped Step Response vs 120596120596119894119894

119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus 119890119890minus120590120590119905119905 cos 120596120596119886119886119905119905 minus120590120590120596120596119886119886

119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71

Un-Damped Step Response vs 120596120596119894119894

119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330

Step Response Characteristics72

K Webb ESE 330

73

Step Response ndash Risetime

Risetime is the time it takes a signal to transition between two set levels Typically 10 to 90

of full transition Sometimes 20 to

80

A measure of the speed of a response

Very rough approximation

119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74

Step Response ndash Overshoot

Overshoot is the magnitude of a signalrsquos excursion beyond its final value Expressed as a

percentage of full-scale swing

Overshoot increases as 120577120577 decreases

120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75

Step Response ndashSettling Time

Settling time is the time it takes a signal to settle finally to within some percentage of its final value Typically plusmn1 or plusmn5

Inversely proportional to the real part of the poles 120590120590

For plusmn1 settling

119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330

In this sub-section wersquoll see that the time-domain output of a system is given by the convolution of its time-domain input and its impulse response

The Convolution Integral76

K Webb ESE 330

77

Convolution Integral

Laplace transform of a system output is given by the product of the transform of the input signal and the transfer function

119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904

Recall that multiplication in the Laplace domain corresponds to convolution in the time domain

119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905

Time-domain output given by the convolution of the input signal and the impulse response

119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution

Time-domain output is the input convolved with the impulse response

119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

Input is flipped in time and shifted by 119905119905

Multiply impulse response and flippedshifted input

Integrate over 120591120591 = 0 hellip 119905119905

Each time point of 119910119910 119905119905 given by result of integral with 119873119873 minus120591120591 shifted by 119905119905

K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330

A few of MATLABrsquos many built-in functions that are useful for simulating linear systems are listed in the following sub-section

Time-Domain Analysis in MATLAB81

K Webb ESE 330

82

System Objects

MATLAB has data types dedicated to linear system models

Two primary system model objects State-space model Transfer function model

Objects created by calling MATLAB functions ssm ndash creates a state-space model tfm ndash creates a transfer function model

K Webb ESE 330

83

State-Space Model ndash ss(hellip)

sys = ss(ABCD)

A system matrix - 119899119899 times 119899119899 B input matrix - 119899119899 times 119898119898 C output matrix - 119901119901 times 119899119899 D feed-through matrix - 119901119901 times 119898119898 sys state-space model object

State-space model object will be used as an input to other MATLAB functions

K Webb ESE 330

84

Transfer Function Model ndash tf(hellip)

sys = tf(NumDen)

Num vector of numerator polynomial coefficients Den vector of denominator polynomial coefficients sys transfer function model object

Transfer function is assumed to be of the form

119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1

Inputs to tf(hellip) are Num = [b1b2hellipbr+1] Den = [a1a2hellipan+1]

K Webb ESE 330

85

Step Response Simulation ndash step(hellip)

[yt] = step(syst)

sys system model ndash state-space or transfer function t optional time vector or final time value y output step response t output time vector

If no outputs are specified step response is automatically plotted

Time vector (or final value) input is optional If not specified MATLAB will generate automatically

K Webb ESE 330

86

Impulse Response Simulation ndash impulse(hellip)

[yt] = impulse(syst)

sys system model ndash state-space or transfer function t optional time vector or final time value y output impulse response t output time vector

If no outputs are specified impulse response is automatically plotted


K Webb ESE 330

87

Natural Response ndash initial(hellip)

[ytx] = initial(sysx0t)

sys state-space system model function x0 initial value of the state - 119899119899 times 1 vector t optional time vector or final time value y response to initial conditions - length(t)times 1 vector t output time vector x trajectory of all states - length(t)times 119899119899 matrix

If no outputs are specified response to initial conditions is automatically plotted


K Webb ESE 330

88

Linear System Simulation ndash lsim(hellip)

[ytx] = lsim(sysutx0)

sys system model ndash state-space or transfer function u input signal vector t time vector corresponding to the input signal x0 optional initial conditions ndash (for ss model only) y output response t output time vector x optional trajectory of all states ndash (for ss model only)

If no outputs are specified response is automatically plotted

Input can be any arbitrary signal

K Webb ESE 330

89

More MATLAB Functions

A few more useful MATLAB functions

Polezero analysis pzmap(hellip) pole(hellip) zero(hellip) eig(hellip)

Input signal generation gensig(hellip)

Refer to MATLAB help documentation for more information

Section 6 Time-Domain Analysis



Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues





Using the Transfer Function to Determine System Response


Impulse response

Step Response


Response of First-Order Systems








Response of Second-Order Systems


Categories of Second-Order Systems


Second-Order Poles - 0le120577le1



Impulse Response ndash Under-Damped

Under-Damped Impulse Response vs 120596 119899



Un-Damped Impulse Response vs 120596 119899



Under-Damped Step Response vs 120596 119899


Un-Damped Step Response vs 120596 119899

Step Response Characteristics




The Convolution Integral


Convolution

Convolution

Convolution

Time-Domain Analysis in MATLAB

System Objects








K Webb ESE 330

This first sub-section of notes continues where the previous section left off and will explore the difference between the forced and natural responses of a dynamic system

Natural and Forced Responses2

K Webb ESE 330

3




K Webb ESE 330

4

Natural Response

119910 + 119887119887119898119898119910 + 119896119896

119898119898119910119910 = 0 (1)



119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 0 (2)


of the mass

K Webb ESE 330

5

Natural Response



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(3)


119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01119898119898119904119904

K Webb ESE 330

6

Natural Response


119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)



119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

3




K Webb ESE 330

4

Natural Response

119910 + 119887119887119898119898119910 + 119896119896

119898119898119910119910 = 0 (1)



119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 0 (2)


of the mass

K Webb ESE 330

5

Natural Response



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(3)


119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01119898119898119904119904

K Webb ESE 330

6

Natural Response


119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)



119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

4

Natural Response

119910 + 119887119887119898119898119910 + 119896119896

119898119898119910119910 = 0 (1)



119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 0 (2)


of the mass

K Webb ESE 330

5

Natural Response



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(3)


119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01119898119898119904119904

K Webb ESE 330

6

Natural Response


119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)



119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

5

Natural Response



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(3)


119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01119898119898119904119904

K Webb ESE 330

6

Natural Response


119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)



119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

6

Natural Response


119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

(4)



119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)

K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

7

Natural Response

119884119884 119904119904 = 015119904119904 +071199041199042+4119904119904+16

= 1199031199031 119904119904+2 +1199031199032 3464119904119904+2 2+ 3464 2 (5)


015119904119904 + 07 = 1199031199031119904119904 + 21199031199031 + 34641199031199032


1199031199031 = 015 1199031199032 = 0115


119884119884 119904119904 = 015 119904119904+2119904119904+2 2+ 3464 2 + 0115 3464

119904119904+2 2+ 3464 2 (6)

K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

8

Natural Response



K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

9




119898119898119910119910 0 + 119896119896

119898119898119884119884 119904119904 = 1

119898119898119865119865119894119894119894119894 119904119904


119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0 +1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(8)

119910 +119887119887119898119898119910 +

119896119896119898119898119910119910 =

1119898119898119865119865119894119894119894119894 119905119905

K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

10



119884119884 119904119904 =119904119904 119910119910 0 + 119910 0 + 119887119887

119898119898119910119910 0

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

+1119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898





K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

11



119884119884 119904119904 =015119904119904 + 07 + 1

1199041199041199041199042 + 119887119887

119898119898 119904119904 + 119896119896119898119898

=0151199041199042 + 07119904119904 + 1

119904119904 1199041199042 + 119887119887119898119898 119904119904 + 119896119896

119898119898


119884119884 119904119904 =1199031199031119904119904 +

1199031199032 119904119904 + 2119904119904 + 2 2 + 3464 2 +

1199031199033 3464119904119904 + 2 2 + 3464 2


1199031199031 = 00625 1199031199032 = 00875 1199031199033 = 00794

119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119910119910 0 = 015 119898119898

119910 0 = 01 119898119898119904119904

119865119865119894119894119894119894 119905119905 = 1119873119873 1 119905119905

K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

12






K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330



K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

14






119901119909 =

minus119887119887119898119898 minus119896119896

1119898119898 0

119901119901119909119909 + 1

0 119865119865119894119894119894119894 119905119905

(1)119910119910 = 0 1

119901119901119909119909


K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

15




119904119904 119875119875 119904119904119883119883 119904119904 minus 119901119901 0

119909119909 0 =minus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 + 1

0 119865119865119894119894119894119894 119904119904


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)

K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

16


119904119904 119875119875 119904119904119883119883 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (2)


119904119904119868119868 minusminus 119887119887

119898119898minus119896119896

1119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 00 119904119904 minus

minus 119887119887119898119898

minus1198961198961119898119898

0119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904

119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)

K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

17


119904119904 + 119887119887119898119898

119896119896

minus 1119898119898

119904119904119875119875 119904119904119883119883 119904119904 = 119901119901 0

119909119909 0 + 10 119865119865119894119894119894119894 119904119904 (3)




K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

18

Cramerrsquos Rule


119808119808119808119808 = 119858119858


119909119909119894119894 =det 119808119808119894119894det 119808119808

=119808119808119894119894119808119808


K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

19



119883119883 119904119904 =

119904119904+119887119887119898119898 119901119901 0 +119865119865119894119894119894119894 119904119904

minus 1119898119898 119909119909 0

119904119904+119887119887119898119898 119896119896

minus1119898119898 119904119904

119883119883 119904119904 =119904119904 119909119909 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(4)



K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

20


119884119884 119904119904 =119904119904 119910119910 0 +119887119887


119898119898119865119865119894119894119894119894 119904119904

1199041199042+119887119887119898119898119904119904+

119896119896119898119898

(5)



119898119898 = 1 119896119896119896119896

119896119896 = 16 119873119873119898119898

119887119887 = 4 119873119873119904119904119898119898

119909119909 0 = 015 119898119898

119901119901 0 = 01 119873119873 119904119904

K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

21




1199041199042+4119904119904+16

119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

(6)


119884119884 119904119904 = 0151199041199042 +07119904119904+1119904119904 1199041199042+4119904119904+16

= 1199031199031119904119904

+ 1199031199032 119904119904+2 +1199031199033 3464119904119904+2 2+ 3464 2 (7)



K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

22



119884119884 119904119904 = 00625119904119904

+ 00875 119904119904+2119904119904+2 2+ 3464 2 + 00794 3464

119904119904+2 2+ 3464 2 (8)





K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

23



119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119904119904119831119831 119904119904 minus 119808119808 0 = 119808119808119831119831 119904119904 + 119809119809119809119809 119904119904

Rearranging

119904119904119831119831 119904119904 minus 119808119808119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)

K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

24


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119809119809 119904119904 + 119808119808 0 (10)


119809119809119809119809 119904119904 119899119899 times 1 119808119808 0 119899119899 times 1


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)


K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

25


119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894119904119904119816119816minus119808119808

(11)





Δ 119904119904 = 119904119904119816119816 minus 119808119808 (12)



K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

26



119883119883119894119894 119904119904 = 119904119904119816119816minus119808119808 119894119894Δ 119904119904

(14)





K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

27






K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

28



119883119883119894119894 119904119904 = 119880119880 119904119904 119904119904119816119816minus119808119808 119894119894lowast

119904119904119904119904minus119860119860= 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904

Δ 119904119904(15)




K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

29


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

(15)


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

119880119880 119904119904 119883119883119894119894 119904119904


119873119873119873119873119898119898119894119894 119904119904Δ 119904119904

K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

31

Transfer Functions


119808 = 119808119808119808119808 + 119809119809119873119873119910119910 = 119810119810119808119808 + D119873119873


119904119904119831119831 119904119904 = 119808119808119831119831 119904119904 + 119809119809119880119880 119904119904 (1)

119884119884 119904119904 = 119810119810119831119831 119904119904 + D119880119880 119904119904 (2)


119904119904119816119816 minus 119808119808 119831119831 119904119904 = 119809119809119880119880 119904119904


119831119831 119904119904 = 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) (3)

K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

32

Transfer Functions


119884119884 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809119880119880(119904119904) + D119880119880 119904119904




119866119866 119904119904 = 119884119884 119904119904119880119880 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)



K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

33


For MIMO systems119808 = 119808119808119808119808 + 119809119809119809119809119858119858 = 119810119810119808119808 + 119811119811119809119809



119814119814 119904119904 = 119832119832 119904119904119809119809 119904119904

= 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + 119811119811


119866119866119894119894119894119894 119904119904 = 119884119884119894119894 119904119904119880119880119895119895 119904119904


K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

34

Transfer Functions


119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904


119883119883119894119894 119904119904 = 119880119880 119904119904 119873119873119873119873119898119898119894119894 119904119904Δ 119904119904



K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

35

Transfer Functions

119866119866 119904119904 = 119810119810 119904119904119816119816 minus 119808119808 minus1119809119809 + D (5)





(6)



response

K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


Eigenvalues36

K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

37


119808119808119808119808 = 120582120582119808119808 (1)





K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

38



Consider the matrix

119808119808 = 1 34 2

And the vectors

1198081198081 = 11 1198081198082 = 1


119858119858 = 119808119808119808119808


K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

39




119808119808 = 1 34 2


1198081198081 = minus07070707 and 1198081198082 = minus06



such that

119808119808119808119808120783120783 = 120582120582120783120783119808119808120783120783 and 119808119808119808119808120784120784 = 120582120582120784120784119808119808120784120784

K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

40




119808119808119808119808 = 120582120582119808119808 (1)rearrange

120582120582119808119808 minus 119808119808119808119808 = 120782120782


120582120582119816119816 minus 119808119808 119808119808 = 0 (2)



K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

41



120582120582119816119816 minus 119808119808 = 0 (3)





K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

43



119884119884 119904119904 = 119880119880 119904119904 119866119866 119904119904 (1)




K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

44

Impulse response

Impulse function

120575120575 119905119905 = 0 119905119905 ne 0



ℒ 120575120575 119905119905 = 1


119884119884 119904119904 = 1 119866119866 119904119904 = 119866119866 119904119904


transfer function

119910119910 119905119905 = 119896119896 119905119905 = ℒminus1 119866119866 119904119904

K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

45

Step Response



ℒ 1 119905119905 = 1119904119904


119884119884 119904119904 = 1119904119904 119866119866 119904119904


119910119910 119905119905 = ℒminus1 1119904119904 119866119866 119904119904


ℒ int0119905119905 119896119896 120591120591 119886119886120591120591 = 1

119904119904 119866119866 119904119904 and ℒminus1 1

119904119904 119866119866 119904119904 = int0

119905119905 119896119896 120591120591 119886119886120591120591


K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

46






K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

48



119866119866 119904119904 = 119860119860119904119904+120590120590

Single real pole at

119904119904 = minus120590120590 = minus1120591120591



119896119896 119905119905 = 119860119860119890119890minus119905119905120591120591

K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

49





K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

50



K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

51


119884119884 119904119904 = 1119904119904 119866119866 119904119904 = 119860119860

119904119904 119904119904+120590120590


119884119884 119904119904 = 119860119860119904119904 119904119904+120590120590

= 1199031199031119904119904

+ 1199031199032119904119904+120590120590

119860119860 = 1199031199031 + 1199031199032 119904119904 + 1205901205901199031199031

1199041199040 1205901205901199031199031 = 119860119860 rarr 1199031199031 = 119860119860120590120590

1199041199041 1199031199031 + 1199031199032 = 0 rarr 1199031199032 = minus119860119860120590120590

119884119884 119904119904 = 119860119860120590120590119904119904minus 119860119860120590120590

119904119904+120590120590



119905119905120591120591

K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

52





K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

53



K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

54



119866119866 119904119904 =119860119860

119904119904 minus 119901119901


119896119896 119905119905 = 119860119860119890119890119901119901119905119905



K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

56



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+1198861198861119904119904+1198861198860

= 119873119873119873119873119898119898 119904119904119904119904+120590120590 2+120596120596119889119889

2 (1)



119866119866 119904119904 = 119873119873119873119873119898119898 1199041199041199041199042+2120577120577120596120596119894119894119904119904+120596120596119894119894

2 (2)


120596120596119886119886 = 120596120596119894119894 1 minus 1205771205772

120577120577 = 120590120590120596120596119894119894


K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

57








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

58


K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

59





120577120577 = 120590120590120596120596119894119894

= sin 120579120579



K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

60



119866119866 119904119904 =119860119860

1199041199042 + 2120596120596119894119894119904119904 + 1205961205961198941198942=

119860119860119904119904 + 120596120596119894119894 2 =

119860119860119904119904 + 120590120590 2


119896119896 119905119905 = 119860119860119905119905119890119890minus120590120590119905119905

K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

61



K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

62


119866119866 119904119904 =119860119860

1199041199042 + 2120577120577120596120596119894119894119904119904 + 1205961205961198941198942


119866119866 119904119904 =119860119860

119904119904 + 120577120577120596120596119894119894 2 + 120596120596119894119894 1 minus 12057712057722 =

119860119860119904119904 + 120577120577120596120596119894119894 2 + 120596120596119886119886

2


119866119866 119904119904 =119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120577120577120596120596119894119894 2 + 1205961205961198861198862 =

119860119860120596120596119886119886

120596120596119886119886

119904119904 + 120590120590 2 + 1205961205961198861198862


119896119896 119905119905 =119860119860120596120596119886119886

119890119890minus120590120590119905119905sin(120596120596119886119886119905119905)

K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

63


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

64


119896119896 119905119905 =119860119860120596120596119886119886


K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

65



119866119866 119904119904 =119860119860

1199041199042 + 1205961205961198941198942


119866119866 119904119904 =119860119860120596120596119894119894

1205961205961198941198941199041199042 + 1205961205961198941198942


119896119896 119905119905 = ℒminus1 119866119866 119904119904


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

66


119896119896 119905119905 =119860119860120596120596119894119894

sin 120596120596119894119894119905119905

K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

67



119884119884 119904119904 =1119904119904119866119866 119904119904


119910119910 119905119905 = ℒminus1 119884119884 119904119904


119910119910 119905119905 = 0

119905119905119896119896 120591120591 119886119886120591120591

K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

68


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205901205902


K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

69


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

70


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942


119890119890minus120590120590119905119905 sin 120596120596119886119886119905119905

K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

71


119910119910 119905119905 = ℒminus11119904119904119866119866 119904119904 =

1198601198601205961205961198941198942

1 minus cos 120596120596119894119894119905119905

K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330


K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

73




80



119905119905119903119903 asymp18120596120596119894119894

K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

74





120635120635 OS

045 20

05 16

06 10

07 5

119874119874119874119874 = 119890119890minus 120577120577120577120577

1minus1205771205772 sdot 100

120577120577 =minus ln 119874119874119874119874

100

1205871205872 + ln2 119874119874119874119874100

K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

75





119905119905119904119904 asymp46120590120590

= 46120577120577120596120596119894119894

K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330



K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

77



119884119884 119904119904 = 119866119866 119904119904 119880119880 119904119904


119910119910 119905119905 = ℒminus1 119866119866 119904119904 119880119880 119904119904 = 119896119896 119905119905 lowast 119873119873 119905119905


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = int0119905119905 119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591

K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

78

Convolution


119910119910 119905119905 = 119896119896 119905119905 lowast 119873119873 119905119905 = 0

119905119905119896119896 120591120591 119873119873 119905119905 minus 120591120591 119886119886120591120591





K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

79

Convolution

K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

80

Convolution

K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330



K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

82

System Objects




K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

83


sys = ss(ABCD)



K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

84


sys = tf(NumDen)



119866119866 119904119904 =1198871198871119904119904119903119903 + 1198871198872119904119904119903119903minus1 + ⋯+ 119887119887119903119903119904119904 + 119887119887119903119903+11198861198861119904119904119894119894 + 1198861198862119904119904119894119894minus1 + ⋯+ 119886119886119894119894119904119904 + 119886119886119894119894+1


K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

85


[yt] = step(syst)




K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

86






K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

87






K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

88






K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








K Webb ESE 330

89









Natural Response

Natural Response

Natural Response

Natural Response

Natural Response










Cramerrsquos Rule












Transfer Functions

Transfer Functions

Transfer Functions


Transfer Functions

Transfer Functions

Eigenvalues

Eigenvalues







Impulse response

Step Response

































Convolution

Convolution

Convolution


System Objects








SECTION 6: TIME-DOMAIN ANALYSIS - College of...

Documents

Transcript of SECTION 6: TIME-DOMAIN ANALYSIS - College of...