Mathematical Tools: Linear Systems Analysis Linear …uprm.edu/id-igert/forms/courses/Mathematical...

Ángel J. Alicea Rodríguez, MEPhD Candidate in Civil Engineering

IGERT Fellow NEU-UPRM

Mathematical Tools: Linear Systems AnalysisA short course on:

Linear State-Space Control Systems

University of Puerto RicoMayagüez Campus

Department of Civil Engineering and Surveying

Friday, June 26, 2015

1. Control System Basic Concepts2. Control Systems Historical Background3. Advantages of Control Systems4. Mathematical Description of Systems5. Linearization of a System’s Description6. Linear Algebra Review7. State Transformation8. Transfer Function Matrix of a System9. Equivalent State Equations10. System’s Realization11. Stability of Linear Systems12. Stability of Linear Control Systems13. Controllability of Linear Control Systems14. Observability of Linear Control Systems15. Control Systems16. State Feedback in Control Systems17. State Feedback using Estimated States

Course Topics

Textbook: Linear State-Space Control Systems, Williams and Lawrence, Wiley 2007

• “A control system consists of subsystems and processes assembled for the purpose of obtaining a desired output with desired performance, given a specific input.”

-- Norman Nise (2008)

• The objective of a control system is to influence in a certain manner a dynamic system in order to make it behave in a desirable manner.

• Control system are typically implemented for regulation and tracking. Tracking: the system is controlled so that its output follows a particular desired trajectory Regulation: the system is controlled in a certain way so that its output is maintained in a certain set point (e.g., stabilization)

• Numerous applications are all around us: Space shuttle lift off to orbit the earth Automatically machined metallic parts Self-guided vehicles Seismic response control in high-rise buildings

Control System Basic Concepts

• The oldest known control system dates back to 1400 B.C. It is known as the Clepsydra and it was a hydraulically regulated device used for time measurement developed by the Babylonians.

It operated by having a water trickle into a measuring container at a constant rate. The level of water in the measuring container could be used to tell time. This concept is known as the Liquid-Level Control.

Later around 270 B.C, Ktesibios became famous for a water clock with moving figures on it.

• Steam pressure and temperature controls began in 1681 with Denis Papin’s invention of the safety valve. Later in the 17th century, Cornelis Drebbel in Holland invented a purely mechanical temperature control system for hatching eggs.

• In 1745 Edmund Lee applied speed control to a windmill. Increasing winds pitched the blades farther back, so that less area was available and vice-versa.

Control Systems Historical Background

• The stability and Steering Concepts of control systems as we know it today began in the latter half of the 19th century.

1868 – James Clerk Maxwell (Stability in control theory)

1874 – Edward John Routh & William Kingdom Clifford (Stability in control theory)

1874 – Henry Bessemer (ships stabilization using a gyro and the ship’s hydraulic system)

1892 – Alexandr Michialovich Lyapunov (Stability in nonlinear systems)

• Some of the most important and remarkable developments of the 20th century were:

Automatic Steering of Ships

Analysis of feedback amplifiers (currently used for feedback control systems)

Bode and Nyquist formed the foundation of linear control systems analysis and design theory

Control Systems Historical Background (cont.)

1. Power Amplification e.g., Radar antennas

2. Remote Control NASA’s Curiosity: Robot for exploration on planet Mars

Rover: Robot designed to work in contaminated areas in Middleton, PA in 1979 nuclear disaster.

3. Convenience of input form Temperature control systems (input: Position on a thermostat & output: Heat)

Structure dynamic assessment (input: Seismic excitation at the base & output: Stories Displacements)

4. Compensation for disturbances Functional antenna systems under windy conditions

Active and passive dynamic control of structures

Advantages of Control Systems

• There are different methods of describing dynamic systems. Those methods can be divided into two groups of systems: Deterministic Systems and Probabilistic Systems.

A. Deterministic Systems1. Differential Equations

2. Difference Equation

3. Fuzzy Equations

B. Probabilistic Systems1. Stochastic Equations

NOTE: In this short course we will only use Differential and Difference Equations to Model Systems.

• We use ‘n’ First Order differential equations to model the system, and arrange them so that they are always in the same form.

• This is called the State Variable Model.

Mathematical Description of Systems

• Definition of the State Variable Model: “The state of an 𝑛𝑡ℎ order system at some time 𝑡0is a set of numbers 𝑥1(𝑡0), 𝑥2(𝑡0), …, 𝑥𝑟(𝑡0) with which we can

determine the outputs for 𝑡 ≥ 𝑡0 given the input for 𝑡 ≥ 𝑡0.”

• Use the following nomenclature:

• For the INPUTS: 𝑢1 𝑡 , 𝑢2 𝑡 ,… , 𝑢𝑟 𝑡

• For the OUTPUTS: 𝑦1 𝑡 , 𝑦2 𝑡 , … , 𝑦𝑟 𝑡

• For all STATE VARIABLES: 𝑥1 𝑡 , 𝑥2 𝑡 ,… , 𝑥𝑟 𝑡

• Arrange the inputs, outputs, and state variables into vector form as follows:

𝑢 𝑡 =

𝑢1 𝑡

𝑢2 𝑡⋮

𝑢𝑟 𝑡

, Ԧ𝑦 𝑡 =

𝑦1 𝑡

𝑦2 𝑡⋮

𝑦𝑛 𝑡

, Ԧ𝑥 𝑡 =

𝑥1 𝑡

𝑥2 𝑡⋮

𝑥𝑛 𝑡

Mathematical Description of Systems (cont.)

• The state variable equations are of the form:

This is a 1𝑠𝑡 order differential equation for each state variable.

• Combining all the equations into vector form:


𝐀 𝒏×𝒏 𝐁 𝒏×𝒓

• The output equations are:

• Combining the output equations into vector form:


𝐂 𝒏×𝒏 𝐃 𝒏×𝒓

• By the state-space arrangement of equations we obtain following two equations for a system’s description:

1. Ԧሶ𝑥 𝑡 = 𝑨 Ԧ𝑥 𝑡 + 𝑩 𝑢 𝑡

2. 𝑦 𝑡 = 𝑪 Ԧ𝑥 𝑡 + 𝑫 𝑢 𝑡

• It is intended to express the equations that describe ANY system in one derivative instead of two.

• The variable should be treated as a basic single variable and not as a derivative of something.

• The price we pay by reducing the system description to half the derivatives is that the equations results in twice the original size.

• The State-Space Matrices are identified as follows: 𝐀 = system’s dynamic matrix

𝐁 = input matrix

𝐂 = output matrix

𝐃 = direct transmission matrix


• In most cases when writing differential equations describing a system, they will be nonlinear (assume not time varying).

• An example of nonlinear equations of a system may be:

ሶ𝑥1 𝑡 = 𝑥1 𝑡 𝑥2 𝑡 + cos 𝑥2 𝑡 + 𝑢2 𝑡

ሶ𝑥2 𝑡 = 𝑥22 𝑡

These equations must be linearized to put them in state variable form.

Let:

ሶ𝑥1 𝑡 = 𝑓𝑖 Ԧ𝑥 𝑡 , 𝑢 𝑡

• The linearization is done as the linear terms in a Taylor Series expansion around some desired Ԧ𝑥 𝑡 .

Let this desired Ԧ𝑥 𝑡 be the operating point (i.e., equilibrium point) Ԧ𝑥0, a constant.

Must also choose a 𝑢0 (constant).

Linearization of a System’s Description

• For some constant input 𝑢, the system remains at 𝑥0. 𝑥0 is called an operating or equilibrium point.

• Therefore, Ԧ𝑥 𝑡 = Ԧ𝑥0 𝑡 + Ԧ𝑥𝛿 𝑡 and 𝑢 𝑡 = 𝑢0 𝑡 + 𝑢𝛿 𝑡 .

Where 𝒙𝜹 𝒕 is the change of 𝒙 𝒕 from the operating point 𝒙𝟎 𝒕

Linearization of a System’s Description (cont.)

The “Tipping Point” concept. Perturbations may push the ball up the slope but as long as the perturbations are relatively small it would always return to its previous state. However, once a perturbation moves the ball beyond the tipping point the ball would, even on its own, continue to a new stable state.

Stability concept depicted by the stable state of a ball.Image Source: Tilo Winkler, and Jose G. Venegas J Applied Physiology 2011; 110: 1482-1486

𝑥0

𝑢0

• For one state variable,

𝑥𝑖 𝑡 = 𝑥0𝑖 + Ԧ𝑥𝛿𝑖 𝑡

ሶ𝑥𝑖 𝑡 = 𝑓𝑖 Ԧ𝑥0 + Ԧ𝑥𝛿 𝑡 , 𝑢0 + 𝑢𝛿 𝑡

• If 𝑓𝑖 𝑡 has partial derivatives, we can write the nonlinear function as a Taylor Series Expansion around the operating point 𝑥0, 𝑢0 .

𝑓 𝑡 =

𝑛=0

∞𝑓𝑖

𝑛𝑎

𝑛!𝑥 − 𝑎 𝑛


• In matrix form:

ሶ𝑥𝑖 𝑡 = 𝑓𝑖 Ԧ𝑥0, 𝑢0 +𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0

𝜕𝑥1

𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0𝜕𝑥2

⋯𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0

𝜕𝑥𝑛

𝑥𝛿1 𝑡

𝑥𝛿2 𝑡⋮

𝑥𝛿𝑛 𝑡

+𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0

𝜕𝑢1

𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0𝜕𝑢2


𝜕𝑚

𝑢𝛿1 𝑡

𝑢𝛿2 𝑡⋮

𝑢𝛿𝑚 𝑡

+ 𝐻. 𝑂. 𝑇.

• By definition: ሶ𝑥𝑖 𝑡 = 𝑓𝑖 Ԧ𝑥 𝑡 , 𝑢 𝑡

So if we choose 𝑓𝑖 Ԧ𝑥0, 𝑢0 = 𝑥0𝑖 = 0 the operating point

𝑥 𝑡 = 𝑥0𝑖 + 𝑥𝛿𝑖 𝑡

ሶ𝑥𝑖 𝑡 = ሶ𝑥𝛿𝑖 𝑡

• Then,

ሶ𝑥𝛿𝑖 𝑡 =𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0

𝜕𝑥1

𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0𝜕𝑥2


𝜕𝑥𝑛

𝑥𝛿1 𝑡

𝑥𝛿2 𝑡⋮

𝑥𝛿𝑛 𝑡

+𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0

𝜕𝑢1

𝜕𝑓𝑖 Ԧ𝑥0, 𝑢0𝜕𝑢2


𝜕𝑚

𝑢𝛿1 𝑡

𝑢𝛿2 𝑡⋮

𝑢𝛿𝑚 𝑡

+ 𝐻. 𝑂. 𝑇.


• Writing for all state variables, 𝑥1, … , 𝑥𝑛, and rearranging in matrix form gives,

IMPORTANT: Discard the H.O.T. to get the linearized model.


Obtain the state-space variable equations that describe the pendulum movement as shown in the figure. Remember to linearize the dynamic nonlinear equations and represent your solution equations in matrix form.

Example: Oscillating Pendulum

Oscillating Pendulum Equations of Motion

𝑚𝑔 sin 𝜃 𝑡 + 𝐿 ሷ𝜃 𝑡 𝑚 = 0

𝑔 sin 𝜃 𝑡 + 𝐿 𝐿 ሷ𝜃 𝑡 =0

ሷ𝜃 𝑡 = −𝑔

𝐿sin 𝜃 𝑡

𝑓 𝑡 = 𝑖𝑛𝑝𝑢𝑡

𝜃 𝑡 = 𝑜𝑢𝑡𝑝𝑢𝑡

Free Body Diagram

𝒇 𝒕

1. State variables

𝒙𝟏 𝒕 = 𝜽 𝒕 𝑓 𝑡 = 𝑢 𝑡

𝒙𝟐 𝒕 = ሶ𝒙𝟏 𝒕 = ሶ𝜽 𝒕

ሶ𝑥2 𝑡 = ሷ𝜃 𝑡 = ሷ𝑥1 𝑡

2. Substituting the state variables

ሶ𝑥1 𝑡 = 𝑥2 𝑡

ሶ𝑥2 𝑡 = −𝑔

𝐿sin 𝑥1 𝑡

3. Linearization

a. Find the operating points:

ሶ𝑥1 𝑡 = 0 = 𝑥2 𝑡 𝒙𝟎𝟏 = 𝟎

ሶ𝑥2 𝑡 = 0 = −𝑔

𝐿sin 𝑥1 𝑡 𝒙𝟎𝟐 = 𝟎

Example: Oscillating Pendulum (cont.)

Red Arrow : Restorative ForceGreen Arrow : Acting Force

b. Obtain the state matrices 𝐴,𝐵, 𝐶, 𝐷:


𝜕𝑓1 𝑥0 , 𝑢0𝜕𝑥2

= 𝟏𝜕𝑓1 𝑥0, 𝑢0𝜕𝑥1

= 𝟎

𝜕𝑓2 𝑥0 , 𝑢0𝜕𝑥1

= ฬ−𝑔

𝐿cos 𝑥1 𝑡

𝑥=0= −

𝒈

𝑳

𝜕𝑓2 𝑥0 , 𝑢0𝜕𝑥2

= 𝟎

𝜕𝑓1 𝑥0, 𝑢0𝜕𝑢

= 𝟎𝜕𝑓2 𝑥0 , 𝑢0

𝜕𝑢= 𝟎

𝑦 𝑡 = 𝜃 𝑡 = 𝑥1 𝑡

𝜕𝑓1 𝑥0, 𝑢0𝜕𝑥1

= 𝟏𝜕𝑓1 𝑥0 , 𝑢0

𝜕𝑥2= 𝟎

𝜕𝑓1 𝑥0, 𝑢0𝜕𝑢

= 𝟎

c. Assemble the State-Space Equations in matrix form:


State Differential Equation

Algebraic Output Equation

Let 𝐴, 𝐵, 𝐶 be matrices and 𝑘, 𝑞, scalars.

• Definitions:

1. 𝐴 is diagonal if only diagonals elements are non-zero.

2. 𝐴 is upper triangular if all elements below the diagonal are zero.

3. 𝐴 is lower triangular if all elements above the diagonal are zero.

4. 𝐴 is a scalar matrix if it is diagonal, and all elements on the diagonal have the same value

5. The Identity matrix 𝐼 is a scalar matrix with non-zero elements equal to one.

Linear Algebra Review

• Basic Operations:

1. Addition: must have same number of rows and columns.

𝑨 + 𝑩 = 𝑪

𝑎11 +𝑏11 =

𝑐11

Properties

𝐴 + 𝐵 = 𝐵 + 𝐴

𝐴 + 𝐵 + 𝐶 = 𝐴 + 𝐵 + 𝐶

𝐴 + ∅ = 𝐴 NOTE:∅ → 𝑧𝑒𝑟𝑜𝑠 𝑚𝑎𝑡𝑟𝑖𝑥

𝐴 + −𝐴 = ∅

Linear Algebra Review (cont.)

2. Multiplication by a constant:

𝑞𝐴 = 𝐴𝑞

𝑞 𝐴 + 𝐵 = 𝑞𝐴 + 𝑞𝐵

𝐶 + 𝑘 𝐴 = 𝐶𝐴 + 𝑘𝐴

𝐶 𝑘𝐴 = 𝐶𝑘 𝐴

1𝐴 = 𝐴

3. Transpose: 𝑨𝑻

𝐴 =1 2 34 5 6

𝐴 =1 42 53 6

NOTE: 𝐴 is symmetric if 𝐴 = 𝐴𝑇


4. Multiplication

𝑪𝒏×𝒎 = 𝑨𝒏×𝒓𝑩𝒓×𝒎

Properties

𝑘𝐴 𝐵 = 𝑘 𝐴𝐵 = 𝐴 𝑘𝐵

𝐴𝐵 𝐶 = 𝐴 𝐵𝐶

𝐴 + 𝐵 𝐶 = 𝐴𝐶 + 𝐵𝐶

𝐶 𝐴 + 𝐵 = 𝐶𝐴 + 𝐶𝐵

𝐴𝐵 ≠ 𝐵𝐴 (in general)

𝐴𝐵 = ∅ only if 𝐴 = 0 𝑜𝑟 𝐵 = 0


Notation: 𝐴 = Ԧ𝑎1 Ԧ𝑎2 ⋯ Ԧ𝑎𝑛

C𝐴 = 𝐶 Ԧ𝑎1 𝐶 Ԧ𝑎2 ⋯ 𝐶 Ԧ𝑎𝑛

5. Linear Space: An n-dimensional real linear space, called ℝ𝑛, is made up of n-dimensional vectors.

𝒙 =

𝒙𝟏𝒙𝟐⋮𝒙𝒏

6. Independence: Let Ԧ𝑥1, Ԧ𝑥2,⋯ , Ԧ𝑥𝑚 be column vectors. They are linearly dependent if ∃ real numbers 𝛼1, 𝛼2,⋯ , 𝛼𝑛, not all zero ∋.

𝜶𝟏𝒙𝟏 + 𝜶𝟐𝒙𝟐 +⋯+ 𝜶𝒎𝒙𝒎 = 𝟎

Ԧ𝑥1, Ԧ𝑥2,⋯ , Ԧ𝑥𝑚 are linearly dependent if at least one of the vectors can be written as a linear combination of the others.

Example #1: If we have vectors: Ԧ𝑥1 Ԧ𝑥2 Ԧ𝑥3 Ԧ𝑥4, a linear combination of vector Ԧ𝑥3 may be as follows:

Ԧ𝑥3 = 𝑘1 Ԧ𝑥1 + 𝑘2 Ԧ𝑥2 + 𝑘4 Ԧ𝑥4

−91

= 311− 2

61


The dimension of a linear space is the number of linearly independent vectors in the space. ℝ𝑛 has at most 𝑛 linearly indepentednt vectors.

A set of 𝑛 linearly independent vectors in ℝ𝑛 is called a basis for that space.

Any vector in the space can be written as a linear combination of the basis vectors.

Example #2: For ℝ2, let Ԧ𝑥1 =10

and Ԧ𝑥2 =01

∴ Any vector in ℝ2 can be written as a linear combination of Ԧ𝑥1 + Ԧ𝑥2.

16−6

= 1610+ −6

01

A basis is not unique: For another case where ℝ2, let Ԧ𝑥1 =11

and Ԧ𝑥2 =3−1

16−6

= −1

211

+11

23−1


Represents any vector

7. Rank: The rank of a matrix is the number of linearly independent rows or columns.

𝒓𝒂𝒏𝒌 𝑨𝒓𝒂𝒏𝒌 𝑨 = 𝟎 ⟹ 𝑨 = ∅

The range space of a matrix 𝐴 is the set of all vectors that can be written as a linear combination of the columns of 𝐴 .

Ԧ𝑥 is a null vector of 𝐴 if 𝐴 Ԧ𝑥 = 0. The null space of 𝐴 is the set of all null vectors

8. Row and Column Operations: If a column of 𝐴 is replaced by a linear combination of that column and one other column of 𝐴 , it is called a column operation. It is similar for row operations. This can be used to determine the rank of a matrix.

𝐴 =3 0 2 2−1 7 4 97 7 0 5

⟹

3 0 2 2−1 7 4 90 7 −14

3−29

3

= 𝐵

𝑟𝑜𝑤 1 = 𝑟𝑜𝑤 2

𝑟𝑜𝑤 2 = 𝑟𝑜𝑤 2

𝑟𝑜𝑤 3 = 𝑟𝑜𝑤 3 −7

3𝑟𝑜𝑤 1


Row Operation

Row or Column operations are written as a multiplication by a matrix.

𝐵 = 𝑃 𝐴

3 0 2 2−1 7 4 90 7 −14

3−29

3

=

1 0 00 1 0−73

0 1

3 0 2 2−1 7 4 97 7 0 5

𝑃 is a lower triangular matrix and has no zeros on the diagonal. The matrix 𝑃 that is equivalent to a row or a column operation is square 𝑛 × 𝑛 and rank 𝑛.


𝑩 𝑷 𝐀

9. Determinants: Useful value used in linear algebra that can be computed from the elements of a usually square matrix.

𝒅𝒆𝒕 𝑨 =

𝒊=𝟎

𝒏

𝒂𝒊𝒋𝒄𝒊𝒋

- 𝑎𝑖𝑗 ⟶ element in row 𝑖, column 𝑗.

- 𝑐𝑖𝑗 ⟶ cofactor of 𝒂𝒊𝒋

⟶ −1 1+𝑗𝑑𝑒𝑡 𝑀𝑖𝑗

- 𝑀𝑖𝑗 ⟶ is the 𝑛 − 1 𝑛 − 1 matrix deleting row 𝑖 and column 𝑗 of matrix 𝐴

Example: Let 𝐴 =3 1 02 2 −10 1 2

. Compute the determinant of matrix 𝐴 .

𝑑𝑒𝑡 𝐴 = 32 −11 2

− 21 01 2

+ 01 02 −1

= 3 2 × 2 − −1 × 1 − 2 1 × 2 − 0 × 1 + 0

𝑑𝑒𝑡 𝐴 = 11


10. Inverse: Also called a ‘reciprocal’ matrix, the inverse of a square matrix 𝐴 is denoted as 𝐴 −1 such that

𝐴𝐴−1 = 𝐴−1𝐴 = 𝐼

If 𝐴 has an inverse it is said to be non-singular.

If it does not have an inverse it is said to be singular.

𝑨−𝟏 =𝒂𝒅𝒋 𝑨

𝒅𝒆𝒕 𝑨=

𝑪𝒊𝒋𝑻

𝒅𝒆𝒕 𝑨

Properties

1. 𝑨−𝟏−𝟏

= 𝑨

2. 𝑨𝑩 −𝟏 = 𝑩−𝟏𝑨−𝟏

3. 𝑨−𝟏𝑻= 𝑨𝑻

−𝟏


NOTE: 𝑪𝒊𝒋 is the cofactors matrix of 𝐴 .

11. Important Theorems:

1. If 𝐴 has an inverse it is unique.

2. If 𝐴 has an inverse ⟺ 𝑟𝑎𝑛𝑘 𝐴 = 𝑛.

3. If 𝐴, 𝑆, and 𝑇 are 𝑛 × 𝑛 matrices and 𝑟𝑎𝑛𝑘 𝐴 = 𝑛, then:

𝐴𝑆 = 𝐴𝑇 ⟹ 𝑆 = 𝑇

4. If 𝐴 is non-singular (𝐴−1 exists) ⟺ 𝑑𝑒𝑡 𝐴 = 0


NOTE: All the above theorems can be mathematically proven, however, their individual proofs are out of the scope of this short course

12. Eigenvalues and Eigenvectors:

Let 𝐴 be a 𝑛 × 𝑛 matrix. The values of 𝜆 that are a solution to:

𝝀𝒙 = 𝑨𝒙

for Ԧ𝑥 = 0, are the eigenvalues of the matrix 𝐴.

The corresponding values of Ԧ𝑥 are the eigenvectors of 𝐴.

In order to find Ԧ𝑥 or 𝜆:

𝜆 Ԧ𝑥 − 𝐴 Ԧ𝑥 = 0

𝜆𝐼 − 𝐴 Ԧ𝑥 = 0

From above, the only way to have 𝜆𝐼 − 𝐴 Ԧ𝑥 = 0, where Ԧ𝑥 ≠ 0, is for 𝜆𝐼 − 𝐴 to be singular, or 𝑟𝑎𝑛𝑘 𝜆𝐼 − 𝐴 < 𝑛. In simple words,

𝒅𝒆𝒕 𝝀𝑰 − 𝑨 = 𝟎

IMPORTANT ⟹ Cayley-Hamilton Theorem: Let 𝒅𝒆𝒕 𝝀𝑰 − 𝑨 = 𝝀𝒏 + 𝜶𝝀𝒏−𝟏 +⋯+ 𝜶𝒏−𝟏𝝀 + 𝜶𝒏


Characteristic Polynomial of 𝐴

• Definitions:

1. Linearity: A system is linear if it has the superposition property. A system is linear if

𝑇 𝑎𝑥1 𝑡 + 𝑏𝑥2 𝑡 = 𝑎𝑇 𝑥1 𝑡 + 𝑏𝑇 𝑥2 𝑡

2. Time-Invariance: A system is time-invariant if the same input always gives the same output.

𝑦 𝑡 − 𝑡0 = 𝑇 𝑥 𝑡 − 𝑡0

• As stated before, the basic model for a linear time-invariant system consists of the state differential equation and the algebraic output equation.

Ԧሶ𝑥 𝑡 = 𝑨 Ԧ𝑥 𝑡 + 𝑩 𝑢 𝑡 𝑥 𝑡0 = 𝑥0

𝑦 𝑡 = 𝑪 Ԧ𝑥 𝑡 + 𝑫 𝑢 𝑡

• The first equation represents a set of 𝑛 coupled first-order differential equations that must be solved for the state vector Ԧ𝑥 𝑡 given the initial state 𝑥 𝑡0 = 𝑥0 and input 𝑢 𝑡 .

• The second equation characterizes an instantaneous dependence of the output on the state and the input.

State Transformation

• The solution of the state equations is an 𝑛 × 𝑛 matrix called the state transformation matrix, 𝜙 𝑡 .

Ԧ𝑥 𝑡 = 𝜙 𝑡 Ԧ𝑥 𝑡0

• For a linear time-invariant system,

𝜙 𝑡 = 𝑒𝐴𝑡

where 𝑒𝐴𝑡 is a power series of the matrix 𝐴. In addition, 𝑒𝐴𝑡 has the following properties:

1. 𝜙 0 = 𝐼

2. ሶ𝜙 𝑡 = 𝐴𝜙 𝑡

• By taking the derivative of the solution we obtain,

𝑑

𝑑𝑡Ԧ𝑥 𝑡 =

𝑑

𝑑𝑡𝜙 𝑡 Ԧ𝑥 𝑡0

= 𝐴𝑒𝐴𝑡 Ԧ𝑥 𝑡0

= 𝐴𝜙 𝑡 Ԧ𝑥 𝑡0ሶԦ𝑥 𝑡 = 𝐴 Ԧ𝑥 𝑡

State Transformation (cont.)

• By solving for 𝑒𝐴𝑡 using Laplace Transforms we obtain,

ሶԦ𝑥 𝑡 = 𝐴 Ԧ𝑥 𝑡

𝑠 Ԧ𝑋 𝑠 − Ԧ𝑥 0 = 𝐴 Ԧ𝑋 𝑠

𝑠 Ԧ𝑋 𝑠 − 𝐴 Ԧ𝑋 𝑠 = Ԧ𝑥 0

𝑠𝐼 − 𝐴 Ԧ𝑋 𝑠 = Ԧ𝑥 0

Ԧ𝑋 𝑠 = 𝑠𝐼 − 𝐴 −1 Ԧ𝑥 0

• Therefore,

Ԧ𝑥 𝑡 = ℒ−1 𝑠𝐼 − 𝐴 −1 Ԧ𝑥 0

State Transformation (cont.)

Note that: 𝛷 𝑠 = 𝑠𝐼 − 𝐴 −1

𝜙 𝑡 = ℒ−1 𝑠𝐼 − 𝐴 −1

Transfer Function: Compact algebraic description of the input/output relation for a linear system. It is the output signal of a system divided by the input signal. This function will allow separation of the input ,system, and output into three separate and distinct parts, unlike the differential equation.

This function allows us to algebraically combine mathematical representations of subsystems to yield a total system representation.

In the s-domain and z-domain, this will be one polynomial divided by another polynomial, and can be expressed as poles and zeros.

The transfer function can be obtained by taking the Laplace transform of the impulse response function of a system.

Transfer Function Matrix of a System

𝑦 𝑡 = න−∞

∞

𝑓 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏

ℒ 𝑦 𝑡 = 𝑌 𝑠 = 𝐻 𝑠 𝑋 𝑠

Dirac Delta Function: 𝛿 𝑡

System’s Impulse Response:

Laplace’s Transform of 𝑦 𝑡 :Transfer Function Schematic Representation in a System

• Lets start with the state variable description:

Ԧሶ𝑥 𝑡 = 𝑨 Ԧ𝑥 𝑡 + 𝑩 𝑢 𝑡

𝑦 𝑡 = 𝑪 Ԧ𝑥 𝑡 + 𝑫 𝑢 𝑡

• Taking the Laplace Transform we obtain:

𝑠𝑋 𝑠 − Ԧ𝑥 0 = 𝐴𝑋 𝑠 + 𝐵𝑈 𝑠

𝑌 𝑠 = 𝐶𝑋 𝑠 + 𝐷𝑈 𝑠

• To get the transfer function, it is necessary to have zero initial conditions (i.e., Ԧ𝑥 0 = 0).

𝑋 𝑠 = 𝑠𝐼 − 𝐴 −1𝐵𝑈 𝑠

𝑌 𝑠 = 𝐶 𝑠𝐼 − 𝐴 −1𝐵 + 𝐷 𝑈 𝑠

𝑌 𝑠 = 𝐺 𝑠 𝑈 𝑠

• Therefore, by inspection the Transfer Function is:

Transfer Function Matrix of a System (cont.)

IMPORTANT: Keep in mind that the state vector Ԧ𝑥 𝑡 of a linear time-invariant system is not unique.

• Let the same system be described by 2 state equations:

Therefore, the two vectors, Ԧ𝑥 𝑡 and Ԧ𝑥 𝑡 , must be related because they describe the same system.

Definition: Let 𝑃 be a 𝑛 × 𝑛 real non-singular matrix. Then Ԧ𝑧 𝑡 = 𝑃 Ԧ𝑥 𝑡 is an equivalence transformation and the state equations are algebraically equivalent.

• The elements of Ԧ𝑧 𝑡 are linear combinations of the elements of Ԧ𝑥 𝑡 :

𝑧1 𝑡 = 𝑃11𝑥1 𝑡 + 𝑃12𝑥2 𝑡 + ⋯+ 𝑃1𝑛𝑥𝑛 𝑡

• In addition, and .

Equivalent State Equations

and

• Therefore, it is possible to obtain the state-space equations with equivalent state description matrices:

𝑃−1 Ԧ𝑧 𝑡 = 𝐴𝑃−1 Ԧ𝑧 𝑡 + 𝐵𝑢 𝑡 ⋮

• Then, by inspection we have,

ഥ𝑨 = 𝑷𝑨𝑷−𝟏

ഥ𝑩 = 𝑷𝑩

ഥ𝑪 = 𝑪𝑷−𝟏

ഥ𝑫 = 𝑫

Equivalent State Equations (cont.)

Remember: It can be demonstrated that since Ԧ𝑥 𝑡and Ԧ𝑧 𝑡 describe the same system, they have the same transfer function.

Realization: It refers to a state-space model implementing a given input-output behavior. It is a quadruple of time varying matrices 𝐴 𝐵 𝐶 𝐷 such that,

Ԧሶ𝑥 𝑡 = 𝐴 Ԧ𝑥 𝑡 + 𝐵𝑢 𝑡

Ԧ𝑦 𝑡 = 𝐶 Ԧ𝑥 𝑡 + 𝐷𝑢 𝑡

with 𝑢 𝑡 and 𝑦 𝑡 describing the input and output of the system at time 𝑡.

• The transfer function, 𝐺 𝑠 , is realizable if ∃ 𝐴, 𝐵, 𝐶, 𝐷 finite so that:

𝐺 𝑠 = 𝐶 𝑠𝐼 − 𝐴 −1𝐵 + 𝐷

and 𝐴, 𝐵, 𝐶, 𝐷 are a realization of 𝐺 𝑠 .

• We usually realize 𝐺 𝑠 using one of the three standard forms:

Controllable Canonical Form

Observable Canonical Form

Diagonal Form

System’s Realization

In order to represent a state-space system in one of the standard forms, it is necessary to find 𝐺 𝑠 and express it as a rational polynomial transfer function:

1. Controllable Canonical Form:

ሶԦ𝑥 𝑡 =

−𝛼1 −𝛼2 −𝛼3 ⋯ −𝛼𝑛−1 −𝛼𝑛1 0 0 ⋯ 0 00 1 0 ⋯ 0 00 0 1 0 ⋯ ⋮⋮ ⋮ ⋮ ⋱ 0 00 0 ⋯ 0 1 0

Ԧ𝑥 𝑡 +

100⋮00

𝑢 𝑡

𝑦 𝑡 = 𝛽1 𝛽2 ⋯ 𝛽𝑛 Ԧ𝑥 𝑡 + 𝐷𝑢 𝑡

System’s Realization (cont.)

2. Observable Canonical Form:

ሶԦ𝑥 𝑡 =

−𝛼1 1 0 0 ⋯ 0−𝛼2 0 1 0 ⋯ 0−𝛼3 0 0 1 ⋯ ⋮⋮ ⋮ ⋮ ⋮ ⋮ 0

−𝛼𝑛−1 0 0 0 ⋯ 1−𝛼𝑛 0 0 0 ⋯ 0

Ԧ𝑥 𝑡 +

𝛽1𝛽2𝛽3⋮

𝛽𝑛−1𝛽𝑛

𝑢 𝑡

Ԧ𝑦 𝑡 = 1 0 ⋯ 0 Ԧ𝑥 𝑡 + 𝐷𝑢 𝑡

Notice that,

- 𝐴𝑜 = 𝐴𝑐𝑇

- 𝐵𝑜 = 𝐵𝑐𝑇

- 𝐶𝑜 = 𝐶𝑐𝑇

- 𝐷𝑜 = 𝐷𝑐


Important: It can be demonstrated that 𝐺𝑜 𝑠 = 𝐺𝑐𝑇 𝑠 .

3. Diagonal Form:

ሶԦ𝑥 𝑡 =

𝜆1 0 ⋯ 00 𝜆2 ⋯ 0⋮ 0 ⋱ 00 ⋯ 0 𝜆𝑛

Ԧ𝑥 𝑡 +

𝛽1𝛽2⋮𝛽𝑛

𝑢 𝑡

Ԧ𝑦 𝑡 = 𝛼1 𝛼2 ⋯ 𝛼𝑛 Ԧ𝑥 𝑡 + 0 𝑢 𝑡

Where: 𝜆1 ≠ 𝜆2 ≠ ⋯ ≠ 𝜆𝑛


Important: In the three cases, the order of the numerator in the transfer function, 𝐺 𝑠 , must be lower than the denominator.

Stability is the base requirement for the design of a control system. In mathematical terms, the concept of stability in linear systems is clarified by the following theorem:

Theorem: Consider the linear system ሶ𝑦 = 𝐴𝑦.

1. If all of the eigenvalues of 𝐴 have negative real part, then every solution is stable.

2. If any eigenvalue of 𝐴 has positive real part, then every solution is unstable.

3. If some but not all of the eigenvalues of 𝐴 have negative real part and the rest have zero real part, then let

𝜆 = 𝑖𝜎1, 𝑖𝜎2, … , 𝑖𝜎𝑚

be the eigenvalues with zero real part (so that the numbers 𝜎1, … , 𝜎𝑚 are real). Let 𝑘𝑗 be the multiplicity of the

eigenvalue 𝑖𝜎𝑗 as a root of the polynomial 𝑑𝑒𝑡 𝐴 − 𝜆𝐼 . If 𝑖𝜎𝑗 has 𝑘𝑗 linearly independent eigenvalues for all 1 < 𝑗 <

𝑚, then all solutions are stable. Otherwise all solutions are unstable.

Stability of Linear Systems

Example: Lets look at several possibilities for 𝐴 and record the stability of all solutions to the system ሶ𝑦 = 𝐴𝑦 as determined by the theorem.

Stability of Linear Systems (cont.)

The stability concept in linear systems involves two types of stability, internal and external stability.

I. Internal Stability: It is an internal notion of stability and involves the qualitative behavior of the zero-input state response (i.e., the response of a related homogeneous state equation that depends solely on the initial state. 1. The system is stable in the sense of Lyapunov if ∀ Ԧ𝑥 0 < 0, Ԧ𝑥 𝑡 ≤ Ԧ𝑥𝑚 < ∞ .

2. The system is asymptotically stable if ∀ Ԧ𝑥 0 < ∞ and Ԧ𝑥 𝑡 → 0 .

II. External Stability: It focuses on external, or input-output, behavior. In particular, we characterize state equations for which the zero-state output response is a bounded signal for every bounded input signal. This is referred as bounded-input, bounded-output stability.

There exists a finite constant 𝜂 such that for any input 𝑢 𝑡 the zero-state output response satisfies

𝑠𝑢𝑝𝑡≥0 Ԧ𝑦 𝑡 ≤ 𝜂𝑠𝑢𝑝𝑡≥0 𝑢 𝑡

Stability of Linear Control Systems

In Simple Words…

1. Stability in the sense of Lyapunov: The eigenvalues of 𝐴 have negative real part or zero.

2. Asymptotically stable: The eigenvalues of 𝐴 have magnitude less than 1.

3. Bounded-Input & Bounded-Output: The poles of the transfer function, 𝐺 𝑠 , have magnitude less than 0for continuous systems. In the case of discrete systems, their magnitude should be less than 1.

Stability of Linear Control Systems (cont.)

Controlabillity is an important property of control systems because it plays a crucial role in stabilization of unstable systems by feedback, or optimal control. The concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations.

• The state equation is controllable if for any initial state Ԧ𝑥 0 and final state Ԧ𝑥 𝑡1 , there is an input to go from Ԧ𝑥 0 to Ԧ𝑥 𝑡1 in finite time. Otherwise the system is uncontrollable.

• For the state equation to be controllable we must be able to solve the below equation for any Ԧ𝑥 0 and Ԧ𝑥 𝑡1 .

Ԧ𝑥 𝑡1 = 𝑒𝐴𝑡1 Ԧ𝑥 0 + න0

𝑡1

𝑒𝐴 𝑡1−𝜏 𝐵𝑢 𝜏 𝑑𝜏

• In conclusion, it can be demonstrated that controllability can be determined if,

𝑟𝑎𝑛𝑘 𝐵 𝐴𝐵 ⋯ 𝐴𝑛−1𝐵 = 𝑛

Controllability of Linear Control Systems

Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. This means that from the system’s outputs it is possible to determine the behavior of the entire system (for any possible sequence of state and control vectors).

If a system is not observable, it means that the current values of some of its states cannot be determined through output sensors, which makes their value unknown to the controller.

• The state equation is said to be observable if ∀ Ԧ𝑥 0 ∃ 0 < 𝑡1 < ∞ ∋ knowing 𝑢 𝑡 and Ԧ𝑦 𝑡 for 𝑡 ∈ 0, 𝑡 , we can determine Ԧ𝑥 0 .

• In conclusion, it can be demonstrated that observability of a state equation can be determined if:

𝑟𝑎𝑛𝑘(

𝐶𝐶𝐴𝐶𝐴2

⋮𝐶𝐴𝑛−1

= 𝑛

Observability of Linear Control Systems

Remember: A control system is a set of devices that interact in order to regulate the behavior of a mechanical or electromechanical system or a process.

Control Systems

Open-loop systems Closed-loop systems

State feedback is a method employed to place the closed-loop poles of a ‘plant or process’ in pre-determined locations in the s-plane. Placing the poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response system.

IMPORTANT: The system must be controllable in order to implement this method.

• State feedback control law features a constant state feedback gain matrix 𝐾 of dimension 𝑚 × 𝑛 and a new external reference input 𝑟 𝑡 necessarily having the same dimension 𝑚 × 1 as the open-loop input 𝑢 𝑡 as well as the physical units.

State Feedback in Control Systems

Closed-loop State Feedback Control block diagram

Problem: How to select 𝐾 so that 𝐴 − 𝐵𝐾 has de desired eigenvalues?

1. Find 𝑃 to transform the system into controllable canonical form:

2. Find ഥ𝐾 so that ҧ𝐴 − ത𝐵ഥ𝐾 has desired eigenvalues.

ഥ𝐾 = ത𝛼1 − 𝛼1 ത𝛼2 − 𝛼2 ⋯ ത𝛼𝑛 − 𝛼𝑛

where ത𝛼1, ത𝛼1, …, ത𝛼𝑛 are the desired eigenvalues to modify the system response.

3. Find 𝐾 from ഥ𝐾.

ഥ𝐾 = 𝐾𝑃−1 → 𝑲 = ഥ𝑲𝑷

State Feedback in Control Systems (cont.)

State Feedback using Estimated States

NOTE: We should use state feedback estimators if we need to know the input but or sensors cannot measure them directly.

• For the case of designing a state feedback controller using estimated states we need to design an observer to the system in order to estimate those states that the sensors cannot measure directly.

Simplified Observer block diagramDetailed Observer block diagram

Problem: How do we select 𝐿 so that 𝐴 − 𝐿𝐶 has de desired eigenvalues?

• We know how to select 𝐾 so that 𝐴 − 𝐵𝐾 has desired eigenvalues.

• 𝐴 − 𝐿𝐶 and 𝐴 − 𝐿𝐶 𝑇 have the same eigenvalues:

𝐴 − 𝐿𝐶 𝑇 = 𝐴𝑇 − 𝐿𝐶 𝑇 = 𝐴𝑇 − 𝐶𝑇𝐿𝑇

Solution: Find 𝐿𝑇 so that 𝐴𝑇 − 𝐶𝑇𝐿𝑇 has the desired eigenvalues, with the same method for controllers.

State Feedback using Estimated States (cont.)

Detailed Observer-based Compensator Block Diagram

State Feedback using Estimated States (cont.)

Thanks for your attention!

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