S. Awad, Ph.D.
M. Corless, M.S.E.E.
E.C.E. Department
University of Michigan-Dearborn
LaplaceTransform
Math Review with Matlab:
Application:Linear Time Invariant (LTI) Systems
U of M-Dearborn ECE DepartmentMath Review with Matlab
2
Laplace Transform:X(s) Linear Time Invariant Systems
Linear Time Invariant (LTI)Systems
Definition of a Linear Time Invariant System
Impulse Response
Transfer Function Simple Systems Simple System Example Pulse Response Example Transient and Steady State Example
U of M-Dearborn ECE DepartmentMath Review with Matlab
3
Laplace Transform:X(s) Linear Time Invariant Systems
System Definition A system can be thought of as a black box with an input
and an output
The signal connected to the input is called the Excitation
The system performs a Transformation, T, (function) on the input
Given an input excitation, the output signal is called the Response
Excitation Response
Output Signal
y(t)
Input Signal
x(t)
System
Transformation
T
U of M-Dearborn ECE DepartmentMath Review with Matlab
4
Laplace Transform:X(s) Linear Time Invariant Systems
Differential Equations
Time domain systems are often described using a Differential Equation
N
N
N
M
M
M
dt
tyd
dt
tdy
dt
tdyty
dt
txd
dt
tdx
dt
tdxtx
)(...
)()()(
)(...
)()()(
2
2
210
2
2
210
Recall that time domain Differentiation corresponds to Laplace Transform domain Multiplication by s with subtraction of Initial Conditions
Output Signal
y(t)
Input Signal
x(t)System
U of M-Dearborn ECE DepartmentMath Review with Matlab
5
Laplace Transform:X(s) Linear Time Invariant Systems
Linear Systems A system is Linear if it satisfies the Superposition Principle
( where and are constants ):
This can be restated given the excitation and response relationships:
)()()( 22113 txtxtx
)()()( 22113 tytyty
)()( 1111 txTty )()( 2222 txTty
)()()()( 22112211 txTtxTtxtxT
Then an Excitation of:
Results in a Response of:
U of M-Dearborn ECE DepartmentMath Review with Matlab
6
Laplace Transform:X(s) Linear Time Invariant Systems
Time Invariance A system is time-invariant if its input-output relationship
does not change as time evolves
0 t0 t
)(tx )(ty
)()( tytxT
0 t 0 t)( ty)( tx
U of M-Dearborn ECE DepartmentMath Review with Matlab
7
Laplace Transform:X(s) Linear Time Invariant Systems
Impulse Response The Impulse Response signal, h(t), of a linear system is
determined by applying an Impulse to the Input, x(t), and determining the output response, y(t)
Due to the properties of a Linear Time Invariant System, the Impulse Response Completely Characterizes the relationship between x and y for all x such that:
Where * denotes the Convolution operation
dthxthtxty )()()(*)()(
U of M-Dearborn ECE DepartmentMath Review with Matlab
8
Laplace Transform:X(s) Linear Time Invariant Systems
Laplace Transform Since Convolution may be Mathematically Intensive,
the Laplace Transform is often used as an aid to analyze the Linear Time Invariant Systems.
Recall the relationship between Convolution in the Time-Domain and Multiplication in the Laplace Transform-Domain
)(*)()( thtxty
)()()( sHsXsY
LT LT LT
U of M-Dearborn ECE DepartmentMath Review with Matlab
9
Laplace Transform:X(s) Linear Time Invariant Systems
Transfer Function The Transfer Function, H(s), of a system is the Laplace
Transform of the Impulse Response, h(t)
)(
)()()(
sX
sYthLTsH
)(*)()( thtxty
)()()( sHsXsY
The Transfer Function completely specifies the relationship between the excitation (input) and response (output) in the Laplace Transform-Domain
U of M-Dearborn ECE DepartmentMath Review with Matlab
10
Laplace Transform:X(s) Linear Time Invariant Systems
Simple Systems Most systems can be created by combining the
following simple system building blocks:
Linear Operations: Multiplication by a Constant Addition of Signals
Time-Domain Differentiation Time-Domain Integration Time-Domain Delay
U of M-Dearborn ECE DepartmentMath Review with Matlab
11
Laplace Transform:X(s) Linear Time Invariant Systems
Linear Operations
Linear operations have a direct correlation between the Time-Domain and Laplace Transform-Domain (s-domain) counterparts
Time-Domain
)(tx )(1 txk1k
Laplace Transform-Domain
)(sX )(1 sXk1k
Time-Domain)(1 tx
)(2 tx
)()( 21 txtx
Laplace Transform-
Domain)(1 sX
)(2 sX
)()( 21 sXsX
U of M-Dearborn ECE DepartmentMath Review with Matlab
12
Laplace Transform:X(s) Linear Time Invariant Systems
Time-Domain Differentiation Time-Domain Differentiation Operation
Equivalent Laplace Transform-Domain Operation
)(ssX)(sXMultiplication
s
dt
tdx )()(txDifferentiation
dt
d
U of M-Dearborn ECE DepartmentMath Review with Matlab
13
Laplace Transform:X(s) Linear Time Invariant Systems
Time-Domain Integration Time-Domain Integration Operation (no initial conditions)
Equivalent Laplace Transform-Domain Operation
dttx )()(txIntegration
dt
s
sX )()(sX
Division
s1
U of M-Dearborn ECE DepartmentMath Review with Matlab
14
Laplace Transform:X(s) Linear Time Invariant Systems
Time-Domain Delay Time-Domain Delay Operation
Equivalent Laplace Transform-Domain Operation
)( 0ttx )(tx Delay by t0
Operation
)(0 sXe st)(sXMultiplication
0ste
U of M-Dearborn ECE DepartmentMath Review with Matlab
15
Laplace Transform:X(s) Linear Time Invariant Systems
Automatic Gain Controls for a radio
Car Mufflers (mechanical filter) Suspension Systems (mechanical low pass filter) Cruise Control (motor speed control)
Examples of LTI Systems
The building blocks described previously can be used to model and analyze real world systems such as:
Audio Equalizers (band pass filters)
U of M-Dearborn ECE DepartmentMath Review with Matlab
16
Laplace Transform:X(s) Linear Time Invariant Systems
System Example Create a system to implement
the differential equation:)(
)()( ty
dt
tdytx
1) Determine the Transfer Function directly from the
Differential Equation
2) Draw the system in the Time-Domain
3) Draw the system in the Laplace Transform-Domain
4) Write the Transfer Function from the System Diagram
5) Determine the Impulse Response
U of M-Dearborn ECE DepartmentMath Review with Matlab
17
Laplace Transform:X(s) Linear Time Invariant Systems
Directly Determine H(s) The Transfer Function H(s) can be directly determined by taking the Laplace
Transform of the differential equation and manipulating terms
)()(
)( tydt
tdytx )()()( sYssYsX
)()1()( sYssX
1
1
)(
)()(
ssX
sYsH
LT
By definition,
H(s) = Y(s) / X(s)
U of M-Dearborn ECE DepartmentMath Review with Matlab
18
Laplace Transform:X(s) Linear Time Invariant Systems
Time-Domain-System Draw time-domain system representation for: )(
)()( ty
dt
tdytx
dt
tdytxty
)()()(
)(tx )(ty
dt
tdy )(
2) Start by drawing Input and Output at far ends
+
-
3) Draw Differentiation Block connected to y(t)
4) Draw Summation Block and its connections
1) Reorder terms to create a Function for y(t)
U of M-Dearborn ECE DepartmentMath Review with Matlab
19
Laplace Transform:X(s) Linear Time Invariant Systems
Laplace Transform-Domain The Laplace Transform-Domain System can be drawn by
leaving the linear summation block and replacing the differentiating block with a multiplication by s
)(tx )(ty
dt
tdy )(
+
-Time-Domain
s
+
-
)(sX )(sY
Laplace Transform
Domain
U of M-Dearborn ECE DepartmentMath Review with Matlab
20
Laplace Transform:X(s) Linear Time Invariant Systems
Verify H(s) The Transfer Function H(s) can also be determined by writing
an expression from the Laplace Transform-Domain System
)(sX )(sY
s
+
-
)()()( ssYsXsY )()()1(
)()()(
sXsYs
sXssYsY
1
1
)(
)()(
ssX
sYsH
Reordering terms gives the same result as taking the Laplace Transform of the Differential Equation
System directly yields:
U of M-Dearborn ECE DepartmentMath Review with Matlab
21
Laplace Transform:X(s) Linear Time Invariant Systems
Impulse Response The Impulse Response of the system, h(t), is simply the
Inverse Laplace Transform of the Transfer Function, H(s)
1
1
)()(
1
1
sLT
sHLTth
)()( tueth t
U of M-Dearborn ECE DepartmentMath Review with Matlab
22
Laplace Transform:X(s) Linear Time Invariant Systems
Pulse Response Example Given a system with an Impulse Response, h(t)=e-2tu(t)
1) Find the Transfer Function for the system, H(s)
2) Find the General Pulse Response,y(t)
3) Plot the Pulse Response for T=1 sec and T=2 sec
)(
)(2 tue
tht
Impulse Response
Pulse Response Output
)()(
)(
txth
ty
0 tT
Input Pulse
)()(
)(
Ttutu
tx
U of M-Dearborn ECE DepartmentMath Review with Matlab
23
Laplace Transform:X(s) Linear Time Invariant Systems
Transfer Function The transfer function of the system is simply the Laplace
Transform of the Impulse Response:
)()( 2 tueth t
The Transfer Function can be used to find the Laplace Transform of the pulse response, Y(s), using:
)()()( sXsHsY
LT2
1)(
s
sH
U of M-Dearborn ECE DepartmentMath Review with Matlab
24
Laplace Transform:X(s) Linear Time Invariant Systems
Laplace Transform of Input Given the equation
for a General Pulse of period T
)()()( Ttututx
sTess
sX 11
)(
s
esX
sT1
)(
The general Laplace Transform is thus:
Combining Terms:
LT
U of M-Dearborn ECE DepartmentMath Review with Matlab
25
Laplace Transform:X(s) Linear Time Invariant Systems
Determine Y(s) Y(s) is found using:
Substituting for H(s) and X(s)
Distributing terms
Rewrite in terms of a new Y1(t)
)()()( 11 sYesYsY sT)2(
1)(1
sssYwhere
)()()( sXsHsY
s
e
ssY
sT1
2
1)(
)2()2(
1)(
ss
e
sssY
sT
U of M-Dearborn ECE DepartmentMath Review with Matlab
26
Laplace Transform:X(s) Linear Time Invariant Systems
Partial Fraction Expansion The Matlab function residue can be used to
perform Partial Fraction Expansion on Y1(s)
[R,P,K] = RESIDUE(B,A) B = Numerator polynomial Coefficient VectorA = Denominator Polynomial Coefficient VectorR = Residues VectorP = Poles VectorK = Direct Term Constant
011
1
011
1
2
2
1
1
asasasa
bsbsbsb
ps
r
ps
r
ps
rk
NN
NN
NN
NN
N
N
U of M-Dearborn ECE DepartmentMath Review with Matlab
27
Laplace Transform:X(s) Linear Time Invariant Systems
» B=[0 0 1];A=[1 2 0];
Expand Y1(s) Use residue to perform partial fraction expansion
A
B
sssssY
02
1
)2(
1)(
21
ss
Ps
R
Ps
RKsY
5.0
2
5.00
)(2
2
1
11
)2(2
1
2
1)(1
ss
sY
» [R,P,K]=residue(B,A)R = -0.5000 0.5000P = -2 0K = []
U of M-Dearborn ECE DepartmentMath Review with Matlab
28
Laplace Transform:X(s) Linear Time Invariant Systems
General Solution y(t) Find y(t) by taking Inverse Laplace Transforms and
substituting y1(t) back into y(t)
)(1)(1)( 22
122
1 Ttuetuety tt
)2(2
1
2
1)(1
ss
sY
)(2
1
2
1)( 2
1 tuety t
LT-1
)()()( 11 sYesYsY sT
)()()( 11 Ttytyty
LT-1
U of M-Dearborn ECE DepartmentMath Review with Matlab
29
Laplace Transform:X(s) Linear Time Invariant Systems
Matlab Declarations The General Pulse Response can be verified using Matlab Variables must be carefully declared using proper syntax
» syms h H t s» h=exp(-2*t)
Assuming the system to be causal, T must be explicitly declared as a positive number
The Heaviside function is equivalent to the unit-step
)()()( Ttututx
)()( 2 tueth t
» T=sym('T','positive')» x=sym('Heaviside(t)-Heaviside(t-T)')
U of M-Dearborn ECE DepartmentMath Review with Matlab
30
Laplace Transform:X(s) Linear Time Invariant Systems
Matlab Verification» H=laplace(h) H = 1/(s+2)
)(1)(1)( 22
122
1 Ttuetuety tt
» X=laplace(x)X =1/s-exp(-T*s)/s» Y=H*XY =1/(s+2)*(1/s-exp(-T*s)/s)
» y=ilaplace(Y)y = -1/2*exp(-2*t)+1/2+ 1/2*Heaviside(t-T)*exp(-2*t+2*T) -1/2*Heaviside(t-T)
U of M-Dearborn ECE DepartmentMath Review with Matlab
31
Laplace Transform:X(s) Linear Time Invariant Systems
The following code recreates the Pulse Response as vectors for T=1 sec and T=2 sec
Matlab Vector Code NOTE as of Matlab 6, ezplot cannot plot functions containing
declarations of Heaviside or Dirac (Impulse)
t=[0:0.01:4]; % Time Vectortmax=size(t,2); % Index to last Time ValueT1=find(t==1); % Index to 1 secondT2=find(t==2); % Index to 2 secondsyexp=0.5*(1-exp(-2*t)); % Base exponential vectory1T=[zeros(1,T1),yexp(1:tmax-T1)];y1=yexp-y1T; % Pulse Response T=1y2T=[zeros(1,T2),yexp(1:tmax-T2)];y2=yexp-y2T; % Pulse Response T=2
U of M-Dearborn ECE DepartmentMath Review with Matlab
32
Laplace Transform:X(s) Linear Time Invariant Systems
Matlab Plots The response for T=1
and T=2 is plotted
subplot(2,1,1);plot(t,y1);title('Pulse Response T=1');grid on;subplot(2,1,2);plot(t,y2);title('Pulse Response T=2');xlabel('Time in seconds');grid on;
)(1)(1)( 22
122
1 Ttuetuety tt
U of M-Dearborn ECE DepartmentMath Review with Matlab
33
Laplace Transform:X(s) Linear Time Invariant Systems
Transient and Steady State Example
Determine an equation for the output of a system, y(t), described by the transfer function H(s) and input x(t)
From the output y(t):1. Identify the Transient Response, ytrans(t), of the system
(portion that goes to zero as t increases)
2. Identify the Steady State Response , yss(t), of the system (portion that repeats for all t)
22
2)(
2
sssH
)()2sin(
)(
tut
tx
)()(
)(
tyty
ty
sstrans
U of M-Dearborn ECE DepartmentMath Review with Matlab
34
Laplace Transform:X(s) Linear Time Invariant Systems
Laplace Transform of Input
Recall the Laplace Transform of a general sine signal with an angular frequency 0
2
2)(
)()2sin()(
2
ssX
tuttxLTLT
Find the Laplace Transform of the input signal x(t)
22
)()sin(o
oo s
tutLT
U of M-Dearborn ECE DepartmentMath Review with Matlab
35
Laplace Transform:X(s) Linear Time Invariant Systems
22
2
2
2)()()(
22 ssssHsXsY
Roots of Y(s) Determine an expression for output signal Y(s)
Determine general form for roots (poles) of denominator of Y(s)
Purely Imaginary Roots
21 jp
Complex Roots
jp 12
))((
2
))((
2)(
*2211 pspspsps
sY
U of M-Dearborn ECE DepartmentMath Review with Matlab
36
Laplace Transform:X(s) Linear Time Invariant Systems
Verify Poles in Matlab
» poles=roots( conv( [1 0 2], [1 2 2]) )
poles = -0.0000 + 1.4142i -0.0000 - 1.4142i -1.0000 + 1.0000i -1.0000 - 1.0000i
))()()((
22
22
2
2
2)(
*2211
22 pspspspsssssY
jp
jp
jp
jp
1
1
2
2
2
2
1
1
U of M-Dearborn ECE DepartmentMath Review with Matlab
37
Laplace Transform:X(s) Linear Time Invariant Systems
4
3
1 21
jejp
2
2
2
2
1
1
1
1*2211 ))()()((
22)(
ps
c
ps
c
ps
c
ps
c
pspspspssY
1
*221
1
111
))()((
22
)()(
pspspsps
c
pssYpsc
2
*211
2
222
))()((
22
)()(
pspspsps
c
pssYpsc
Partial Fraction Expansion Note that since poles are complex conjugates, coefficients
will also be complex conjugates
21 jp jp 12
U of M-Dearborn ECE DepartmentMath Review with Matlab
38
Laplace Transform:X(s) Linear Time Invariant Systems
Find Coefficients in Matlab
» syms s t» p1=j*2^0.5; p1c=conj(p1); p2=(-1+j); p2c=conj(p2);» c1=(2*2^0.5)/(s-p1c)/(s-p2)/(s-p2c);» c1=subs(c1,'s',p1)c1 = 0.3536 + 0.0000i
» c2=(2*2^0.5)/(s-p1)/(s-p1c)/(s-p2c);» c2=subs(c2,'s',p2)c2 = 0.3536 - 0.3536i
4
23536.01 c
3536.03535.02 jc
1
*221
1 ))()((
22
pspspsps
c
2
*211
2 ))()((
22
pspspsps
c
U of M-Dearborn ECE DepartmentMath Review with Matlab
39
Laplace Transform:X(s) Linear Time Invariant Systems
Inverse Laplace Take Inverse Laplace Transform of Y(s)
2
2
2
2
1
1
1
111 )()(ps
c
ps
c
ps
c
ps
cLTsYLTty
)()()(*22
*11 *
22*11 tuecectuececty tptptptp
)(Re2)(Re2)( 2121 tuectuecty tptp
Reduce terms by combining complex conjugates
U of M-Dearborn ECE DepartmentMath Review with Matlab
40
Laplace Transform:X(s) Linear Time Invariant Systems
)(2
1Re2)(
4
2Re2
)(
)1(42 tueetuee
ty
tjjtjj
)(Re2)(Re2)( 2121 tuectuecty tptp
Substitute Values When substituting coefficients, it is useful to use the
polar representation to simplify cosine conversions
42 2
13536.03535.0
j
ejc
21 jp jp 12
jec4
23536.01
U of M-Dearborn ECE DepartmentMath Review with Matlab
41
Laplace Transform:X(s) Linear Time Invariant Systems
Steady State and Transient Responses
The complex signal can be converted into a function of cosines
)(2
1Re2)(
4
2Re2)( )1(42 tueetueety tjjtjj
)(4
cos)(2cos2
2)( tutetutty t
)(2cos2
2)( tuttyss )(
4cos tutey t
trans
Transient Response(Goes to 0 at t increases)
Steady State Response(Repeats as t increases)
U of M-Dearborn ECE DepartmentMath Review with Matlab
42
Laplace Transform:X(s) Linear Time Invariant Systems
Matlab Verification Matlab can be used to determine Inverse Laplace Transform Result will have transient and steady state component Result will appear different but be mathematically equivalent
» X=(2^0.5)/(s^2+2); H=2/(s^2+2*s+2);» Y=X*H; y=ilaplace(Y);» y=simplify(y); pretty(y)
1/2 1/2 - 1/2 2 cos(2 t) + 1/2 1/2 1/2 2 exp(-t) cos(t) + 1/2 2 exp(-t) sin(t)
22
2
2
2)(
22 ssssY
Steady State
Transient
U of M-Dearborn ECE DepartmentMath Review with Matlab
43
Laplace Transform:X(s) Linear Time Invariant Systems
Verify Equivalence The Hand and Matlab steady state results are equivalent
because a phase shift of is the same as negating the cosine
The Hand and Matlab transient results are equivalent by applying the relationship: 4cos2)sin()cos( xxx
)(2cos2
2)( tuttyHss
)(4
cos tutey tHtrans
)(2cos2
2)( tuttyMss
)()sin()cos(2
2tuttey t
Mtrans
U of M-Dearborn ECE DepartmentMath Review with Matlab
44
Laplace Transform:X(s) Linear Time Invariant Systems
Summary Laplace Transform is a useful technique for analyzing
Linear Time Invariant Systems
Impulse Response and its Laplace Transform, the Transfer Function, are used to describe system characteristics
Simple System Blocks for multiplication, addition, differentiation, integration, and time shifting can be used to describe many real world systems
Matlab can be used to determine the Transient and Steady-State Responses of a complex system
Top Related