1. Laplace Transform Math Review with Matlab:
3/31/2017
Math Review with Matlab:
Laplace Transform
Application: Linear Time Invariant (LTI) Systems
S. Awad, Ph.D.
M. Corless, M.S.E.E.
E.C.E. Department
University of Michigan-Dearborn

2. 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

3. System Definition
A system can be thought of as a black box with an input and an output
Output Signal
y(t)
Input Signal
x(t)
Excitation
Response
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

4. Differential Equations
Output Signal
y(t)
Input Signal
x(t)
Time domain systems are often described using a Differential Equation
Recall that time domain Differentiation corresponds to Laplace Transform domain Multiplication by s with subtraction of Initial Conditions

5. Linear Systems
A system is Linear if it satisfies the Superposition Principle
( where a and b are constants ):
This can be restated given the excitation and response relationships:
Then an Excitation of:
Results in a Response of:

6. Time Invariance
A system is time-invariant if its input-output relationship does not change as time evolves t T t T

7. 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

8. 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 LT

9. Transfer Function
The Transfer Function, H(s), of a system is the Laplace Transform of the Impulse Response, h(t)
The Transfer Function completely specifies the relationship between the excitation (input) and response (output) in the Laplace Transform-Domain

10. 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

11. Multiplication by a Constant
Linear Operations
Multiplication by a Constant
Addition of Signals
Time-Domain
Time-Domain
Laplace Transform-Domain
Laplace Transform-
Domain
Linear operations have a direct correlation between the Time-Domain and Laplace Transform-Domain (s-domain) counterparts

12. Time-Domain Differentiation
Time-Domain Differentiation Operation
Equivalent Laplace Transform-Domain Operation

13. Time-Domain Integration
Time-Domain Integration Operation (no initial conditions)
Equivalent Laplace Transform-Domain Operation

14. Time-Domain Delay Time-Domain Delay Operation
Equivalent Laplace Transform-Domain Operation

15. 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)
Automatic Gain Controls for a radio
Car Mufflers (mechanical filter)
Suspension Systems (mechanical low pass filter)
Cruise Control (motor speed control)

16. System Example Create a system to implement the differential equation:
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

17. 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 LT
By definition,
H(s) = Y(s) / X(s)

18. Time-Domain-System + - Draw time-domain system representation for:
1) Reorder terms to create a Function for y(t)
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 + -

19. 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 + -
Time-Domain + -
Laplace Transform
Domain

20. Verify H(s)
The Transfer Function H(s) can also be determined by writing an expression from the Laplace Transform-Domain System + -
System directly yields:
Reordering terms gives the same result as taking the Laplace Transform of the Differential Equation

21. Impulse Response
The Impulse Response of the system, h(t), is simply the Inverse Laplace Transform of the Transfer Function, H(s)

22. Pulse Response Example
Given a system with an Impulse Response, h(t)=e-2tu(t) t T
Input Pulse
Impulse Response
Pulse Response Output
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

23. Transfer Function
The transfer function of the system is simply the Laplace Transform of the Impulse Response: LT
The Transfer Function can be used to find the Laplace Transform of the pulse response, Y(s), using:

24. Laplace Transform of Input
Given the equation for a General Pulse of period T LT
The general Laplace Transform is thus:
Combining Terms:

25. 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)
where

26. 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 Vector
A = Denominator Polynomial Coefficient Vector
R = Residues Vector
P = Poles Vector
K = Direct Term Constant

27. Expand Y1(s) Use residue to perform partial fraction expansion
3/31/2017
Expand Y1(s)
Use residue to perform partial fraction expansion
» B=[0 0 1];A=[1 2 0];
» [R,P,K]=residue(B,A)
R =
0.5000
P = -2
K = []

28. General Solution y(t)
Find y(t) by taking Inverse Laplace Transforms and substituting y1(t) back into y(t)
LT-1
LT-1

29. 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)
» T=sym('T','positive')
» x=sym('Heaviside(t)-Heaviside(t-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

30. Matlab Verification » H=laplace(h) H = 1/(s+2) » X=laplace(x) X =
1/s-exp(-T*s)/s
» Y=H*X
Y =
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)

31. Matlab Vector Code
NOTE as of Matlab 6, ezplot cannot plot functions containing declarations of Heaviside or Dirac (Impulse)
The following code recreates the Pulse Response as vectors for T=1 sec and T=2 sec
t=[0:0.01:4]; % Time Vector
tmax=size(t,2); % Index to last Time Value
T1=find(t==1); % Index to 1 second
T2=find(t==2); % Index to 2 seconds
yexp=0.5*(1-exp(-2*t)); % Base exponential vector
y1T=[zeros(1,T1),yexp(1:tmax-T1)];
y1=yexp-y1T; % Pulse Response T=1
y2T=[zeros(1,T2),yexp(1:tmax-T2)];
y2=yexp-y2T; % Pulse Response T=2

32. 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');

33. 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)

34. Laplace Transform of Input
Recall the Laplace Transform of a general sine signal with an angular frequency w0
Find the Laplace Transform of the input signal x(t)

35. Purely Imaginary Roots
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
Complex Roots

36. Verify Poles in Matlab » poles=roots( conv( [1 0 2], [1 2 2]) )

37. Partial Fraction Expansion
Note that since poles are complex conjugates, coefficients will also be complex conjugates

38. 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 = i
» c2=(2*2^0.5)/(s-p1)/(s-p1c)/(s-p2c);
» c2=subs(c2,'s',p2)
c2 = i

39. Inverse Laplace Take Inverse Laplace Transform of Y(s)
Reduce terms by combining complex conjugates

40. Substitute Values
When substituting coefficients, it is useful to use the polar representation to simplify cosine conversions

41. Steady State and Transient Responses
The complex signal can be converted into a function of cosines
Steady State Response
(Repeats as t increases)
Transient Response
(Goes to 0 at t increases)

42. Matlab Verification Steady State Transient
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 2 cos(2 t) +
1/ /2
1/2 2 exp(-t) cos(t) + 1/2 2 exp(-t) sin(t)
Steady State
Transient

43. Verify Equivalence
The Hand and Matlab steady state results are equivalent because a phase shift of p is the same as negating the cosine
The Hand and Matlab transient results are equivalent by applying the relationship:

44. 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