S. Awad, Ph.D.

1
LaplaceTransform
Math Review with Matlab
ApplicationLinear 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

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

• 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

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Time Invariance

• A system is time-invariant if its input-output
  relationship does not change as time evolves

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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

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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

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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

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Linear Operations

• Linear operations have a direct correlation
  between the Time-Domain and Laplace
  Transform-Domain (s-domain) counterparts

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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

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Time-Domain Delay

• Time-Domain Delay Operation

• Equivalent Laplace Transform-Domain Operation

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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)

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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
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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)

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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
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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

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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

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Impulse Response

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

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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 T1 sec and T2 sec
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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

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Laplace Transform of Input

• Given the equation for a General Pulse of period T

LT

• The general Laplace Transform is thus

• Combining Terms

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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)

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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
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Expand Y1(s)

• Use residue to perform partial fraction expansion

B0 0 1A1 2 0
R,P,Kresidue(B,A) R -0.5000
0.5000 P -2 0 K
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General Solution y(t)

• Find y(t) by taking Inverse Laplace Transforms
  and substituting y1(t) back into y(t)

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Matlab Declarations

• The General Pulse Response can be verified using
  Matlab
• Variables must be carefully declared using proper
  syntax

syms h H t s hexp(-2t)
Tsym('T','positive')
xsym('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

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Matlab Verification
Hlaplace(h) H 1/(s2)
Xlaplace(x) X 1/s-exp(-Ts)/s YHX Y
1/(s2)(1/s-exp(-Ts)/s)
yilaplace(Y) y -1/2exp(-2t)1/2
1/2Heaviside(t-T)exp(-2t2T)
-1/2Heaviside(t-T)
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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 T1 sec and T2 sec

t00.014 Time Vector tmaxsize(t,2)
Index to last Time Value T1find(t1) Index
to 1 second T2find(t2) Index to 2
seconds yexp0.5(1-exp(-2t)) Base
exponential vector y1Tzeros(1,T1),yexp(1tmax-T1
) y1yexp-y1T Pulse Response
T1 y2Tzeros(1,T2),yexp(1tmax-T2) y2yexp-y2T
Pulse Response T2
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Matlab Plots

• The response for T1 and T2 is plotted

subplot(2,1,1)plot(t,y1) title('Pulse Response
T1') grid on subplot(2,1,2)plot(t,y2) title('
Pulse Response T2') xlabel('Time in
seconds') grid on
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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)

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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)

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Roots of Y(s)

• Determine an expression for output signal Y(s)

• Determine general form for roots (poles) of
  denominator of Y(s)

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Verify Poles in Matlab
polesroots( conv( 1 0 2, 1 2 2) ) poles
-0.0000 1.4142i -0.0000 - 1.4142i
-1.0000 1.0000i -1.0000 - 1.0000i
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Partial Fraction Expansion

• Note that since poles are complex conjugates,
  coefficients will also be complex conjugates

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Find Coefficients in Matlab
syms s t p1j20.5 p1cconj(p1) p2(-1j)
p2cconj(p2) c1(220.5)/(s-p1c)/(s-p2)/(s-p2c
) c1subs(c1,'s',p1) c1 0.3536
0.0000i c2(220.5)/(s-p1)/(s-p1c)/(s-p2c)
c2subs(c2,'s',p2) c2 0.3536 - 0.3536i
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Inverse Laplace

• Take Inverse Laplace Transform of Y(s)

• Reduce terms by combining complex conjugates

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Substitute Values

• When substituting coefficients, it is useful to
  use the polar representation to simplify cosine
  conversions

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Steady State and Transient Responses

• The complex signal can be converted into a
  function of cosines

Transient Response (Goes to 0 at t increases)
Steady State Response (Repeats as t increases)
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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(20.5)/(s22) H2/(s22s2) YXH
yilaplace(Y) ysimplify(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)
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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