Slide 1

Report
Chapter 3
Laplace Transforms
1. Standard notation in dynamics and control
(shorthand notation)
2. Converts mathematics to algebraic operations
3. Advantageous for block diagram analysis
1
Laplace Transforms
Chapter 3
• Important analytical method for solving linear ordinary
differential equations.
- Application to nonlinear ODEs? Must linearize first.
• Laplace transforms play a key role in important process control
concepts and techniques.
- Examples:
• Transfer functions
• Frequency response
• Control system design
• Stability analysis
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Definition
The Laplace transform of a function, f(t), is defined as
Chapter 3

F ( s)  L  f (t )   f  t  e st dt
0
(3-1)
where F(s) is the symbol for the Laplace transform, L is the
Laplace transform operator, and f(t) is some function of time, t.
Note: The L operator transforms a time domain function f(t)
into an s domain function, F(s). s is a complex variable:
s = a + bj,
j
1
3
Inverse Laplace Transform, L-1:
Chapter 3
By definition, the inverse Laplace transform operator, L-1,
converts an s-domain function back to the corresponding time
domain function:
f  t   L1  F  s  
Important Properties:
Both L and L-1 are linear operators. Thus,
L  ax  t   by  t   aL  x  t   bL  y  t 
 aX  s   bY  s 
(3-3)
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where:
- x(t) and y(t) are arbitrary functions
Chapter 3
- a and b are constants
- X  s  L  x  t  and Y  s  L  y  t 
Similarly,
L1  aX  s   bY  s    ax  t   b y  t 
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Laplace Transforms of Common
Functions
Chapter 3
1. Constant Function
Let f(t) = a (a constant). Then from the definition of the
Laplace transform in (3-1),

L  a    ae
0

 st
a  st
dt   e
s
0
 a a
 0  
 s s
(3-4)
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2. Step Function
Chapter 3
The unit step function is widely used in the analysis of process
control problems. It is defined as:
S t 
0 for t  0

1 for t  0
(3-5)
Because the step function is a special case of a “constant”, it
follows from (3-4) that
1
L  S  t   
s
(3-6)
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3. Derivatives
Chapter 3
This is a very important transform because derivatives appear
in the ODEs we wish to solve. In the text (p.53), it is shown
that
 df 
L    sF  s   f  0 
 dt 
(3-9)
initial condition at t = 0
Similarly, for higher order derivatives:
dn f
L n
 dt

n
n 1
n  2 1

s
F
s

s
f
0

s
f  0 






n2
n 1
...  sf   0  f   0
(3-14)
 
 
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where:
- n is an arbitrary positive integer
Chapter 3
- f  k   0
dk f
dt k
t 0
Special Case: All Initial Conditions are Zero
Suppose
1
n1
f  0   f    0   ...  f    0  . Then
dn f
L n
 dt
 n
  s F s

In process control problems, we usually assume zero initial
conditions. Reason: This corresponds to the nominal steady state
when “deviation variables” are used, as shown in Ch. 4.
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4. Exponential Functions
Consider f  t   ebt where b > 0. Then,


Chapter 3
 b s t
L ebt    ebt e st dt   e   dt

 0
0
1   b s t  
1

e

0
bs 
sb
(3-16)
5. Rectangular Pulse Function
It is defined by:
0 for t  0

f  t   h for 0  t  t w
0 for t  t
w

(3-20)
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Chapter 3
6. Impulse Function (or Dirac Delta Function)
The impulse function is obtained by taking the limit of the
rectangular pulse as its width, tw, goes to zero but holding
1
the area under the pulse constant at one. (i.e., let h  )
tw
Let,
  t  impulse function
Then,
L   t   1
11
h
Chapter 3
f t 
tw
Time, t
The Laplace transform of the rectangular pulse is given by
F s 

h
1  e t w s
s

(3-22)
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Other Transforms
e jt  cos t  j sin t
Note:
e jwt  cos t  j sin t
Chapter 3
j  1
 e-jt  e  jt 

L(cosωt) = L
2


1 1
1 

= 

2  s  jω s  jω 
1  s  jω
s  jω 
=  2


2  s  ω2 s2  ω2 
s
= 2
s  ω2
 e  jt - e  jt 
L(sin ωt) = L 

2j


ω
= 2
s  ω2
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Difference of two step inputs S(t) – S(t-1)
(S(t-1) is step starting at t = h = 1)
Chapter 3
By Laplace transform
1 e s
F ( s)  
s s
Can be generalized to steps of different magnitudes
(a1, a2).
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Table 3.1. Laplace Transforms
Chapter 3
See page 54 of the text.
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