Laplace Transforms

1
Laplace Transforms
1. Standard notation in dynamics and control
(shorthand notation) 2. Converts mathematics to
algebraic operations 3. Advantageous for block
diagram analysis
Chapter 3
2
Laplace Transform (L.T.)
For t 0,
Chapter 3
The Laplace transform is a linear operation.
3
L.T. of some simple functions
Constant Function
Chapter 3
Exponential Function
4
L.T. of the derivative of f(t)
Chapter 3
L.T. of the derivative of any order n
5
L.T. of trigonometric functions
Recall the Euler formula
Chapter 3
6
Unit step function
If tgta
If tlta
Chapter 3
u(t-a)
1
0
a
t
7
L.T. of unit step function
Chapter 3
for a0
8
Effects of unit step function
switching off and on
t-shifting
Chapter 3
f(t)
0
0
0
t
t
t
a
a
f(t)
f(t)u(t-a)
f(t-a)u(t-a)
9
L.T. of t-shifting function (time delay)
If tlta
If tgta
Chapter 3
or
10
The unit rectangular pulse function
The rectangular pulse could be represented as the
sum of two step functions
if
otherwise
u(t-a)
1/k
fk(t-a)
1/k
Chapter 3
Area1
a
t
ak
0
a
t
ak
0
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The unit impulse function (Dirac delta function)
d(t-a)
if
otherwise
Chapter 3
a
t
0
and
L.T .of the unit impulse function
for a0,
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Laplace transforms can be used in process control
for
1. Solution of differential equations
(linear) 2. Analysis of linear control systems
(frequency response) 3. Prediction of
transient response for different inputs
Chapter 3
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Chapter 3
14
Example 1
Solve the ODE,
First, take L of both sides of (3-26),
Chapter 3
Rearrange,
Take L-1,
From Table 3.1,
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Example 2
system at rest (s.s.)
Chapter 3
To find transient response for u(t) unit step
at t gt 0 1. Take Laplace Transform (L.T.) 2.
Use partial fraction expansion 3. Take inverse
L.T.
Step 1 Take L.T. (note zero initial
conditions)
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Rearranging,
Step 2a. Factor denominator of Y(s)
Chapter 3
Step 2b. Use partial fraction expansion
Multiply by s, set s 0
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For a2, multiply by (s1), set s-1 (same
procedure for a3, a4)
Step 3. Take inverse of L.T.
Chapter 3
You can use this method on any order of ODE,
limited only by factoring of denominator
polynomial (characteristic equation)
Must use modified procedure for repeated roots,
imaginary roots.
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One other useful feature of the Laplace transform
is that one can analyze the denominator of the
transform to determine its dynamic behavior.
For example, if
the denominator can be factored into
(s2)(s1). Using the partial fraction technique
Chapter 3
The step response of the process will have
exponential terms e-2t and e-t, which indicates
y(t) approaches zero. However, if
We know that the system is unstable and has a
transient response involving e2t and e-t. e2t
is unbounded for large time. We shall use this
concept later in the analysis of feedback system
stability.
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Other applications of L.T.
A. Final value theorem
B. Initial value theorem
Chapter 3
by initial value theorem
by final value theorem