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Process Control
Lecture1
Laplace Transformations
Introduction
Laplace transform techniques provide powerful tools in numerous fields of technology such as
Control Theory where knowledge of the system transfer function is essential and where the
Laplace transform comes into its own.
Definition : Let f(t) be defined for t ≥ 0. The Laplace transform of f(t), denoted by F(s) or
L{f(t)}, is an integral transform given by the Laplace integral
The parameter s is assumed to be positive and large enough to ensure that the integral converge.
In more advanced applications s may be complex and in such cases the real part of s must be
positive and large enough to ensure convergence.
In determining the transform of an expression, you will appreciate that the limits of the integral
are substituted for t, so that the result will be an expression in s.
Note:
The Laplace transform is an operation that transforms a function of t (i.e., a function of time
domain), defined on [0, ∞], to a function of s (i.e., of frequency domain). F(s) is the Laplace
transform, or simply transform, of f(t). Together the two functions f(t) and F(s) are called a
Laplace transform pair.
Example(1): Find the Laplace transform of f(t)=1
L{f(t)}=F(s)= ∫
=- ∫
=∫
[
]=- [
]=-
Example: Find the Laplace transform of f(t) = a, where a is a constant.
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Example(3): let f(t)=t
L{f(t)}=F(s)= ∫
=∫
By using integral by part
∫
∫
Let u=t
dv=
du=dt
v=-
∫
∫
=
∫
∫
{
+ ∫
=0
=
So L{f(t)}= =
Example(4) : let f(t)=
L{f(t)}=F(s)= ∫
=∫
=∫
=
=
∫
{
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Process Control
Lecture1
Example(4): let f(t)=
L[
]=
∫
=
∫
∫
)
∫
=
=0-
∫
[
={
]
[
=-
H.w( :
∫
]
=
find laplace for
Some properties of the Laplace Transform
1.L{0}=0
2.L{f(t)
=L{f(t)}+L{g(t)}
3. L{cf(t)}=c L{f(t)} c is constant
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Theorem 1:Shifting Theory:
L{f(t)
=F(s+a)
T
Example(5): find the Laplace transform for f(t)=
; then {
Since the cos bt=
Example(6): find the Laplace transform for f(t)= t
L{
H.W: find LT for:
1.
2.
,
3.
Theorem 2: Multiplying by t and tn
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Process Control
Lecture1
H.W: Determine the Laplace transform of( t2sint )
Laplace transform of derivative
]
L[
L[
]
L[
]
Example(7):Find th L.T of the function x(t) which satisfy the following differential equation and
initial condition :
x(0) –s
Sol.:[
̇
]
[
̈
̇ ]+5[sx(s)-
x(0)]+2x(s)=
[
x(s) +4 x(s)+5sx(s)+2x(s)]=
X(s)=
H.w : Find th L.T of the function y(t) which satisfy the following differential equation and
initial condition
̈
̇
;y(0)=1.
̇
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Process Control
Laplace transform of integral
L∫
Example(8): find x(s) for the following
X(t)=∫
L{cosat}=
L{∫
=
H.W: Find the L.T of the following:
[1] ∫
[2] ∫
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Process Control
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