Laplace Transforms: Definition

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Laplace Transforms Definition The Laplace
transform of a function, f(t), is defined as
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
Observations
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, The integral is
performed only for positive time there is no
past information
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Property 1 Superposition Principle
Similarly,
Both L and L-1 are linear operators very
important!
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Laplace Transforms of Common Functions 1
Constant Function Let f(t) a (a constant) Then
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Laplace Transforms of Common Functions 2
Unit Step Function S(t)
The operator L is defined for t?0
L S ( t ) L 1
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Laplace Transforms of Common Functions 3
Derivative of a Function f(t)
Working it out
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Laplace Transforms of Derivatives This is a very
important transform because derivatives appear in
the ODEs we wish to solve. We showed that
initial condition at t 0
Similarly, for higher order derivatives
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where - n is an arbitrary positive integer -


Special Case All Initial Conditions are
Zero Suppose Then 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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• Exponential Functions
• Consider where b gt 0. Then,

• Rectangular Pulse Function
• It is defined by

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Time, t
The Laplace transform of the rectangular pulse is
given by
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Table 3.1. Laplace Transforms, p. 54

• 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
• the area under the pulse constant at one. (i.e.,
  let )
• Let,
• Then,

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Example 3.2
Boundary conditions
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Example 3.2
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Example 3.2
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Example 3.2
We need to break the denominator into simpler
terms
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Example 3.2
We need to break the denominator into simpler
terms
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Example 3.2
Why?
Table 3.1, entry 2
Table 3.1, entry 5
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Example 3.2
Thus, once we determine ?1, ?2, ?3, and ?4, we
get
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Example 3.2
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Partial Fraction Expansions
Basic idea Expand a complex expression for Y(s)
into simpler terms, each of which appears in the
Laplace Transform table. Then you can take the
L-1 of both sides of the equation to obtain y(t).
Example
Perform a partial fraction expansion (PFE)
where coefficients and have to be
determined.
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A General PFE Consider a general expression,
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Here D(s) is an n-th order polynomial with the
roots all being real numbers
which are distinct so there are no repeated
roots. The PFE is
Note D(s) is called the characteristic
polynomial.
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• Special Situations
• Two other types of situations commonly occur when
  D(s) has
• Complex roots e.g.,
• Repeated roots (e.g., )
• For these situations, the PFE has a different
  form. See SEM
• text (pp. 61-64) for details.

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Important Properties of Laplace Transforms

• Final Value Theorem

It can be used to find the steady-state value of
a closed loop system (providing that a
steady-state value exists. Statement of FVT
providing that the limit exists (is finite) for
all where Re (s)
denotes the real part of complex variable, s.
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Example Suppose,
Then,
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2. Time Delay
Time delays occur due to fluid flow, time
required to do an analysis (e.g., gas
chromatograph).
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2. Time Delay
Time delays occur due to fluid flow, time
required to do an analysis (e.g., gas
chromatograph).
t0
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Example
What is the Laplace transform of the following
function?
?0
?-1
?1
?1