EE448 2015 Lec2 LaplaceTransform
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Transcript of EE448 2015 Lec2 LaplaceTransform
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EE 448
Control Systems, Sensors and
Actuators
Instructor : Huỳnh Việt Thắng <[email protected]>
TA : Lại T. Kim Phụng
LA : Vũ Vân Thanh
Lecture 2. Laplace transform review
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Overview
• Issues
– Review of Laplace Transform (LT)
– Use Laplace transform to model the control systems in the frequency domain
– Learn to use MATLAB to find LT and inverse LT.
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Motivation
• The analysis and design sequence includes obtaining the
system's schematic and developing mathematical models of
the physical reality
• Developing mathematical models from schematics normally
leads to the LTI differential equations
3
which relates the output, c(t), to the input, r(t), by way of
the system parameters, ai, and bj.
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Motivation (2)
• Problem: the system parameters (the coefficients), the output
c(t) and the input r(t) all appear throughout the equation
• We prefer a mathematical representation in which the input,
output and system are distinct and separate parts
• and cascaded interconnections
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Laplace transform review
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Laplace Transform Review
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Laplace Transform Table
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Laplace Transform Theorems
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Inverse Laplace Transform
Partial-Fraction Expansion:
• To find the inverse Laplace transform of a complicated
function, we can convert the function to a sum of simpler
terms for which we know the Laplace transform of each term
� the result is called a partial-fraction expansion.
• Example:
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Inverse Laplace Transform (2)
Partial-Fraction Expansion:
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Inverse Laplace Transform (3)
Partial-Fraction Expansion:
• Case 1: Roots of the Denominator F(s) are real and
distinct
• Case 2: Roots of the Denominator F(s) are real and
repeated
• Case 3: Roots of the Denominator F(s) are complex or
imaginary
Textbook pages 37-44
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• Solve for f1(t)?
=>> Roots of the Denominator of F(s) are complex or
imaginary
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• Roots of the Denominator of F(s) are real and repeated
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Example 2.3 (page 39)
• The Laplace transform of (2.14) is
• Solving for the response, yields
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Example 2.3 (cont.)
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Example 2.3
• After finishing Example 2.3, students should also try
with Problems 5, 6, 7, 8 with the support of Matlab!
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MATLAB examples (1)
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MATLAB examples (2)
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MATLAB examples (3)
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MATLAB examples (4): Example 2.3
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MATLAB Example: Symbolic Toolbox
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Exercises
Problems: 1, 2, 5, 6, 7, 8 in page 98 (textbook)
– Problems 1, 2, 7, 8: in handwritten form!
– Problems 5 and 6: use MATLAB.
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