EE 448

Control Systems, Sensors and

Actuators

Instructor : Huỳnh Việt Thắng <thang143@gmail.com>

TA : Lại T. Kim Phụng

LA : Vũ Vân Thanh

Lecture 2. Laplace transform review

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

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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