Instructor: Chen-Hsiung Yang
Dynamic System Modeling Analysis and Control
動態系統建模分析與控制
Lecture2
The Laplace Transform
Outline
2-1 Introduction
2-2 Complex numbers, complex variables, and complex functions
2-3 Laplace transformation
2-4 Inverse laplace transformation
2-5 Solving linear, time-invariant differential equations
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2-1 Introduction
Section 2-2 reviews complex numbers
Section 2-3 defines the Laplace transformation and gives Laplace transforms of several
common functions of time. Also examined are some of the most important Laplace transform theorems that apply to linear system analysis.
Section 2-4 deals with the inverse Laplace transformation .
Section 2-5 presents the Laplace transform
approach to the linear, time-invariant differential equation.
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2-2 Complex Numbers, Complex Variables, and Complex Functions
Complex number z = x + jy
In converting complex numbers to polar form rectangular
To convert complex numbers to rectangular form from polar
Complex number Complex conjugate
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2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.)
Euler’s theorem →
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2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.)
Complex algebra
Equality of complex numbers
Addition
Subtraction
Power and roots
Comments ： &
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2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.)
Complex algebra
Equality of complex numbers
Multiplication
∵ Counterclockwise rotation by 90 ° ∴
∵ ∴
In polar form
∵ if ∴
∵ if ∴
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2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.)
Complex algebra
Equality of complex numbers
Division
∵ ∴
∵
∴
Counterclockwise rotation by 90 °
∴ or
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2-3 Laplace Transformation
Laplace Transformation
Define
Laplace transform ：
Inverse laplace transform ：
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2-3 Laplace
Transformation(Cont.)
Laplace Transformation-Example-Example
Exponential function
Step function
Unit Step function
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2-3 Laplace
Transformation(Cont.)
Laplace Transformation-Example-Example
Ramp function
Sinusoidal function
<note>
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2-3 Laplace
Transformation(Cont.)
Laplace Transformation-Example-Example
Pulse function
Impulse function
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2-3 Laplace
Transformation(Cont.)
Laplace Transformation-Theorem-Theorem
Differentiation theorem
Integration theorem
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2-3 Laplace
Transformation(Cont.)
Laplace Transformation-Theorem-Theorem
Final-value theorem
Initial-value theorem
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2-4 Inverse Laplace Transformation
Partial-fraction expansion
F(s) involves distinct poles only
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Partial-fraction expansion
F(s) involves distinct poles only
2-4 Inverse Laplace Transformation (Cont.)
<Note>
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Laplace Transformation
F(s) involves multiple poles
2-4 Inverse Laplace Transformation (Cont.)
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2-5 Solving linear, time-
invariant differential equations
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