Instructor: Chen-Hsiung Yang 動態系統建模分析與控制 Lecture2 The Laplace Transform Presentation slide for courses, classes, lectures et al. Instructor: Chen-Hsiung Yang Dynamic System Modeling Analysis and Control

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 Beginning course details and/or books/materials needed for a class/project.

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.

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

2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) Euler’s theorem →

2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) Complex algebra Equality of complex numbers Addition Subtraction Power and roots Comments： &

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 ∴

2-2 Complex Numbers, Complex Variables, and Complex Functions(Cont.) Complex algebra Equality of complex numbers Division ∵ ∴ ∵ ∴ Counterclockwise rotation by 90 ° or

2-3 Laplace Transformation Define Laplace transform： Inverse laplace transform ：

2-3 Laplace Transformation(Cont.) Laplace Transformation-Example Exponential function Step function Unit Step function

2-3 Laplace Transformation(Cont.) Laplace Transformation-Example Ramp function Sinusoidal function <note>

2-3 Laplace Transformation(Cont.) Laplace Transformation-Example Pulse function Impulse function

2-3 Laplace Transformation(Cont.) Laplace Transformation-Theorem Differentiation theorem Integration theorem

2-3 Laplace Transformation(Cont.) Laplace Transformation-Theorem Final-value theorem Initial-value theorem

2-4 Inverse Laplace Transformation Partial-fraction expansion F(s) involves distinct poles only

2-4 Inverse Laplace Transformation (Cont.) Partial-fraction expansion F(s) involves distinct poles only <Note>

2-4 Inverse Laplace Transformation (Cont.) F(s) involves multiple poles

2-5 Solving linear, time-invariant differential equations

2-5 Solving linear, time-invariant differential equations(Cont.)