Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Automatic Control Theory Excises(4) 2 — 4, 5, 6, 7 Additional excises : F(s) is know. Obtain f(t).

Mathematical Model of Control Systems Review (1) Modeling in Time Domain — Differential Equation Differential equations for linear components and systems Properties of linear time-invariant differential equations Linearization of nonlinear equations Solving differential equations

Review (2) 2 Definition of Laplace transform （ 2 ） Unit Step 3 The Laplace transform of Typical function （ 5 ） Exponential function （ 1 ） Unit Pulse （ 3 ） Unit Ramp （ 4 ） Unit Parabolic （ 6 ） Sine Function （ 7 ） Cosine function

Summary (3) （ 2 ） Differential theorem 4 Important theorems of Laplace transform （ 5 ） S hifting in complex domain （ 1 ） Linear property （ 3 ） Integration theorem （ 4 ） S hifting in time domain （ 6 ） Initial-value theorem （ 7 ） Finial Value Theorem

Automatic Control Theory （ Lecture 4 ） Chapter 2 Mathematical Model of Control Systems Review: Laplace Transform 2.3 Complex Domain Model of Control Systems

Tasks and the Structure of the Course General Concepts Mathematical Model Time-Domain Method Performance Specifications Complex-Domain Method Frequency-Domain Method Analysis Design The structure of the course

Review - Laplace Transform （ 12 ） 5 Inverse Laplace Transformation （ 1 ） Inverse Laplace transform integral （ 2 ） Partial-fraction expansion （分解部分分式法） Trial and error method Coefficients comparison method Residue method Solution: Example 1. Ifis given, obtain

Review - Laplace Transform （ 13 ） Solving linear differential equation by Laplace transform Zero initial condition n>m : Characteristic roots ( Pole ) :The modes corresponding to

Review - Laplace Transform （ 14 ） Partial-fraction expansion by residue method Generally where ： Let I. When has no multiple roots

Review - Laplace Transform （ 15 ） Solution. Example 2. If is given, determine Example 3. If is known, determine Solution.

Review - Laplace Transform （ 16 ） Example 4. If is known, determine Solution 1. Solution 2.

Review - Laplace Transform （ 17 ） II. When has multiple roots (Suppose are mth- multiple roots and others are single root )

Review - Laplace Transform （ 19 ） Example 5. If is known, determine Solution 1.

Solution of differential equations Example 6 R-C Circuit

(1) Input u r (t) Factors affecting system response (2) Initial condition (3) Structure and parameters 系统的结构参数 —— 规定 r(t) = 1(t) —— 规定 0 初始条件 —— 自身特性决定系统性能 Factors affecting system response 影响系统响应的因素

§2.3 控制系统的复域模型 — 传递函数 §2.3.1 Transfer Function 传递函数的定义 The ratio of the Laplace transforms of the output and the input. 在零初始条件下，线性定常系统输出量拉氏变换与输入量拉氏变换之比。 §2.3.2 General Form of Transfer Function 传递函数的标准形式 General form of differential equation 微分方程一般形式： 拉氏变换： 传递函数： ⑴ 首 1 标准型： ⑵ 尾 1 标准型： §2.3 Transfer function (1)

§2.3 Transfer function (2) Example 7 If Obtain the highest 1 form and the lowest 1 form. Determine the gain. Solution. 首 1 标准型 尾 1 标准型 增益 is known

§2.3.3 Properties of transfer functions (1) G(s) is a complex function. 是复函数； (2) G(s) only depends on the structure and parameters of the system. 只与系统自身的结构参数有 关； (3) G(s) has a direct relation with the differential equation. 与系统微分方程直接关联； (4) G(s) = L[ k(t) ] ； (5) G(s) corresponds to the zero-pole plot on s- plane. 与 s 平面上的零极点图相对应。 §2.3 Transfer function (3)

Example 8 Suppose the step response of a system under zero initial condition is Obtain ：（ 1 ） Transfer function of system ； （ 2 ） Gain of the system ； （ 3 ） Characteristic roots and corresponding modes of the system （ 4 ） Plot the zero-pole diagram （ 5 ） Unit pulse response （ 6 ） Differential equation （ 7 ） When c(0)=-1, c’(0)=0; r(t)=1(t) ， determine the system’s response Solution. （ 1 ） §2.3.3 Properties of transfer function (1)

§2.3.3 Properties of transfer function (2) (2) (4) as shown in figure (3) (5) (6)

§2.3.3 Properties of transfer function (3) （7）（7） The free response part which is caused by the initial condition is

（ 1 ） It does not reflect the entire information of system which under non-zero initial condition （ 2 ） It only fits for SISO systems. （ 3 ） It can only describe linear time-invariant system §2.3.4 Limitations of transfer function Example 8 Determine linear/nonlinear and time-invariant/ time-varing system § Limitations of transfer function

Summary §2.3.3 Properties of transfer functions §2.3.1 Definition of Transfer Functions §2.3.2 Standard form of transfer functions §2.3.4 Limitations of Transfer Function Mathematical Model of Control Systems Differential Equation (Time-Domain) Transfer Function (Complex Domain) (1) G(s) is a complex function ； (2) G(s) only depends on the structure and parameters of the system (3) G(s) has direct relation with differential equation. (4) G(s) = L[ k(t) ] ； (5) G(s) corresponds to zero-pole plot on s-plane.

Automatic Control Theory Excises(4) 2 — 4, 5, 6, 7 Extra excises : If F(s) is known ， determine f(t)