Introduction to Automatic Control The Laplace Transform Li Huifeng Tel:

Introduction to Automatic Control The Laplace Transform Module objectives When you have completed this module you should be able to: –Apply the Laplace transform to differential equations. –Solve linear differential equations. –Apply the main theorems of the Laplace transform. –Know how useful this techniques is to handle dynamical systems

Introduction to Automatic Control The Laplace Transform Subsections Definition Correspondences of the Laplace transform Main theorems of the Laplace transform The inverse Laplace transform Solving linear differential equations using the Laplace transform

Introduction to Automatic Control The Laplace Transform How to understand LT The Laplace transform is conceptually similar to multiplication via logarithms log(axb) = log(a) + log(b) To multiply a by b –Compute logarithms of a and b –Add these logarithms –Inverse logarithm of sum gives product of a and b.

Introduction to Automatic Control The Laplace Transform Linear differential equation Time domain solution Laplace Solution Laplace transformed Equation Convolution Time Domain Laplace Domain Laplace Transform Inverse Laplace Transform Algebraic manipulation

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform Definition 函数 f(t) ， t 为实变量，如果线性积分 存在，则称其为函数 f(t) 的拉普拉斯变换。变换后的 函数是复变量 s 的函数，记作 F(s) 或 L[f(t)] ，即： 称 F(s) 为 f(t) 的变换函数或象函数，而 f(t) 为 F(s) 的原 函数。

Introduction to Automatic Control The Laplace Transform Correspondences of the Laplace transform Table 2.1: Corresponding elements of the Laplace transform or Textbook Page 21Table 2.1: Corresponding elements of the Laplace transform

Introduction to Automatic Control The Laplace Transform 几种典型函数的拉氏变换

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform Euler equation

Introduction to Automatic Control The Laplace Transform Equating real & imaginary parts yields:

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform 典型函数的拉氏变换形式

Introduction to Automatic Control The Laplace Transform Main theorems of the Laplace transform ① Superposition theorem( 叠加定理 ): 各函数和的拉氏变换＝各函数拉氏变换的和

Introduction to Automatic Control The Laplace Transform ② Similarity theorem: ( 比例尺改变 )

Introduction to Automatic Control The Laplace Transform ③ Real Shifting theorem( 延时定理 ):

Introduction to Automatic Control The Laplace Transform 提示： f(t) 相当于 t· 1(t) 在 时间上延迟了一个  值。 解:解: 例1例1

Introduction to Automatic Control The Laplace Transform ④ Complex Shifting theorem( 衰减定理） :

Introduction to Automatic Control The Laplace Transform 解:解: 例2例2

Introduction to Automatic Control The Laplace Transform ⑤ Derivative theorem: )0(f)0(fs)0(fs)s(Fs] dt )t(fd [L )0(f)0(sf)s(Fs] dt )t(fd [L )0(f)s(sF] dt )t(df [L )1n('2n1nn n n '2 2 2      拉氏变换将原函数求导数的运算转换为 “ 象函数 乘s后减初值 ” 的代数运算。

Introduction to Automatic Control The Laplace Transform ⑥ Complex differentiation theorem: ( 不要求掌握 )

Introduction to Automatic Control The Laplace Transform ⑦ Integral theorem: 拉氏变换将原函数求积分的运算转换为 “ 象函数除 以s后加初值 ” 的代数运算。

Introduction to Automatic Control The Laplace Transform ⑧ Initial value theorems:

Introduction to Automatic Control The Laplace Transform Given: Find f(0) 解:解:

Introduction to Automatic Control The Laplace Transform ⑨ final value theorems:

Introduction to Automatic Control The Laplace Transform Given: Find. 解:解:

Introduction to Automatic Control The Laplace Transform ⑩ Convolution in the time domain: ( 不要求掌握 )

Introduction to Automatic Control The Laplace Transform 11Convolution in the frequency domain: ( 不要求掌握 ) 拉氏变换性质的证明

Introduction to Automatic Control The Laplace Transform The inverse Laplace transform 定义： 由象函数 F(s) 求其原函数 f(t) 的运算称 为拉氏反变换。

Introduction to Automatic Control The Laplace Transform Partial Fraction Expansion 思路： 将 F(s) 分解成一些简单的有理分式函数之和， 然后由拉氏变换表一一查出对应的反变换函 数，即得所求的原函数 f(t) 。

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform 将 F(s) 的分母多项式 A(s) 进行因式分解， 即写为： A(s) = (s - s 1 )(s - s 2 )…(s - s n ) 式中， s 1, s 2, …s n 为 A(s) = 0 的根。 分两种情况讨论： 1.A(s) = 0 无重根 2.A(s) = 0 有重根

Introduction to Automatic Control The Laplace Transform 1. A(s)=0 无重根情况 可将 F(s) 换写为 n 个部分分式之和，每个分式 的分母都是 A(s) 的一个因式。 关键问题 : 确定每个部分分式中的待定常数 C i 。

Introduction to Automatic Control The Laplace Transform 确定待定常数 C i

Introduction to Automatic Control The Laplace Transform 代入 C i 即可求得 f(t):

Introduction to Automatic Control The Laplace Transform 确定待定系数 所以 例3例3

Introduction to Automatic Control The Laplace Transform 要点： 明确四个步骤，语言表述清晰

Introduction to Automatic Control The Laplace Transform 分子分母同阶的例子： 所以 例4例4

Introduction to Automatic Control The Laplace Transform 分母有复数根的例子： 例5例5

Introduction to Automatic Control The Laplace Transform 求Ci求Ci 所以 欧拉方程

Introduction to Automatic Control The Laplace Transform 欧拉方程

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform 2. A(s)=0 有重根情况

Introduction to Automatic Control The Laplace Transform 确定待定常数 C i

Introduction to Automatic Control The Laplace Transform 例6例6

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform Solving linear differential equations using the Laplace transform Linear differential equation Time domain solution Laplace Solution Laplace transformed Equation Convolution Time Domain Laplace Domain Laplace Transform Inverse Laplace Transform Algebraic manipulation 1 2 3

Introduction to Automatic Control The Laplace Transform 1 2 3

Introduction to Automatic Control The Laplace Transform 三个步骤 : ①代入初始条件对微分方程进行拉氏变换； ②解变换方程 ( 代数方程 ) ，求出响应函数的拉 氏变换式； ③用部分分式法求拉氏反变换，得到微分方程 的解。

Introduction to Automatic Control The Laplace Transform 分析 : 例7例7

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform

Introduction to Automatic Control The Laplace Transform 解:解: Step1: For a ODE w/initial conditions, apply Laplace transform to each term. Step2: Solve for Y(s)

Introduction to Automatic Control The Laplace Transform Step3: Apply partial fraction expansion to obtain inverse Laplace transform.

Introduction to Automatic Control The Laplace Transform l ODE w/initial conditions l Apply Laplace transform to each term l Solve for Y(s) l Apply partial fraction expansion l Apply inverse Laplace transform to each term 例8例8

Introduction to Automatic Control The Laplace Transform OVERVIEW This module is a mathematical section to establish a base for the theory of control systems. This is a tool and it is indispensable as most of linear system dynamics are described in a mapped space that can only be understood when the main theorems of the Laplace transform are known. Special focus is put on the solution of differential equations using the Laplace transform.

Introduction to Automatic Control The Laplace Transform Homework 见 WORD 文件：拉氏变换作业