上海交通大学：《系统模型、分析与控制 Modeling、Analysis and Control》课程教学资源[04]Lecture30-why in class 拉普拉氏变换——引子及预备

ME369-系统模型、分折与控制 3.0拉普拉氏变换-引子及预备 School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 引例]输入输出模型 o dx@-f0-f。-f dr A6. M d'x,(t) M:dr =fa+fk-fk: 么-0] M f=K[x()-x] f=K() (t) f() f) School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 1

1 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 lecture 2.3 Fall 2012 lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 3.0 拉普拉氏变换 –引子及预备 ME 369– 系统模型、分析与控制 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [引例]输入输出模型 1 1 2 2 1 1 2 2 2 2 2 ( ) ( ) ( ) B K B K K d x t M f t f f dt d x t M f f f dt             1 2 ( ) ( ) B dx t dx t f B dt dt           1 1 1 2 ( ) ( ) K f K x t x t   2 2 2 ( ) K f K x t 

函数变换 对数变换(Logarithmic transformation) x=ab →lgx=lga+lgb x=a"bmIgx=nlga+mlgb 查表 School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 复数(Complex) 5=0-j0 r=G2+0 Magnitude模 B=arctan-,0∈(-π，π]Phase相角 S=σ+j0 s=r(cos0+jsine) j=万 joimaginary part o=rcos0 rectangular form 虚部 o=rsine 直角坐标形式 Phasor(Euler)form 相位矢量形式 real part 0 实部 s=reio 6 e1=cos0+jsin0 Polar form Euler's theorem 极坐标形式 欧拉公式 School of Mechanical Engineering ME369-lecture 3.0 Shanghal Jiao Tong University Fall 2015 2

2 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University x  ab n m x  a b lg x  lga  lgb lg x  nlga  mlgb 对数变换（Logarithmic transformation) 查表 函数变换 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 复数（Complex） s    j 2 2 arctan , ( , ] r                   Magnitude 模 Phase 相角 s  r(cos  jsin) cos sin j e j      Euler’s theorem 欧拉公式 j s re   rectangular form 直角坐标形式 cos sin r r          Phasor(Euler) form 相位矢量形式 * 1 j   Polar form 极坐标形式 s    j real part 实部 imaginary part 虚部 r 

复数运算 s=G+jo=re Addition/,subtraction::S,±S2=(o,±o2)+j(o,±o2) 加/减 Multiplication: 乘 52=rH5e9+8) Division: S=上e8-%) 除 S23 (G≠0) nth roots: j( 0+2k江) n次方根 (k=0,±1，+2，…） School of Mechanical Engineering ME369-lecture 3.0 Shanghal Jiao Tong University Fall 2015 复函数 F(s)=U(s)+jV(s)=U(o,@)+jv(o,@) F(s) Magnitude模 F(s=2+ Phase相角 ∠F(s）=arctan U Complex conjugate 共轭复数 F(s)=U-jV School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 3

3 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 复数运算 1 2 1 2 1 2 s s j      ( ) ( )     1 2 1 ( ) 1 2 2 j s s e r r    1 2 2 1 ) 2 s 1 j( e s r r    2 ( 0) r  ( 2 j ) k n n n s er     ( 0, 1, 2, ) k    Addition/subtraction: 加/减 Multiplication: 乘 Division: 除 nth roots: n次方根 j s j re       BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University F s U s jV s U jV ( ) ( ) ( ) ( , ) ( , )           2 2 F s U V     arctan U F s V   F s U jV ( )   复函数 Magnitude 模 Phase 相角 Complex conjugate 共轭复数

复函数 s=G+jo=re Exponential function: ee(coso+jsin@) 指数函数 ests=ee (e)'=e Logarithmic function: lns=lnr+i(0+2kπ 对数函数 (s≠0)(k=0,±1，+2，… In()=In(s)-In(s,) In(s s2)=In(s)+In(s) Power function: 幂函数 saealns (s≠0) (a:复常数) School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 复数积分性质 ∫kF(s)ds=kF(s)ds (k:constant) 「[E(s)+E(sd=∫F(s)d+∫F(s)ds 「F(s)ds=∫F(s)d+∫F(s)ds Curve section c=c+cz 曲线线段c由c,c2组成 School of Mechanical Engineering ME369-lecture 3.0 Shanghai Jiao Tong University Fall 2015 4

4 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University ( ) cos s ni s e e j     1 2 1 2 s s s s e e e   ( ) ' s s e e  ln ln ( ) s i  r k     2 ( 0, 1, 2, ) k    j s j re       1 2 1 1 ln( ) ln( ) ln( ) s s s s   1 1 1 2 ln( ) ln( ) ln( ) s s s s   ( 0) s  a aln s s e ( 0) s  ( : ) a 复常数 Exponential function: 指数函数 Logarithmic function: 对数函数 Power function: 幂函数 复函数 BE315-Lecture 3.0 Fall 2011 ME369-lecture 3.0 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 复数积分性质 ( ) ( ) c c kF s ds k F s ds    ( : k constant) 1 2 1 2 [ ( ) ( )] ( ) ( ) c c c F s F s ds F s ds F s ds       1 2 ( ) ( ) ( ) c c c F s ds F s ds F s ds      Curve section c=c1+c2 曲线线段c由c1,c2组成