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[17]Lecture91-state space 控制系统的状态空间分析

ME369-系统棋型、分析与控制 9.1控制系统的状态空间分析 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 引例1] R ☐000m duc=i dt di +Ri+uc =u School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 1

1 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 9.1控制系统的状态空间分析 ME369-系统模型、分析与控制 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [引例 1]            Ri u u dt di L i dt du C C C

状态空间表达 D ()=Ax(1)+Bu(t) y(t)=Cx(t)+Du(t) School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 例1] M d +b+e=0) dt x(t)=x(t) M x2(t)=x(t) L白b y=x School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 2

2 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 状态空间表达 x Ax Bu ( ) ( ) ( ) t t t   y Cx Du ( ) ( ) ( ) t t t   ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [例1] 2 2 ( ) d x dx M b kx u t dt dt    1 x t x t ( ) ( )  2 x t x t ( ) ( )  1 y x 

[例2] (0 「M血=K,0,-y+B,_)-Ky-B血 =f-K0-y)B,(座-$ M2 dt dt dt School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 模型转化一能控式、 能观式状态空间 ym+ay-l+…an-少+any=btm+b4im-l+…+bni+b,u Y(s) bs”+bs"-1++bn-1S+ba U(s) s"+as+.+ars+an 0 1 0 07 能控式(Controllable canonical form) 0 0 1… 0 x+ 0 0:0 0 0… -0。-01-an-2… -a, 1 y=[b-a,bo b-ambo ···b2-a2b b-abo]x+bou 「00… 0 -a. 能观式(Observable canonical form) 10 0 -an-l -a bo 龙= 1+ 00.… 0 -4 b,-a,bo 00…1 a L点-ab y=[00,..0]x+u School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 3

3 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 2 2 1 2 2 2 1 2 ( ) ( ) dv dy dy M f K y y B dt dt dt      1 2 1 1 1 2 2 1 2 1 1 ( ) ( ) dv dy dy dy M K y y B K y B dt dt dt dt       [例2] ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 模型转化—能控式、能观式状态空间 ( ) ( 1) ( ) ( 1) 1 1 0 1 1 n n n n n n n n y a y a y a y b u b u b u b u             1 0 1 1 1 1 1 ( ) ... ( ) ... n n n n n n n n Y s b s b s b s b U s s a s a s a              1 2 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 n n n u a a a a                                         x x y b a b b a b b a b b a b b u           n n n n 0 1 1 0 2 2 0 1 1 0 0    x 0 1 1 1 0 2 2 2 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 n n n n n n a b a b a b a b u a b a b a b a b                                              x x   0 y b u   0 0 . . . 0 1 x 能控式（Controllable canonical form） 能观式（Observable canonical form）

模型转化一对角式状态空间 Y(s)bos"+bs"+..+bs+b U(s) s"+as"+...+an-is+an -P -P2 i- x+ y=[9g2··clnx+bu 6o+ C +9 (s+P)3(s+p)2s+台s+ School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 例3]模型转化 Y(s) 52+8s+15 U(s)s3+7s2+145+8 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 4

4 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 1 0 1 1 1 1 1 ( ) ... ( ) ... n n n n n n n n Y s b s b s b s b U s s a s a s a              1 2 1 1 n 1 n n p p x x u p                                1 2 0 1 ... n n y c c c x b u    1 1 1 1 2 3 0 3 2 ( ) ( ) 4 n i i i c c c c b s p s p s p s p            0 1 n i i i c b  s p    模型转化— 对角式状态空间 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [例 3]模型转化 2 3 2 ( ) 8 15 ( ) 7 14 8 Y s s s U s s s s      

模型转换一状态空间toT℉ I文=Ax+Bu sX(s)-x(0)=AX(s)+BU(s) LT y=Cx+Du Y(s)=CX(s)+DU(s) x(0)=0 X(s)=(sI-A)BU(s) G(S)=Y()=C(SI-AB+D U(s) Ys)=「C(sl-A)B+DU(s) D=0 G(s)= Y(s)= adi(sI-A)B U(s) sI-A School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 例4]模型转化 卧 0 0 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 5

5 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 模型转换—状态空间 to TF x = Ax + Bu y Du  Cx + LT s s s U s X x AX B ( ) (0) ( ) ( )    Y s s DU s ( ) ( ) ( )   CXx(0) 0  1 ( ) ( ) ( ) s s U s  X I A B   1 Y s s D U s ( ) ( ) ( )         C I A B 1 ( ) ( ) ( ) ( ) Y s G s s D U s      C I A B D  0 ( ) ( ) ( ) ( ) Y s adj s G s U s s     I A C B I A ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 1 1 2 2 0 1 0 1 x x k b u x x M M M                                   1 2 1 0 x y x        [例 4]模型转化

复习]矩阵的线性变换 -PAPHPP-P-APP(I-A)P P-1I(I-A)川P曰P-1IPI(-A)I P-PI(I-A)月(2I-A)川 文=Ar+Bu x=Pz y=Cx+Du School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fal12015 能控式的对角化 0 1 0 0 0 0 0 =Ax+Bu= x+ y=Cx 0 0 0 1 -a.-a.1-an-2 4 -41 1 1… 11 P= 房 店 … 无- 「0… 0 A=PAP= 0 B=PB C=CP 0 0 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 6

6 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [复习]矩阵的线性变换 x   Ax  Bu y  Cx  Du x  Pz | | | | | ( ) | -1 -1 -1 -1 I  P AP  P P  P AP  P I - A P | || ( ) || | | || || ( ) | -1 -1  P I - A P  P P I - A | || ( ) | | ( ) | -1  P P I - A  I - A ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 能控式的对角化 1 2 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 n n n u u a a a a                                           x Ax B x 1 2 2 2 2 1 2 1 1 1 1 2 1 1 1 n n n n n n                              P y  Cx               n          0 0 0 0 0 0 ~ 2 1 -1 A P AP B P B ~ -1  C  CP ~

[例4]能控式的对角化 0 1 01 001x+0u y=[00 -6-11-66 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 齐次状态方程&状态转移矩阵 文=Ax+Bu (t)=ax(t) u(t)=0 x(t)=e"x(0) x(t)=Ax(t)） x(t)=e4x(0) e=1+a++…+a+…=2a 21 k! e=I+At+ 21 k! 该无穷级数在有限时间是绝对收敛的 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 7

7 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University [例4 ] 能控式的对角化 u                          6 0 0 6 11 6 0 0 1 0 1 0 x x y  1 0 0x ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 齐次状态方程& 状态转移矩阵 x Ax B   u u t( ) 0  x Ax ( ) ( ) t t  ( ) (0) at x t e x  x t ax t ( ) ( )  ( ) (0) t t e  A x x         2 2 ! 1 1 1 2! ! ! At k k k k k e I At A t A t A t k k 2 2 0 1 1 1 1 2! ! ! at k k k k k e at a t a t a t k k          该无穷级数在有限时间是绝对收敛的

齐次状态方程&状态转移矩阵（续） e40=I If AB=BA e+)=e·e4 lev]"=err e4-)e4-)=e-) If AB+BA de“=Ae d [e"=e e 0 0 School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fal12015 齐次状态方程&状态转移矩阵（续） (t)=Ax(t) |r sX(s)-x(0)=AX(s) X(s)=(sI-A)x(0) IT x(0=L[X(s=E[(sl-A)x(0)] x(①)=e"xo =[(sl-A1x(0) e4=L-[(sI-A)] School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 8

8 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University d t t e e dt  A A A 1 2 1 2 t t t t e e e    A A A ( )1 [ ]t t e e    A A If AB=BA If AB≠BA -1 e e  P AP A P P -1                 1 2 0 0 n A                 1 2 0 0 n t t At t e e e e e  A0 I t n n t [ ] e e  A A t t t t t t e e e     2 1 1 0 2 0 A( ) A A ( ) ( ) 齐次状态方程& 状态转移矩阵（续） ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University x Ax ( ) ( ) t t  s s s X x AX ( ) (0) ( )   LT ( ) ( ) (0) s s  X I A x   1   -1 -1 1 ( ) ( ) ( ) (0) - t L s = L s -      x X I A x ILT -1 L s - ( ) (0)      1 I A x 0 ( ) t t e  A x x [( ) ] t e L s     A I A 1 1 齐次状态方程& 状态转移矩阵（续）

[例5-1]状态转移矩阵求取 4[ e4 =L[(SI-A)] School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 [例5-2]状态转移矩阵求取 n！ School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 9

9 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 0 1 = -2 -3       A [( ) ] t e L s     A I A 1 1 [例5-1] 状态转移矩阵求取 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 0 1 = -2 -3       A 1 1 2 2 2 ! t n n e t t t n      A I A A A ! [例5-2] 状态转移矩阵求取

[例5-3]状态转移矩阵求取 A=pAp 对角线形式 4[ ePAP=plep e==pe p School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 [例5-4]状态转移矩阵求取* 凯莱-哈密顿定理(Cayley-Hamilton theorem) e"=a,0)1+a0）A+.…+an0A [1元 a(t) a(t) School of Mechanical Engineering ME369-lecture 9.1 Shanghai Jiao Tong University Fall 2015 10

10 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 0 1 = -2 -3       A -1 e e  P AP A P P -1 1 t t t 1 1 e e e       A p Ap A p p p p ˆ 1 A p Ap   对角线形式 [例5-3] 状态转移矩阵求取 ME369-lecture 9.1 Fall 2015 School of Mechanical Engineering Shanghai Jiao Tong University 0 1 = -2 -3       A 凯莱-哈密顿定理（Cayley-Hamilton theorem） 0 1 ( ) ( ) ( ) t n n e a t a t a t       A I A A 1 1 1 2 1 2 1 0 1 1 1 2 1 1 2 2 2 2 1 1 ( ) 1 ( ) 1 ( ) 1 n n t n t n t n n n n a t e a t e a t e                                                         [例5-4] 状态转移矩阵求取*