控制系統(tǒng)狀態(tài)空間表達(dá)式的學(xué)習(xí)教案

1、會計(jì)學(xué)1控制系統(tǒng)狀態(tài)控制系統(tǒng)狀態(tài)(zhungti)空間表達(dá)式的空間表達(dá)式的第一頁，共77頁。第1頁/共77頁第二頁，共77頁。0 0k kk k2 22 20 0)x)xt ta ak!k!1 1t ta a2!2!1 1atat(1(1x(t)x(t)則解則解, ,x xx(0)x(0)ax,ax,x x設(shè)標(biāo)量微分方程設(shè)標(biāo)量微分方程證明證明 ! :00kkkattnaex簡單到復(fù)雜簡單到復(fù)雜(fz)(fz)的處理方法的處理方法：在在t t時(shí)時(shí)的的狀狀態(tài)態(tài)？x x) )x x( (t tA Ax x, ,x x0 00 0第2頁/共77頁第三頁，共77頁。對于對于 ，解在形式上的推廣：，解在形

2、式上的推廣：0 00 0 x x) )x x( (t tA Ax x, ,x xk k0 0k k3 30 03 32 20 02 20 00 0t tx xA Ak k! !1 1t tx xA A3 3! !1 1t tx xA A2 2! !1 1t tA Ax xx xx x( (t t) )第3頁/共77頁第四頁，共77頁。0 0A At t0 00 0k kk kk k0 0k kk k2 22 2k k0 0k k3 30 03 32 20 02 20 00 0 x x e e) )x xt tA Ak k! !1 1( () )x xt tA Ak k! !1 1t tA A2

3、 2! !1 1A At t( (I It tx xA Ak k! !1 1t tx xA A3 3! !1 1t tx xA A2 2! !1 1t tA Ax xx x即即x x( (t t) ) 狀態(tài)轉(zhuǎn)移狀態(tài)轉(zhuǎn)移(zhuny)矩陣矩陣關(guān)鍵問題：狀態(tài)關(guān)鍵問題：狀態(tài)(zhungti)轉(zhuǎn)移矩陣轉(zhuǎn)移矩陣 eAt ?第4頁/共77頁第五頁，共77頁。矩矩陣陣, ,間間轉(zhuǎn)轉(zhuǎn)移移. .稱稱為為狀狀態(tài)態(tài)轉(zhuǎn)轉(zhuǎn)移移在在狀狀態(tài)態(tài)空空意意味味x x( (t t) )隨隨t t不不斷斷即即為為時(shí)時(shí)變變函函數(shù)數(shù)陣陣, ,為為t t的的函函數(shù)數(shù), ,一一般般, ,或或e e轉(zhuǎn)轉(zhuǎn)移移陣陣為為e e) )t tA A(

4、(t tA At t0 0) )x(t)x(tt t(t(t或x(t)或x(t)(t)x(0)(t)x(0)x(t)x(t)e e) )t t(t(te e記記(t)(t)0 00 0) )t tA(tA(t0 0AtAt0 0量量轉(zhuǎn)轉(zhuǎn)移移, ,) )到到終終態(tài)態(tài)x x( (t t) )的的向向或或x x( (t t) )反反映映從從初初態(tài)態(tài)x x( (0 0) )x x( (t te ex x( (t t) )x x( (0 0) )或或e e) )齊齊次次狀狀態(tài)態(tài)方方程程解解x x( (t t: :一一. .狀狀態(tài)態(tài)轉(zhuǎn)轉(zhuǎn)移移矩矩陣陣0 00 0) )t tA A( (t tA At t0

5、0第5頁/共77頁第六頁，共77頁。) )x x( (0 0) ) )( (t tt t( (t t) ) )x x( (t tt t( (t t) )則則x x( (t t) )t t) )和和( (t t若若知知x x( (t t1 11 12 21 11 12 22 21 12 21 1 ) )x x( (0 0) ); ;( (t tx xx x) )x x( (t t) )則則和和( (t tx xx x已已知知x x( (0 0) )例例: :1 12 21 11 11 11 11 12 20 01 10 0) )x x( (0 0) ); ;( (t tx xx x) )則則x

6、x( (t t) ), ,若若已已知知( (t t2 22 22 21 12 22 22 22 21 11 12 2A At tA At t) )t tA A( (t t2 21 11 12 2e ee e或或e e) )( (t t) ) )( (t tt t( (t t第6頁/共77頁第七頁，共77頁。B B) )t t( (A AB Bt tA At tn nn nn nn ne ee e有有e eB BA A時(shí)時(shí), ,當(dāng)當(dāng)且且僅僅當(dāng)當(dāng)A AB B, ,B B, ,5 5. .A A) )A A( (t tA AA At te ee ee e) ); ;( (t t1 1. .( (t

7、t) )( () )I Ie eI I; ;( (0 0) )t t) )2 2. .( (t tt t) )A A( (t tA At t1 1A At t1 1e ee et t) ); ;( ( (t t) ) 3 3. .A A; ;A A( (0 0) )( (0 0) )0 0時(shí)時(shí), ,t t A A; ;e eA Ae ee ed dt td d( (t t) )A A; ;A A( (t t) )( (t t) )4 4. .A At tA At tA At t狀態(tài)轉(zhuǎn)移狀態(tài)轉(zhuǎn)移(zhuny)矩陣的基本性質(zhì)：矩陣的基本性質(zhì)：第7頁/共77頁第八頁，共77頁。n n2 21 1 A

8、 A即即若若A A為為對對角角線線陣陣, ,0 00 0. 1t t t t t t A At tn n2 21 1e ee ee ee e則則 ( (t t) )0 00 0第8頁/共77頁第九頁，共77頁。1 1t t1 1t t t t t t A At t1 1T TT Te eT Te ee ee eT Te e( (t t) )則則 , ,A AT T即即T T變變換換對對角角線線化化, ,2 2. .若若A A能能通通過過非非奇奇異異n n2 21 10 00 01 1t tA At t1 1T TT Te ee e: :即即得得左左乘乘T T右右乘乘T TT T, ,e eT

9、T ) )T Tt tA A2 2! !1 1A At t( (I IT TT Tt tA AT T2 2! !1 1A AT Tt tT TI It t2 2! !1 1t tI Ie eA AT T, ,T T: :證證明明A At t1 12 22 21 12 22 21 11 12 22 2t t1 1 第9頁/共77頁第十頁，共77頁。121111211212ttttnttn0!)!(! t tJtJtAtAte ee ee e(t)(t)為約當(dāng)陣若A3.0011JA1)(第10頁/共77頁第十一頁，共77頁。1 1t tt t2 2! !1 1t t1 1t t1 1) )! !(

10、 (m m1 1t t2 2! !1 1t t1 1e ee e式式中中; ;e ee ee ee ee e則則( (t t) )2 21 1m mi i2 2t tt tA At tA At tA At tA AJ Jt tA At ti ii ii il l2 21 10 00 00 0i ii im mm mi ii ii ii i2 21 10 01 11 1其其中中A A, ,A A0 0A A0 0A AJ J( (2 2) )A A0 00 0lil,213 3. .若若A A為為約約當(dāng)當(dāng)陣陣第11頁/共77頁第十二頁，共77頁。第12頁/共77頁第十三頁，共77頁。3 32 2

11、3 32 23 32 23 32 22 22 2A At tt t2 25 5t t2 27 73 3t t1 1t t3 37 73 3t t2 2t tt t6 67 7t t2 23 3t tt tt t1 12 2! !t t3 32 21 10 0t t3 32 21 10 01 10 00 01 1e e, ,3 32 21 10 01 1. .A A例例2 2k kk k2 22 2A At tt tA Ak k! !1 1t tA A2 2! !1 1A At tI Ie e定義求解. 1第13頁/共77頁第十四頁，共77頁。1 1J Jt tA At t1 1T TT Te

12、ee e A AT T; ;T T則則J J, ,立立特特征征向向量量的的重重特特征征值值( (2 2) )A A有有只只對對應(yīng)應(yīng)一一獨(dú)獨(dú); ;T TT Te ee e, ,e ee ee ee eA AT TT T則則 ( (1 1) )A A特特征征值值互互異異, ,1 1 t tA At tt t t t t t t t1 1n n2 21 10 00 0變變換換A A為為約約當(dāng)當(dāng)標(biāo)標(biāo)準(zhǔn)準(zhǔn)型型. 2第14頁/共77頁第十五頁，共77頁。2 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt t1 1t tA At t1 12 21 12 2e ee e2

13、2e e2 2e ee ee ee e2 2e e1 11 11 12 2e e0 00 0e e2 21 11 11 1T TT Te ee e1 11 11 12 2T T, ,2 21 11 11 11 11 1T T 2 21 1, ,0 0, ,2 2) )1 1) )( ( (2 23 3A AI I為為友友矩矩陣陣3 32 21 10 02 2. .A A例例2 22 21 12 2第15頁/共77頁第十六頁，共77頁。2 2t t2 2t t2 2t t2 2t t2 2t t2 2t t2 2t t1 1J Jt tA At t2 2t te ee e2 2t te e2 2

14、t te e2 2t te ee e2 22 22 21 11 10 0t t1 1e e1 1/ /2 21 11 11 1T TT Te ee e. .只只對對應(yīng)應(yīng)一一獨(dú)獨(dú)立立特特征征向向量量2 2, ,0 0, ,4 44 44 42 22 2A AI I; ;4 42 22 20 0補(bǔ)補(bǔ)例例A A1 1, ,2 22 22 20 01 12 2J J2 22 22 21 1T T , ,1 1/ /2 21 11 11 1p pp pT T: :求求出出p pA A) )p pI I( (0 0A A) )p pI I( (由由1 12 21 11 12 22 21 11 1,第16頁

15、/共77頁第十七頁，共77頁。由;2 20 00 00 01 10 00 01 11 1A AT TT T. .J J應(yīng)應(yīng)一一獨(dú)獨(dú)立立特特征征向向量量只只對對故故1 1, ,n n秩秩為為2 2A A) )I I( ( 1 11 11 1, ,2 2重重特特征征值值352110011為為友友矩矩陣陣, ,4 45 52 21 10 00 00 01 10 0A A1211322524112011111 13 32 21 13 33 31 12 21 11 11 1T Tp pp pp p求求T T0 0A A) )p pI I( ( p pA A) )p pI I( ( 0 0A A) )p

16、pI I( ( ,2 2, , 1 1, , 3 31 1, ,2 2, 0 02 25 54 4A AI I2 23 3則2.2 矩陣指數(shù)函數(shù)矩陣指數(shù)函數(shù)(zh sh hn sh)-狀態(tài)轉(zhuǎn)移矩陣狀態(tài)轉(zhuǎn)移矩陣 求法求法2第17頁/共77頁第十八頁，共77頁。1 12 21 11 13 32 22 25 52 2e e0 00 00 0e e0 00 0t te ee e4 41 11 12 20 01 11 11 11 1T TT Te ee e2 2t tt tt tt t1 1J Jt tA At t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt t

17、t t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt t4 4e e3 3e et te e8 8e e8 8e e3 3t te e4 4e e4 4e e2 2t te e2 2e e2 2e et te e4 4e e5 5e e3 3t te e) )e ee e2 2( (t te ee ee et te e2 2e e2 2e e3 3t te ee e2 2t te e2.2 矩陣矩陣(j zhn)指數(shù)函數(shù)指數(shù)函數(shù)-狀態(tài)轉(zhuǎn)移矩陣狀態(tài)轉(zhuǎn)移矩陣(j zhn) 求法求法2第18頁/共77頁第十九頁，共77頁。 A A) )

18、 ( (s sI Ie e1 1A At t1 1L L0 00 00 0 x xA A) )X X( (s s) )( (s sI IA AX X( (s s) )x xs sX X( (s s) ): :兩兩邊邊取取拉拉氏氏變變換換x xx x( (0 0) )A Ax x( (t t) ), ,( (t t) )x x0 01 10 01 1 x xA A) ) ( (s sI Ix x( (t t) )x xA A) )( (s sI IX X( (s s) )1 1L L第19頁/共77頁第二十頁，共77頁。2 2t tt t2 2t tt t2 2t tt t2 2t tt t1

19、1A At t2 2e ee e2 2e e2 2e ee ee ee e2 2e e A A) ) ( (s sI Ie e1 1L L2 2s s2 21 1s s1 12 2s s2 21 1s s2 22 2s s1 11 1s s1 12 2s s1 11 1s s2 22 2) )1 1) )( (s s( (s ss s2 2) )1 1) )( (s s( (s s2 22 2) )1 1) )( (s s( (s s1 12 2) )1 1) )( (s s( (s s3 3s ss s2 21 13 3s s2 23 3s ss s1 1A A) )a ad dj j( (

20、s sI IA As sI I1 1A A) )( (s sI I2 21 13 3s s2 21 1s sA A) )( (s sI I, ,3 32 21 10 04 4. .A A例例2 2第20頁/共77頁第二十一頁，共77頁。I I的的線線性性組組合合. .A A, , , ,也也是是A A, ,A A, ,A A同同理理, ,I I的的線線性性組組合合, ,A A, , , ,為為A AI Ia aA Aa aA Aa aA Aa a由由以以上上定定理理得得：A A1 1n n2 2n n1 1n n1 1n n0 01 12 2n n2 2n n1 1n n1 1n nn n(

21、(t t) )I I( (t t) )A A( (t t) )A A( (t t) )A At tA A1 1) )! !( (n n1 1t tA An n! !1 1t tA A1 1) )! !( (n n1 1t tA A2 2! !1 1A At tI Ie e0 01 12 2n n2 2n n1 1n n1 1n n1 1n n1 1n nn nn n1 1n n1 1n n2 22 2A At t : :哈哈密密頓頓定定理理4 4. .用用凱凱萊萊0 0I Ia aA Aa aA Aa aA A則則f f( (A A) )0 0, ,a aa aa aA AI I若若0 01

22、11 1n n1 1n nn n0 01 11 1n n1 1n nn n1 1n n0 0j jj jj j(t)(t)A A問題問題(wnt)：如何確定系數(shù)：如何確定系數(shù)n-1, , 0？第21頁/共77頁第二十二頁，共77頁。第22頁/共77頁第二十三頁，共77頁。t tt tt t1 11 1n nn n2 2n nn n1 1n n2 22 22 22 21 1n n1 12 21 11 11 1n n1 10 0j jn n2 21 1e ee ee e1 11 11 1( (t t) )( (t t) )( (t t) ): :) )( (t t) )的的計(jì)計(jì)算算公公式式( (1

23、 1同同時(shí)時(shí)，方方程程組組有有唯唯一一解解( (1 1) )當(dāng)當(dāng)A A特特征征值值互互相相第23頁/共77頁第二十四頁，共77頁。2 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt t1 10 0A At t2 2e ee e2 2e e2 2e ee ee ee e2 2e e3 32 21 10 0) )e e( (e e1 10 00 01 1) )e e( (2 2e e( (t t) )A A( (t t) )I Ie e, ,e ee ee e2 2e ee ee e1 11 11 12 2e ee e2 21 11 11 1e

24、 ee e1 11 1( (t t) )( (t t) )2 2t tt t2 2t tt t2 2t tt t2 2t tt t1 1t tt t1 12 21 11 10 02 21 12 21 1, , ,3 32 21 10 0A A2 21 1第24頁/共77頁第二十五頁，共77頁。t tt tt t2 2t t2 2n nt t1 1n n1 11 1n n1 12 2n n1 12 21 11 12 2n n1 13 3n n1 11 13 3n n1 11 11 1n n2 2n n3 3n n1 10 01 11 11 11 11 1e et te ee et t2 2! !

25、1 1e et t2 2) )! !( (n n1 1e et t1 1) )! !( (n n1 11 11 1) )( (n n2 2) )( (n n2 21 12 2! !2 2) )1 1) )( (n n( (n n1 1) )( (n n1 11 1( (t t) )( (t t) )( (t t) )( (t t) )( (t t) )0 0為為A A的的n n重重特特征征值值時(shí)時(shí)( (2 2) )當(dāng)當(dāng)1 1tnntetntt2121)() 1(.)(2)( 一階導(dǎo)數(shù)tnnettnn11)()!1( 階導(dǎo)數(shù)第25頁/共77頁第二十六頁，共77頁。2 2t tt tt t2 2t

26、 tt tt t2 2t tt t2 2t tt tt t2 2t tt tt t1 1e ee et te e2 2e e2 2e e3 3t te ee e2 2t te ee ee et te e1 11 11 12 22 23 31 10 02 2e ee et te e4 42 21 11 11 11 12 21 10 0; ;2 21 1, ,4 45 52 21 10 00 00 01 10 0A A3 31 1, ,2 2t tt tt t1 12 23 33 32 21 11 11 12 21 10 03 31 11 1e ee et te e1 11 12 21 10 0(

27、 (t t) )( (t t) )( (t t) )例例第26頁/共77頁第二十七頁，共77頁。2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt tt t2 2t tt t4 4e e3 3e et te e8 8e e8 8e e3 3t te e4 4e e4 4e e2 2t te e2 2e e2 2e et te e4 4e e5 5e e3 3t te e) )e ee e2 2( (t te ee ee et te e2 2e e2 2e e3 3

28、t te ee e2 2t te e2 2t tt tt t2 2t tt tt t2 2t tt t2 21 10 0e ee et te e2 2e e2 2e e3 3t te ee e2 2t te e( (t t) )( (t t) )( (t t) )2 22 21 10 0A At t( (t t) )A A( (t t) )A A( (t t) )I Ie e4 45 52 21 10 00 00 01 10 04 45 52 21 10 00 00 01 10 0) )e ee et te e( (4 45 52 21 10 00 00 01 10 0) )2 2e e2 2

29、e e( (3 3t te e1 10 00 00 01 10 00 00 01 1) )e e2 2t te e( (2 2t tt tt t2 2t tt tt t2 2t tt t 第27頁/共77頁第二十八頁，共77頁。t tt t) )A(tA(t0 0) )t tA(tA(tt tt tA A0 0AtAtAtAt0 00 00 00 0Bu(Bu()d)de e) )x(tx(te ex(t)x(t)Bu(Bu()d)de e) )x(tx(te ex(t)x(t)e eB Bu u( (t t) )e ex x e ed dt td d即即 B Bu u( (t t) ), ,

30、e eA Ax x) )( (x xe eB Bu uA Ax xx xA At tA At tA At tA At tt tt tA A0 0A At tt tA At tt tA A0 00 00 0B Bu u( () )d de et tt tx xe e 則則 B Bu u( () )d de ex x d d e ed dd d兩兩邊邊取取積積分分Bu的解Bu的解AxAxx x（1 1）積分法）積分法第28頁/共77頁第二十九頁，共77頁。BU(s)BU(s)A)A)(sI(sIx(0)x(0)A)A)(sI(sIBU(s)BU(s)A)A)(sI(sIx(0)x(0)A)A)(s

31、I(sI X(s)X(s)x(t)x(t)1 11 11 11 11 11 11 11 1L LL LL LL L B BU U( (s s) )x x( (0 0) )A A) )X X( (s s) )( (s sI I B BU U( (s s) )A AX X( (s s) )x x( (0 0) )s sX X( (s s) ): :拉拉氏氏變變換換法法B BU U( (s s) )A A) )( (s sI Ix x( (0 0) )A A) )( (s sI IX X( (s s) )1 11 1x x( (0 0) ) )0 0已已知知x x( (t t若若t tB Bu u,

32、 ,A Ax xx x0 00 0（2 2）拉氏變換）拉氏變換(binhun)(binhun)第29頁/共77頁第三十頁，共77頁。 A A) ) ( (s sI Ie e1 1A At t1 1L Ldettt0)()()0()(BuxxAA At te e第30頁/共77頁第三十一頁，共77頁。2 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2e ee e2 2e e2 2e ee ee ee e2 2e e( (t t) )1 1( (t t) )u u( (t t) )u u, ,1 10 0 x x3 32 21 10 0 x x2 2t tt t2 2t

33、 tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt tt t0 0) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t2 21 12 2t tt t2 2t tt t2 2t tt t2 2t tt te ee ee e2 21 1e e2 21 1) )x x( (0 0) )2 2e ee e( () )x x( (0 0) )2 2e e2 2e e( () )x x( (0 0) )e e( (e e) )x x( (0 0) )e e( (2

34、 2e e1 1( () )d d1 10 02 2e ee e2 2e e2 2e ee ee ee e2 2e e( (0 0) )x x( (0 0) )x x2 2e ee e2 2e e2 2e ee ee ee e2 2e ex x( (t t) )1、積分法、積分法第31頁/共77頁第三十二頁，共77頁。2 2t t2 21 1t t2 21 12 2t t2 21 1t t2 21 11 1 e e( (0 0) )2 2x x( (0 0) ) 2 2x x1 1 e e( (0 0) )x x( (0 0) ) 2 2x x e e2 21 1( (0 0) )x x( (

35、0 0) ) x x1 1 e e( (0 0) )x x( (0 0) ) 2 2x x2 21 12 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt t2 2t tt tt t0 0) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t) )2 2( (t t) )( (t t2 21 12 2t tt t2 2t tt t2 2t tt t2 2t tt te ee ee e2 21 1e e2 21 1) )x x( (0 0) )2 2e ee e( () )x x( (0 0

36、) )2 2e e2 2e e( () )x x( (0 0) )e e( (e e) )x x( (0 0) )e e( (2 2e e1 1( () )d d1 10 02 2e ee e2 2e e2 2e ee ee ee e2 2e e( (0 0) )x x( (0 0) )x x2 2e ee e2 2e e2 2e ee ee ee e2 2e ex x( (t t) )第32頁/共77頁第三十三頁，共77頁。B BU U( (s s) )A A) )( (s sI Ix x( (0 0) )A A) )( (s sI IX X( (s s) )s s2 21 13 3s s2

37、 23 3s ss s1 1A A) )( (s sI I; ;3 3s s2 21 1s sA A) )( (s sI I: :2 2. .拉拉氏氏變變換換法法1 11 12 21 11 11/s1/s2 23s3ss s1 1(0)(0)sxsx(0)(0)2x2x(0)(0)x x(0)(0)3)x3)x(s(s2 23s3ss s1 1s s1 11 10 0s s2 21 13 3s s2 23s3ss s1 1(0)(0)x x(0)(0)x xs s2 21 13 3s s2 23s3ss s1 12 22 21 12 21 12 22 22 21 12 2結(jié)結(jié)果果同同積積分分法

38、法 X X( (s s) ) x x( (t t) ), ,2 2s s1 1( (0 0) )2 2x x( (0 0) )2 2x x1 1s s1 1( (0 0) )x x( (0 0) )2 2x x2 2s s1 1/ /2 2( (0 0) )x x( (0 0) )x x1 1s s1 1( (0 0) )x x( (0 0) )2 2x xs s1 1/ /2 22 2s s1 11 1s s1 12 2s s1 1/ /2 21 1s s1 1s s1 1/ /2 22 2s s( (0 0) )2 2x x( (0 0) )2 2x x1 1s s( (0 0) )x x

39、( (0 0) )2 2x x2 2s s( (0 0) )x x( (0 0) )x x1 1s s( (0 0) )x x( (0 0) )2 2x x2 21 12 21 12 21 12 21 12 21 12 21 12 21 12 21 11 1L L第33頁/共77頁第三十四頁，共77頁。一.解的特點(diǎn):一.解的特點(diǎn):) )x x( (t t ) )a a( () )d de ex xp p( (x x( (t t) ) a a( () )d d) )l ln nx x( (t tl ln nx x( (t t) ) ) )l ln n( (x x( () )x x( () )d

40、dx x( (a a( (t t) )d dt tx x( (t t) )d dx x( (t t) )a a( (t t) )x x( (t t) )d dt td dx x( (t t) )x x: :標(biāo)標(biāo)量量系系統(tǒng)統(tǒng)0 0t tt tt tt t0 0t tt tt tt t0 00 00 00 0 ) ) )d dA A( (e ex xp p( () )t t( (t t, , ) ), ,) )x x( (t tt t( (t t, ,x x( (t t) )A A( (t t) )x x( (t t) )( (t t) )x x: :推推廣廣到到向向量量方方程程t tt t0 0

41、0 00 00 0是否是否(sh fu)可可推廣？推廣？第34頁/共77頁第三十五頁，共77頁。) ) x x( (t tA A( () )d de ex xp p ) ) )x x( (t tt t( (t t, ,才才有有x x( (t t) )時(shí)時(shí), ,0 0t tt t0 00 00 0A A( (t t) )A A( () )d dd dA A( () )A A( (t t) )滿滿足足乘乘法法交交換換條條件件即即A A( () )d d但但只只有有A A( (t t) )和和t tt tt tt tt tt t0 00 00 0第35頁/共77頁第三十六頁，共77頁。. .成成封封

42、閉閉形形式式，而而是是級級數(shù)數(shù)一一般般很很難難滿滿足足，不不能能寫寫此此條條件件很很苛苛刻刻, ,3 3t tt t2 2t tt tt tt tt tt t0 00 00 00 0) )d dA A( (3 3! !1 1) )d dA A( (2 2! !1 1) )d dA A( (I I) )d dA A( (e ex xp pA A( (t t) )成成立立. .) )d dA A( () )d dA A( (A A( (t t) )須須式式成成立立, ,使使要要比比較較上上兩兩式式, ,t tt tt tt t 0 0 0 00 0 2 2t tt tt tt tt tt t )

43、)d dA A( (A A( (t t) ) 2 2! !1 1) )d dA A( (A A( (t t) )A A( (t t) ) )d dA A( (A A( (t t) )e ex xp p: :) )對對上上式式兩兩邊邊左左乘乘A A( (t t0 00 00 0 A A( () )d dA A( (t t) )e ex xp p A A( () )d d e ex xp pd dt td dt tt tt tt t0 00 0則則須須) )是是方方程程解解, , x x( (t tA A( () )d d若若e ex xp p : :證證明明0 0t tt t0 0A A( (t

44、 t) ) ) )d dA A( (2 21 1) )d dA A( (A A( (t t) )2 21 1A A( (t t) ) ) )d dA A( ( e ex xp pd dt td dt tt tt tt tt tt t0 00 00 0第36頁/共77頁第三十七頁，共77頁。) )t tA A( (t t) )( (t t, ,) )t t( (t t, ,( (4 4) ). .t t) ), ,( (t t) )t t( (3 3) ). .( (t t, ,I It t) )( (2 2) ). .( (t t, ,) )t t, ,( (t t) )t t, ,) )(

45、(t tt t, ,( (1 1) ). .( (t t) )的的基基本本性性質(zhì)質(zhì)t t三三. .( (t t, ,0 00 00 01 10 00 02 20 01 11 12 20 0I I) )t t, ,) )和和( (t tt tA A( (t t) )( (t t, ,) )t t( (t t, , 滿滿足足n n非非奇奇異異陣陣, ,n n) ), ,t tt t) )類類似似定定常常系系統(tǒng)統(tǒng)的的( (t t式式中中( (t t, , ) ), ,) )x x( (t tt t( (t t, ,x x( (t t) ) ) )的的解解為為：x x( (t tx x( (t t)

46、)A A( (t t) )x x( (t t) ), ,x x方方程程二二. .線線性性時(shí)時(shí)變變其其次次狀狀態(tài)態(tài)0 00 00 00 00 00 00 00 00 0t tt t0 0(t)A(t)AA A(t)(t)(t)(t)4.4.t)t)( (t)(t)3.3.I I(0)(0)t)t)2.2.(t(t) )(t(t1.1.(t)(t)( () ): :定常系統(tǒng)狀態(tài)轉(zhuǎn)移矩陣定常系統(tǒng)狀態(tài)轉(zhuǎn)移矩陣1 1第37頁/共77頁第三十八頁，共77頁。t tt t0 00 00 00 0d d) )B B( () )u u( () )( (t t, ,) ) )x x( (t tt t( (t t,

47、 ,則則x x( (t t) )分分段段連連續(xù)續(xù), ,t t 內(nèi)內(nèi), ,B B( (t t) )的的元元在在 t t設(shè)設(shè)A A( (t t) ), ,B B( (t t) )u u( (t t) ), ,A A( (t t) )x x( (t t) )( (t t) )x x次次狀狀態(tài)態(tài)方方程程的的解解四四. .線線性性時(shí)時(shí)變變系系統(tǒng)統(tǒng)非非齊齊 1 1 u u0 00 0u u0 00 00 0( (t t) ) _ _ _x x) ) ) x x( (t tt t( (t t, , ( (t t) ) )x xt t( (t t, ,) ) )x x( (t tt t( (t t, ,則則x

48、 x( (t t) ), ,線線性性系系統(tǒng)統(tǒng)滿滿足足疊疊加加原原理理B B( (t t) )u u( (t t) ), ,A A( (t t) )x x( (t t) )( (t t) )x xB B( (t t) )u u( (t t) )A A( (t t) )x x( (t t) )( (t t) )x x) )t t( (t t, ,( (t t) ) x x) ) ) x x( (t tt tA A( (t t) )( (t t, ,B B( (t t) )u u( (t t) )A A( (t t) )x x( (t t) )( (t t) )x x) )t t( (t t, ,(

49、 (t t) ) x x) ) ) x x( (t tt t( (t t, ,u u0 0u u0 00 0u u0 0u u0 00 0第38頁/共77頁第三十九頁，共77頁。d d) )B B( () )u u( () )( (t t, ,) ) )x x( (t tt t( (t t, ,d d) )B B( () )u u( () ), ,( (t t) )t t( (t t, ,) ) )x x( (t tt t( (t t, ,x x( (t t) )t tt t0 00 0t tt t0 00 00 00 00 00 0t t) )B B( (t t) )u u( (t t) )

50、, ,( (t t) )B B( (t t) )u u( (t t) )t t( (t t, ,( (t t) )x xB B( (t t) )u u( (t t) )A A( (t t) )x x( (t t) )( (t t) )x x) )t t( (t t, ,A A( (t t) )x x( (t t) )0 00 01 1u uu u0 00 0) )( (t t得得x x代代入入t t用用t t) ), ,( (t tx xd d) )B B( () )u u( () ), ,( (t t( (t t) )x x0 0u u 1 1 0 00 0u ut tt t0 0u u0

51、0第39頁/共77頁第四十頁，共77頁。t tt t0 0t t1 1t t2 22 21 10 0t tt t0 0t t1 11 10 0t tt t0 00 00 00 01 10 00 00 00 00 0d dd d) )d dA A( () )A A( () )A A( (d d) )d dA A( () )A A( (A A( () )d dI I) )t t( (t t, ,則則不不滿滿足足該該條條件件, ,一一般般, , t tt t3 3t tt t2 2t tt tt tt t0 00 0t tt t0 00 00 00 00 00 00 0A A( () )d d 3

52、3! !1 1A A( () )d d 2 2! !1 1A A( () )d dI I A A( () )d de ex xp p ) )t t即即( (t t, ,) ) x x( (t tA A( () )d de ex xp p ) ) )x x( (t tt t( (t t, ,才才有有x x( (t t) )A A( (t t) )時(shí)時(shí), ,A A( () )d dd dA A( () )交交換換條條件件即即A A( (t t) )滿滿足足乘乘法法A A( () )d d只只有有A A( (t t) )和和t tt tt tt tt tt t0 00 00 0第40頁/共77頁第四

53、十一頁，共77頁。2 22 2t t0 0t t0 0t t2 21 1t tt tt t2 21 1d dt t1 11 1t tA A( () )d d: :求求( (t t, ,0 0) )t t1 11 1t tA A( (t t) ) 例例1 1?t t0 0t t0 0A A( () )d dA A( (t t) )A A( (t t) )A A( () )d dt t1 11 1t tt t2 21 1t tt tt t2 21 1t tt t2 21 1t t2 23 3t t2 23 3t tt t2 21 1t t2 21 1t tt tt t2 21 1t t1 11 1

54、t t2 22 23 32 22 23 32 22 2可可寫寫為為封封閉閉形形式式A A( () )d dA A( (t t) )A A( (t t) )A A( () )d d即即t t0 0t t0 0,第41頁/共77頁第四十二頁，共77頁。4 42 23 33 34 42 22 22 22 22 22 2t t0 0t t8 81 1t t1 1t t2 21 1t tt t2 21 1t tt t8 81 1t t1 1t t2 21 1t tt tt t2 21 12 21 1t t2 21 1t tt tt t2 21 11 10 00 01 1 A A( () )d de ex

55、 xp p ( (t t, ,0 0) )t tt t3 3t tt t2 2t tt tt tt t0 00 00 00 00 0A A( () )d d 3 3! !1 1A A( () )d d 2 2! !1 1A A( () )d dI I A A( () )d de ex xp p ) )t t即即( (t t, ,第42頁/共77頁第四十三頁，共77頁。表表達(dá)達(dá). .( (t t, ,0 0) )只只能能級級數(shù)數(shù)A A( (t t) ), ,A A( () )d dA A( () )d dA A( (t t) )t t0 0t t0 0a at t2 2t t0 0a at tt

56、 t0 0e ea a1 10 02 2t t0 0d de e0 0t t0 0A A( () )d d: :求求( (t t, ,0 0) )x x, ,e e0 0t t0 0 x xa at t2 2a at ta at ta at t2 2a at tt t0 0e ea a1 10 0t te ea a1 10 0e ea a1 10 02 2t t0 0e e0 0t t0 0A A( () )d dA A( (t t) )?t t0 0t t0 0A A( () )d dA A( (t t) )A A( (t t) )A A( () )d d第43頁/共77頁第四十四頁，共77

57、頁。2 2a at t2 2a at t3 32 22 2a at t2 23 3a at t2 20 0a a2 20 0t t0 0a a0 0a at t2 2e e2 2a a1 1e ea a1 11 10 01 1) )( (a at ta a1 1t t2 21 11 1e e2 2a a1 10 01 1) )( (a at ta a1 10 0e ea a1 10 02 2t t0 01 10 00 01 1d de ea a1 10 02 20 0e e0 00 0e ea a1 10 02 2t t0 01 10 00 01 1( (t t, ,0 0) )0 00 0t

58、 tt t0 0t t1 1t t2 22 21 10 0t tt t0 0t t1 11 10 0t tt t0 00 00 00 01 10 00 00 00 00 0d dd d) )d dA A( () )A A( () )A A( (d d) )d dA A( () )A A( (A A( () )d dI I) )t t( (t t, , 第44頁/共77頁第四十五頁，共77頁。0 00 01 12 22 21 13 30 01 11 12 22 23 31 12 22 23 32 23 33 3b ba aa aa ab ba aa ab ba ab bu ux xy yu ux

59、 xa ax xa ax xa ax xu ux xx xu ux xx x3 31 10 03 32 22 21 11 10 03 31 13 32 22 22 21 1u ub bu ub bu ub bu ub by ya ay ya ay ya ay y0 01 12 23 30 01 12 2u u. .x xy yu ux xa aa aa ax x3 30 01 12 21 10 00100010201第45頁/共77頁第四十六頁，共77頁。u u. .x x0 00 01 1y yu ux xa aa aa aI I0 00 0 x xn n0 01 11 1n n1 1n n

60、1 10 01 1n n0 01 12 2n n1 1n nn n0 01 12 2n n1 1n nn n1 1n n2 21 10 03 32 21 11 1n n2 2n n1 1n nb bb bb bb bb b1 1a aa aa aa a0 01 1a aa aa a0 00 01 1a aa a0 00 00 01 1a a0 00 00 00 01 1: :關(guān)系為關(guān)系為與b的與b的式中,式中,0 01 11 1n n1 1n nn n0 01 11 1n n1 1n nn nn na as sa as sa as sb bs sb bs sb bs sb bW W( (s s