现代控制理论基础：第三章线性控制系统的运动分析

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1、现代控制控制理论基础现代控制控制理论基础/Modern Control Theory第三章第三章 线性线性School of Information and Electrical Engineering教学要求：教学要求：重点内容：重点内容：School of Information and Electrical EngineeringSchool of Information and Electrical Engineering000()()(),tttttt xAxBuxxu0 xSchool of Information and Electrical Engineering 20120,

2、kkktbbtb tb tbbn幂级数法 方程的解为式中，均为 维向量。xx自由运动，控制输入为零，xAxSchool of Information and Electrical Engineering)(210121kkkktbtbbAtkbtbb可得()tx xAx代入方程 将方程两边系数必相等，即方程两边系数必相等，即1022103320011221133 21kkbAbbAbA bbAbA bbA bk！School of Information and Electrical Engineering 00b且x2200002 200011()2!11()2!1(0)(0)!kkk kk

3、 kktbb tb tb tktttbkbtk故xAAAIAAAxAx2 2112tkketttk！AIAAASchool of Information and Electrical Engineering()(0)tteAxx于是齐次状态方程的解为于是齐次状态方程的解为0!,1,AtkkkketA义称为矩阵指数 状态转移矩阵定定0)()()t ttttet000若初始时刻，对应的初始状态为则齐次状态方程的解为0()，A(xxxSchool of Information and Electrical Engineering112201231(0)0 xxxx x 例例3.1 线性定常系统齐次状

4、态方程为线性定常系统齐次状态方程为 求齐次状态方程的解求齐次状态方程的解 解：解：2 23 3232323232323112!3!1001010111012323232!3!371267752313322tettttttttttttttttt AIAAASchool of Information and Electrical Engineering232323232323()(0)3711267750231332217233ttetttttttttttttttt AxxSchool of Information and Electrical Engineering0()0()()A ttx t

5、ex t0()x ttt()(0)Atx tex0()x tnn0()A t teSchool of Information and Electrical Engineering1x2xO1t2tt0()x t1()x t2()x t0()x t2()x t10()A tte1()x t0()A t te21()A tteSchool of Information and Electrical Engineering()(0)()tttt 0对于线性定常系统，当初始时刻=0时，满足以下矩阵微分方程和初始条件的解，即定义为系统的状态转移矩阵。I 0()0()t ttte对于线性定常系统，有以下对

6、应关系并具有如下基本性质。A School of Information and Electrical Engineering0100121221211020()()(0)()()()()()()()()()()()()()()()()()nttttetttttttttttnt ntttttttttee ee初始状态可逆性线性关系(1)(2)(3)(4)(5)(6)(7)可分阶段转为正整数移AA BABA BIAAAB=BA()tte eABABBAAtet )(School of Information and Electrical EngineeringSchool of Informat

7、ion and Electrical Engineering12n00A12()ntttteetee00A ASchool of Information and Electrical Engineering1111m m11100A则有：则有：12121112!1!1012!0001mmttm mtttmtteemA约当块约当块 m mA若若 为为（2 2）School of Information and Electrical Engineering则有：则有：12e0ee0ejttttAAAA具有约当块的矩阵具有约当块的矩阵 A若若 为为（3 3）12j00AAAJA其中：其中：12,jA

8、 AA为约当块为约当块School of Information and Electrical Engineering有则：有则：cossinsincostttteettA-A(5)(5)若若 为为A则有：则有：1tteAAPe P1P AP=A(4)(4)若矩阵若矩阵 通过非奇异变换矩阵通过非奇异变换矩阵 化为对角线矩阵化为对角线矩阵APSchool of Information and Electrical Engineering2 2011()2!ktk kkkttetttkkAIAAAASchool of Information and Electrical Engineering01

9、23 xx 例例3.2 已知已知系统状态系统状态方程为方程为 试求其状态转移矩阵试求其状态转移矩阵。解：解：2 222222212!10010110123232!(102)(03)2!2!(026)(137)2!2!tetttttttttttt AIAASchool of Information and Electrical Engineering2222222222222(1)(124)2!2!2(1)2(124)2!2!(1)(124)2!2!(1)2(124)2!2!2222tttttttttttttttttttttttteeeeeeee School of Information an

10、d Electrical Engineering 0ttsss xAxXxAX 11100sstLsXIAxxIAx11()tteLsAIA 有：有：School of Information and Electrical Engineering110011()01232331()1()2(1)(2)31(1)(2)(1)(2)2(1)(2)(1)(2)sssssadj ssssssssssssssssIAIAIAIA0123 xx 例例3.3 已知系统状态方程为已知系统状态方程为 试用拉氏反变换法求其状态转移矩阵。试用拉氏反变换法求其状态转移矩阵。解：解：School of Informat

11、ion and Electrical Engineering22112221111212221212122()()222tttttttttsssssssseeeeteLseeee 由拉氏反变换得到AIA School of Information and Electrical Engineering121()nttttteeee00其中，是化 为对角线标准形的线性变换矩阵。APPPAnnSchool of Information and Electrical Engineering123123111026149 (1)求特征值矩阵 的特征值为(2)计算特征向量A=PPP01161166115

12、A 例例3.4 已知系统矩阵已知系统矩阵 试用对角线标准形法求其状态转移矩阵。试用对角线标准形法求其状态转移矩阵。解：解：School of Information and Electrical Engineering 12232323232111adj026149533342322668ntttttttttttttttteeteeeeeeeeeeeeee 00 (3)构造变换阵则 有：APPPPPPP323232323966527312916212922tttttttttttteeeeeeeeeeeeSchool of Information and Electrical Engineeri

13、ng111111111211(1)!1(2)!()tttntttntttetetenetetentetee0AQQ School of Information and Electrical Engineering121111111111122222211221(1)21(2)1(1)2(2)2()()12!(1)!0(2)!000(1)!0(2)!mmntmttttmtttttmtttmtmmttetetet eemteteemteeteteemteem0若 具有重特征值重特征值互异特征值则状态转移矩阵为，AAQ 2211200mmntttteee0Qm1m1m2m2School of Inf

14、ormation and Electrical Engineering111()0nnnnfaaaAAAAI12121nnnnnaaaaAAAAIt2 23 3210121e111()2!3!()()()()(2)tk knntettttka ta ta tat化为 的有限项AAAIAAAAIAAA(),0,1,(1)ia t in均是时间的标量函数School of Information and Electrical Engineering121210111211222211()1()1()1nntntntnnnntetete 12,n 011(),(),()nttt()it的计算Scho

15、ol of Information and Electrical Engineering1111111(1)!02111(2)!(1)(2)n-321312!2!(1)n-2112111!1!2n-2111111()00001()0001(1)()001()012()1tnntnnnntnnnnntettntett ett et1tte1011(),(),()ntttSchool of Information and Electrical Engineering1120112n-2111112n-21111112n-211()()00001()0001(1)()1()1()1mmninmnmm

16、mmmnnnnnnmtttnttt设 的特征值中 为 重，为单特征值。,的计算式为，A1111111(1)!21(2)!mntnmtnmtttteteeeeSchool of Information and Electrical Engineering0100016116A 例例3.5 已知系统矩阵已知系统矩阵 试用化有限项法求其状态转移矩阵试用化有限项法求其状态转移矩阵 解：解：230231232(033534221122tttttttttateeea teeeateee 矩阵 的特征多项式为特征值为AIASchool of Information and Electrical Engine

17、ering 2012232323232323232323531133422225133638222225273129167291722tttttttttttttttttttttttttttteattateeeeeeeeeeeeeeeeeeeeeeeeeee 由此可得：AIAA这种情形计算较繁杂!School of Information and Electrical Engineering412102113A 例例3.6 已知系统矩阵已知系统矩阵 试用化有限项法求其状态转移矩阵试用化有限项法求其状态转移矩阵 解：解：113012(0101tttatea tteate 矩阵 的特征多项式为其二重

18、特征值为，单特征值，则AIASchool of Information and Electrical Engineering 33333333201201253942413933016222111111424100412156120101026001113ttttttttttttteteeeteeteeeeteeeattatata tat则系统状态转移矩阵为：AIAA33333333333333146252222222tttttttttttttttttteteteteteeteeeteeteeteeeteeSchool of Information and Electrical Enginee

19、ring()()()ttt xAxBu 0()(-)0()(-)0ede0d0tt tttttttttt0非齐次状态方程的解为当时，=有0 xxeBuxxeBuSchool of Information and Electrical Engineering 0000()()d()0()dttttttttttt若用状态转移矩阵表示，非齐次状态方程的解为xxBuxxBuSchool of Information and Electrical Engineering 00()(-)0dtt ttttetAxxeBu()u t()x tSchool of Information and Electri

20、cal Engineering 111111()(-)00()0()()0()0()0()e0d()0()dtttttttssssssssssstLsLsstt即xAxBuXxAXBUIA XxBUXIAxIABUxIAxIABUxeBuxBuSchool of Information and Electrical Engineering0101(0)()1()2310utt xxxu 22222022222113.3()022tttttttttttttttteeeeeteeeeeeetttBudee系统状态转移矩阵已在例中求得AAxx 例例3.7 已知线性定常系统的状态方程为已知线性定常系统

21、的状态方程为 求系统状态方程的解。求系统状态方程的解。解：解：School of Information and Electrical Engineering()()()()()()()()()()tttttttttt xAxBuyCxDumyxnur(),(),()tttABC()tD,nn,nr mnmr(),(),(),(t)tttABCD0,)t 0()tx()tuSchool of Information and Electrical Engineering()()()ttt xAx00()(,)()tt ttxx 00000(,)(,)()(,)(,)t tt ttt tt t其中

22、，为时变系统的状态转移矩阵，是矩阵微分方程的解。AI 10001200012211233210ddddd()d,tttttttttt tIAAAAAA School of Information and Electrical Engineering 1 2 3100(,)(,)t tt t00(,)t t I 211020(,)(,)(,)t tt tt t 4(,)(,)()tt A School of Information and Electrical Engineering 1121221000123321000(,0)ddddddttttIAAAAAA 例例3.8 线性时变系统齐次状

23、态方程为线性时变系统齐次状态方程为 计算系统状态转移矩阵计算系统状态转移矩阵(t,0)。解：解：01()()()()0ttttt xAxx200001()002tttAddtSchool of Information and Electrical Engineering1112210032140012()()0010160008ttAAddtddt121212332100032100031256()()()010101000040048ttAAAddddddttSchool of Information and Electrical Engineering352463524600010640(

24、,0)0100028481640012848ttttttttttttt School of Information and Electrical Engineering()()()()()ttttt xAxBu00()()t tttxx000()(,)()(,)()()tttt tttdxxBu 000()(,)()()(,)()()()()ttttt ttttdttyCxCBuDuSchool of Information and Electrical Engineering 10,1,2,kkkkkkkxGxHuyCxDu110(0)(0),(1),()(0)(),()kkkjjkjk 当

25、给定初始条件和输入信号序列即可求得xuuxxG xGHu 11000,1,2,kkkjjkkkjkk 输出响应：yCxDu=CG xCGHuDuSchool of Information and Electrical Engineering 11111100kkkjjkZzzZzzZzzjZzz 解得：由解的唯一性可得：xIGxIGHUGIGGHuIGHU 110kkkZzzZzzk输出响应：yCxDu=CIGxCIGHUDuSchool of Information and Electrical Engineering211221()(1)()(0)()()()()()kkkkkkkkkkk

26、kkkknn为线性定常离散系统的状态转移矩阵。是满足如下矩阵差分方程和初始条件的唯一解，(1)是方阵，且具有如下性质(：义2)GGI 定定School of Information and Electrical Engineering 10110()()(0)(1)()()()()()(1)()()0(1)0,1,2,kjkj hkjkkkjjkhhkhkhkjjkkkkkjjkk线性定常离散系统的状态转移方程：若初始时刻系统的初始状态为则线性定常离散系统的状态转移方程为：离散时间系统的输出响应为：，xxHuxxxHuyCxDu=CxCHuDuSchool of Information and

27、 Electrical Engineering 例例3.9 求线性定常离散系统求线性定常离散系统 的解。的解。解：解：1122(1)()011()(1)()0.161110()10,1,2,1x kx ku kx kx ku kk x 01111000.161110 1.84 xGxHu 01012110.1611.8412.84 0.84 xGxHuSchool of Information and Electrical Engineering386.116.01184.084.2116.010)2()2()3(HuGxxSchool of Information and Electrica

28、l Engineering1(0.2)(0.8)0.1611110.16(0.2)(0.8)4111515130.230.830.230.80.810.81114130.230.830.230.8zzzzzzzzzzzzzzzzzz IGIGzGzIZk11)(School of Information and Electrical Engineering 22110-211zzzzzzzzzzzzz 于是xHU()1,()(1)zu kzz又U=-kaazzZ)(1kkkkkkkk11)(-0.834)(-0.231)(-0.838.0)(-0.238.0)(-0.835)(-0.235)(

29、-0.831)(-0.234)(zGzIZkSchool of Information and Electrical Engineering 017222560.290.81813.417.6760.290.8181zzzzzzzzzzzzzzzz XIGxHU 187)(-0.896.17)(-0.264.31825)(-0.8922)(-0.2617)(kkkk1zXZkxSchool of Information and Electrical Engineering386.116.084.084.284.1011)()(代入上式得0,1,2,3k令21，kxkxSchool of Inf

30、ormation and Electrical Engineering 10,1,2,kkkkkkkkkkkxGxHuyCxDu011()()(,)()(,1)(,)(1,)(,)(,)(,)()(,1)()()()()()()()()()(ki hki hthThkkkkkk hhk iiik hkhk hh hkkhiiikhk设初始时刻，初始状态为已知，并且为非奇异矩阵，则方程的解为其中，为状态转移矩阵，它满足如下矩阵差分方程及初始条件系统的输出方程：xGxHuGHuxIyCxD+C )()kuSchool of Information and Electrical Engineerin

31、g 连续系统的离散化（矩阵微分方程）（矩阵差分方程）连续时间系统的状态方程离散系统方程假设：假设：(1)等采样周期等采样周期T；,(12)u tu kTkTtkT(3)采样周期的选择满足采样周期的选择满足shannon采样定理采样定理School of Information and Electrical Engineering对线性定常连续系统：对线性定常连续系统：()()()()()ttttttyCxDBuuxAx 00()()1ddTTTtkTTkTTkTkTkTkTTeTtTtte 其中 AAxGxHuyCxDuGHBB School of Information and Electr

32、ical Engineering 例例3.10 试求线性定常连续系统的状态方程试求线性定常连续系统的状态方程 的离散化方程。的离散化方程。解：解：1122010021xxuxx 11111e021111(1 e)(2)210e02tttsLsLsss sLsAIASchool of Information and Electrical Engineering11202011(1 e)()=e20e1111e01(1 e)244()=edd21110ee2211(1)()(1 e)2)e1)(0tt TtTTttTTTt=t=xkTx kTxkTx kT 系统离散化状态方程为AAGHB1e1()

33、221(1 e)2()Tu kTSchool of Information and Electrical Engineering对线性定常连续系统：对线性定常连续系统：()()()()()()()()()ttttttttttyxAxBxuuCD离散化后的状态空间表达式为：离散化后的状态空间表达式为：(1)11,1,dkTkTt kTt kTkTkTkTkTkTkTkTkTkTkTkTkT kTkTkTkTtkTt 中 其xGxHuyCxDuGHBCCDD School of Information and Electrical Engineering 1111kTxkTx kTTTtttttt

34、kTxkTx kTkTkTkTkTT其中，为采样周期。将上式代入中，并令，可得xxAxBuAxBu当采样周期当采样周期 T 较小，在满足所要求精度的前提较小，在满足所要求精度的前提下，用近似的离散化方程，计算就容易得多，近下，用近似的离散化方程，计算就容易得多，近似方法就是用差商代替微商，即令似方法就是用差商代替微商，即令School of Information and Electrical Engineering 1(0,1,2,)kTTkTkTTkTkTkTkTkTkTkTTkTkTTkTkT kT即其中仅能描述系统在虚拟采样时刻的状态方程的近似特性。采样周期 越小，近似的离散化状态方程

35、越精确。IAxBuGxHuGBxIAHSchool of Information and Electrical Engineering(1)近似离散化是一般离散化的特例近似离散化是一般离散化的特例(2)定常系统离散化是时变系统离散化的特例定常系统离散化是时变系统离散化的特例(3)一般说来，没有精确离散化一般说来，没有精确离散化(4)离散化是有条件的，离散化是有条件的，“连续化连续化”是无条件的是无条件的(5)连续系统的结论可以在离散系统中找到对应，连续系统的结论可以在离散系统中找到对应，反之则未必反之则未必School of Information and Electrical Enginee

36、ring小结1()()=ettLs常用公式拉氏变换法AIA 0()0()0()dttttttxxxxBu School of Information and Electrical Engineering000()(,)()(,)()()tttt tttdxxBu00()(,)()tt ttxx 10001200012211233210ddddd()d,tttttttttt tIAAAAAA School of Information and Electrical Engineering 1010()()(0)(1)()()0(1)kjkjkkkjjkkkjjkxxHuyCxCHuDu11()()()()()(,)()(,1)(,)()(,1)()()()()()ki hki hk hhk iiik hhkkkkkkiiikxyCxCDxHuHu+uSchool of Information and Electrical Engineering 00()()1ddTTTtkTTkTTkTkTkTkTTeTtTtte 其中 AAxGxHuyCxDuGHBB