线性系统的能控性和能观测性

1、( )u t( ) tx( ) tx( )u t1( )x t2( )x t( )u t1( )x t2( )x t( )u t2C2( )x t2( )x t( )u t1( )x t2( )x t1111222111xxuRCRCxxRC 2( )x t2C( ) ( )( )tttxAxBu( )( ) ( )( ) ( )tttttyCxDu00( )txxdtTxnurdT,A Bn nn rt0dtT0 x1dtT10tt( )u t01,ttt0 x1( )0 x t0 x0t0t0t0t0dtT0dtT0dtT0t0t0( )0tx11( )txx01( ), , t tt t

2、u1x0t1x1x0t0t0tdT0t, (0),0t0 x = Ax+ Bu xx10t 110(0, )dTttTtcteetAAWBBx = Ax+ Bum nn r2-1ncQ = BABA BLAB2-1rankrankncnQBABA BAB321001000101aaa xxu212121001011AAaaaa bbb2121210010131rankQrank bAbA brankancaaa12,a a1L1i1C1cu111 12 1111ddddddccciLRiutuR cuutuicit1121,cxi xuyi11111222121122210110111RLLx

3、xuxxR CR CxyuxRR111122121111()cRLL LR CR CQBAB12111RR CLcQ( , )A B1122102025xxuxx 112233110001040023xxxxuxx 1122133244552100010210000230510000521xxxxuxxuxxxx1122102020 xxuxx 11122233110420100000230 xxuxxuxx112233445521004021200215130050 xxxxxxuxxxx , (0),0t0 x = Ax+ Bu xxA(1, ),iin,irankIABn1,in,ran

4、k sIABnsC ()sIAB判据判定方法特点格拉姆矩阵判据的各行函数线性独立需要求矩阵指数函数并判定函数相关，计算复杂秩判据满秩1.计算简便可行。2.缺点为不知道状态空间中哪些变量(特征值/极点)能控约当标准形判据约当标准形中同一特征值对应的B矩阵分块的最后一行线性无关1.易于分析状态空间中哪些变量(特征值/极点)能控。2.缺点为需变换成约当标准形PBH 判据1.易于分析哪些特征值(极点)能控。2.缺点为需求系统的特征值x = Ax+ Buy = Cx+ Du( )u t01 ,t t0( )ty1( )tyx = Ax+ Buy = Cx+ Dun-1mQ = CBCABCABDmran

5、krankmn-1QCBCABCABD0010011 10 xxuyxu 2 0 01rank CB CAB Drankm 11221241123210 xxuxxxyx 4221ABB 120CBCAB )()()()()(tutBtxtAtx10TTC0100( , )( , ) ( )( )( , )dttW t ttBBt0( , )t t0t1t10tt( ( ),( )A tB t01 , t t111( )( )( )( )( )( )2,3,.,iiiB tB tB tA t BtBtin 12( )( )( ).( )cnQ tB tB tB t( )cQ t12rank(

6、)rank( )( ).( )cnQ tB tB tB tn0t1122233100001001xtxxtxuxtx 00( )( )11M tB t 10021d( )( )( )( )dM tA t M tM tttt 2221144202d( )( )( )( )11d22ttMtA t M tM tttttttt 012( )( )( )MtM tMt( )y t( )u t00, ( ),dxAxttTyCxxx,nmn nm nxRyRARCR0dtT1dtT10tt01 , tt t( ) ty0 x0 x0t0dtT1dtT10tt01 , tt t( ) ty0t0 x0t0

7、dtT0dtT1dtT10tt01 , tt t( ) ty0t0 x0t01 , tt t10tt( ) ty-1( )( ) ( )x tCt y t( )0t u0, (0),0 xAxtxxyCxn nm m1o10(0, )dTttTtteetAAWC C1TTTTTnToQCA CAC （）TorankQn4510 11xxyx11rankrankrank155oCQCA1212ddddiiRiLi RLutt20122120d()d()iR iii RLtyii R1122xixi、00112200100210RRRxxLLuLxxRRRLLxyRRx00220000022RRR

8、RRRRRLLCQCAorank12n Q( ,)A C70(1)050 xxy3x410(2)040003 121 xxyxA(1, )iniiIAranknC1,2,insIAranknCsC ()sIAC-AteB-1 nQcB ABA B- C rankIAn判据判据判定方法判定方法特点特点格拉姆矩阵判据 的各行函数线性独立需要求矩阵指数函数并判定函数相关，计算复杂秩判据 满秩1.计算简便可行。2.缺点为不知道状态空间中哪些变量(特征值/极点)能观测约当标准形判据约当标准形中同一特征值对应的C矩阵分块的第一列线性无关1. 易于分析状态空间中哪些变量(特征值/极点)能观测。2.缺点为需变

9、换成约当标准形PBH 判据1.易于分析哪些特征值(极点)能观测。2.缺点为需求系统的特征值( )0u t ( ) txAx00( )txx0,dt tT( ) tyCx0t11,0dtT t10o0100( , )( , )( ) ( )( , )dtTTtt tt tttt ttWCC( )0u t ( ) txAx00( )txx0,dt tT( ) tyCx0t11,0dtT t011111( )( )rank( )nttntNNN1101111001122( )( )d( )( ) ( )( )dd( )( ) ( )( )dt tnnnt tttttttttttttNCNNANNNA

10、N(1)( ) ( )( ) ( )kk x kkkxGHu( ,)G H( ,)G H2n-1ckQ= HGHG HLGHrankcknQ nckckrankrankGQQ( ,)G H-1 nckHGHGHQckranknQ( ,)G H1- ,rankI G HnC ( ,)G H(1)( )( )( )kGkkCkxxyx0(1)( ), (0),0,1,2,kx kkxGxx( )( )kkyCx1oknCCGQCG1rankrankoknnCCGQCG( ,)G CrankIGnC203(1)120( )012100( )( )010kkkk xxyx2100010rankrank

11、rank3203.okCQCGCGx = Ax+ Buy = Cx12,l ijij lni(1)( )( )( )( )kkkkkxGxHuyCx0ee dTATAtGHtB() A,B,C() G,H,CRe-0ij2/Im- 1, 2,ijTkk ABCxxuyx12100011000010000010nnnaaABaa 1101.00.00.10-.-1nnABaaa ( ,)A B1cT-11 nccTQBABAB1cxT x ABxxu1ccTQ111111ncccT TTBABABI1111110.00.010.02,3,.,ciciTBTABin第 列1121111121012

12、10001000100001ncccnicn iinnnATATTABA BA BTABA BaA Baaaa11100cBTB1ccTQ( ,)A B2cT111211cnTT ATT A-1-110 0 1 nTB ABAB2cxT x ABxxuAB( , ,)A B C1210100001000011000nnnAaaaaC( , ,)A B C12100010001000010001nnnaaAaaC( ,)A B C1oT( ,)A B C111oonCCATQCA1oxT x ABCxxuyx( ,)A B C2oT-12111 noTRARAR1111000011onCCARQ

13、CA -1 nrank B ABABn-1 ()nrank CA CACn- ,rankIA Bnrank I-A B =n,能控性能观测性意义输入 状态状态 输出秩判据约当标准形判据同一特征值的约当块对应B的分块的最后一行是否相关同一特征值的约当块对应C的分块的最后一行是否相关PBH判据ABABCCxxuxxuyxyxAABBCC( , ,)A B C( , ,)A B C( , ,)A B C( , ,)A B C( , ,)A B C( , ,)A B CABCxxuyx-1 nccrankQrank BABABnncPxx 11112112222112200BAAACCxxuxxxyx

14、cn11111221AABxxxu-cn n2222AxxABCxxuyx1rankrank.oonCCAQnnCAoxP x11111222212211200BABAACxxuxxxyx1111111ABCxxuyxonn22112222AABxxxuon1111112212211111122211111( )( )()0000*()G sG sC sIABBACsIBAAsIABCBsIAC sIABABCxxuyx111112131412222242333334444424123400000000000AAAABAABAAACC xxxxuxxxxyxxxx,11111221331441

15、:c noAAAABxxxxxu22222442,222:c oAABC xxxuyx,3333344:nc noAAxxx4444,444:nc oAC xxyxcoPcoP,000000c oc ococcnc onc oIPPPPPPPIcP, c oP,nc oPcoP,000000o co ccooono cno cIPPPPPPPIoP, o cP,no cP( ,)A B CuyxAxBCx1( )adj()( )( )()( )det()( )Y ssN sG ssU ssD sCIA BC IABIA( ,)A B Cx = Ax+ Buy = Cxadj()( )det()sssCI - A BGI - Adet()sI - Aadj()sCI - A BABCDxxuyxu-1( )(-)G sC sIABD( , , ,)A B C Dlim ( )ssDG( , , )A B C1111.( ).nnnnnb sbG ssa saia(1,2, )ib in12112100011000010000100nnnnnaaABaaCD 1211210100