chap11

Modern Control Theory Chapter 11:Analysis of Control Systems in State Space Dr.Y. X. Liu Automation Department Inner Mongolia University 2004-3-16Y. X. Liu, Inner Mongolia University2 Lecture Content ? About author ? The basic concepts about state space ? State-space representations ? Diagonalization of nn matrix ? Properties of state-transition matrices ? Results in vector-matrix analysis ? Controllability and observability 2004-3-16Y. X. Liu, Inner Mongolia University3 About author ?Professor ?Department of Mechanical Engineering University of Minnesota ?B.S., 1947, Mechanical Engineering, University of Tokyo ?M.S., 1953, Mechanical Engineering, University of Illinois at Urbana-Champaign ?Ph.D., 1956, Engineering Science, University of California - Berkeley ?Telephone: (612) 625-9374 ?E-mail: Katsuhiko Ogata 2004-3-16Y. X. Liu, Inner Mongolia University4 Research ? Major research interests are in the field of discrete-time control systems, including optimal control of complex plants. ? Work includes mathematical modeling of complex plants and developing design techniques of discrete-time control systems and optimization problems, such as an optimal stopping problems, via a dynamic programming approach. 2004-3-16Y. X. Liu, Inner Mongolia University5 Selected Publications ?Systems Dynamics. 3rd ed. Upper Saddle River: Prentice Hall, 1998. ?Modern Control Engineering. 3rd ed. Upper Saddle River: Prentice Hall, 1997. ?Discrete-Time Control Systems, 2nd. ed. Upper Saddle River: Prentice Hall, 1995. ?Solving Control Engineering Problems with MATLAB. Upper Saddle River: Prentice Hall, 1994. ?Engenharia de Control Moderno, 2nd ed. Brazil: Prentice Hall, 1993. ?“Metody Przestrzeni Stanow w Teorii Sterowania.“ Wydawnictwa, Naukowo, Techniczne, Warsaw, Poland, 1974. ?Dynamic Programming. Tokyo. Japan: Baifukan Co., 1973. 2004-3-16Y. X. Liu, Inner Mongolia University6 The concepts about state space ? States and state variables ?表征系统运动的信息称为状态。确定系统的状态的 一组独立（数目最少）变量称为状态变量记为： ? State vectors ?把描述系统状态的n个状态变量x1(t),x2(t),xn(t)看 作向量x(t)的分量，则x(t)称为n维状态向量。 ? State space ?以n个状态变量作为坐标轴所组成的n维空间称为状 态空间。 )(),.,(),(21txtxtxn 2004-3-16Y. X. Liu, Inner Mongolia University7 State-space equations 1 ? Single input ?assume state variables: x1(t),x2(t),xn(t)，and the common form of state equation is as follow: ( )( )( )( )( ) ( )( )( )( )( ) ( )( )( )( )( ) += += += tubtxatxatxatx tubtxatxatxatx tubtxatxatxatx nnnnnnn nn nn L& M L& L& 2211 222221212 112121111 ( )( )( )tuttbAxx+=& = = = = nnnnn n n nnb b b aaa aaa aaa x x x x x x M L MMM L L & M & & & M 2 1 21 22221 11211 2 1 2 1 ,bAxx 2004-3-16Y. X. Liu, Inner Mongolia University8 State-space equations 2 ? Multiple input ?For the multiple input (suppose p) the state variables are: x1(t),x2(t), xn(t)，and the common form of state equation is as follow: ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) += += += tubtubtubtxatxatxatx tubtubtubtxatxatxatx tubtubtubtxatxatxatx pnpnnnnnnnn ppnn ppnn LL& M LL& LL& 22112211 222212122221212 121211112121111 BuAxx+=& = = = = pnpnn p p nnnn n n nu u u bbb bbb bbb aaa aaa aaa x x x M L MMM L L L MMM L L M 2 1 21 22221 11211 21 22221 11211 2 1 ,uBAx 2004-3-16Y. X. Liu, Inner Mongolia University9 Output equations ? Signal： ( )( )( )( )( ) ( )( )( )tdutcty tdutxctxctxcty nn += += x L 2211 ?Multiple： ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) ( )( )( )( )( )( )( ) += += += tudtudtudtxctxctxcty tudtudtudtxctxctxcty tudtudtudtxctxctxcty pqpqqnqnqqq ppnn ppnn LL M LL LL 22112211 222212122221212 121211112121111 DuCxy+= = = = = pqpqq p p nqq n n nu u u ddd ddd ddd cqcc ccc ccc y y y M L MMM L L L MMM L L M 2 1 21 22221 11211 21 22221 11211 2 1 ,uDCy 2004-3-16Y. X. Liu, Inner Mongolia University10 State-space representations ? Single input single output duy u += += cx bAxx& ?Multiple input multiple output DuCx BuAxx += += y & x=n vector，u=p1 vector，y=q1vector， A=nn matrix，B=np matrix，C=qn matrix， D=qp matrix. x=n vector，u and y are scalars，A=nn matrix， b=n1 matrix，c=1n matrix，d is scalar. + + D dt A BC )(tu)(ty)(tx&)(tx 2004-3-16Y. X. Liu, Inner Mongolia University11 State-space representations of transfer function systems ( ) ( ) nn nn nn nn asasas bsbsbsb sU sY + + = 1 1 1 1 1 10 L L ?Consider a system defined by: ? Controllable canonical forms: ?1. Define state variables as follows (P701, 11.17): ()()()ubxbabxbabxbaby uxaxaxax xx xx xx nnnnn nnnn nn 0011201110 1211 1 32 21 += += = = = L L& & M & & 2004-3-16Y. X. Liu, Inner Mongolia University12 Controllable canonical form 1 ? State equation: ?Output equation: u x x x x aaaax x x x n n nnnn n + = 1 0 0 0 1000 0100 0010 1 2 1 121 1 2 1 MM L L MMMM L L & & M & & ub x x x babbabbaby n nnnn0 2 1 0110110 + = M L 2004-3-16Y. X. Liu, Inner Mongolia University13 Block diagram of the system defined by controllable canonical form + b2-a2b0 b0 u x1 y x2xn-1xn a2a1an-1an bn-1-an-1b0bn-anb0b1-a1b0 + + + + + + - + + + + + + + + 2004-3-16Y. X. Liu, Inner Mongolia University14 Controllable canonical form 2 u x x x xaaaa x x x x n n nn n n + = 0 0 0 1 1000 0100 0001 1 2 1121 1 2 1 MM L L MMMM L L & & M & & ?2. Define state variables as follows: 1 21 12 12111 = = = += nn nn nnn xx xx xx uxaxaxax & & M & L& ?State equation: 2004-3-16Y. X. Liu, Inner Mongolia University15 Observable canonical form ? State equation: ?Output equation: u bab bab bab x x x a a a x x x x nn nn n n n n n + = 011 011 0 2 1 1 1 1 2 1 100 001 000 MM L MMMM L L & & M & & ub x x x x y n n 0 1 2 1 1000+ = L 2004-3-16Y. X. Liu, Inner Mongolia University16 Block diagram of the system defined by observablecanonical form + b0 u x1 y x2xn-1xn a1an-1an bn-1-an-1b0bn-anb0b1-a1b0 + + + + + - 2004-3-16Y. X. Liu, Inner Mongolia University17 Diagonal canonical form 1 ? Denominator polynomial of transfer function involves only distinct roots ( ) ( )()()() n n n nn nn ps c ps c ps c b pspsps bsbsbsb sU sY + + + + + += + + = L L L 2 2 1 1 0 21 1 1 10 u x x x p p p x x x nnn + = 1 1 1 0 0 2 1 2 1 2 1 MO & M & & ub x x x cccy n n0 2 1 21 + = M L ?State equation: ?Output equation: 2004-3-16Y. X. Liu, Inner Mongolia University18 Block diagram of the system defined by diagonalcanonical form pi: poles of G(s) ci: residues of related poles b0 ux1y x2 xn c1 c2 cn + + + + 1 1 ps+ 2 1 ps+ n ps+ 1 + 2004-3-16Y. X. Liu, Inner Mongolia University19 Diagonal canonical form 2 ? State equation: u c c c x x x p p p x x x nnnn + = MO & M & & 2 1 2 1 2 1 2 1 0 0 ub x x x y n 0 2 1 111+ = M L ?Output equation: 2004-3-16Y. X. Liu, Inner Mongolia University20 Jordan canonical form ? Denominator polynomial of transfer function involves only one three-multiple roots ( ) ( )() ()()() () n n n nn nn ps c ps c ps c ps c ps c b pspsps bsbsbsb sU sY + + + + + += + + = L L L 4 4 1 3 2 1 2 3 1 1 0 4 3 1 1 1 10 u x x x x x p p p p p x x x x x nn + = 1 1 1 0 0 000 000 0000 10 0001 4 3 2 1 4 1 1 1 1 1 1 1 1 MM L MM L L MM L & M & & & & ub x x x cccy n n0 2 1 21 + = M L ?State equation: ?Output equation: 2004-3-16Y. X. Liu, Inner Mongolia University21 Block diagram of the system defined by Jordancanonical form b0 u x1 y x2 xn c1 c4 cn + + + + 1 1 ps+ 4 1 ps+ n ps+ 1 1 1 ps+ 1 1 ps+ c3 c2 x3 x4 + + + + + 2004-3-16Y. X. Liu, Inner Mongolia University22 Diagonalization of nn matrix 1 ? 1,n= n distinct eigenvalues of any nnmatrix A, will transform P-1AP into the diagonal matrix with a linear nonsingular matrix P. = n 0 0 2 1 1AP P ?P = n real eigenvectors A ()nippp ppp iiiii n L L , 10or 21 = = AIA P 2004-3-16Y. X. Liu, Inner Mongolia University23 Diagonalization of nn matrix 2 ? 1,n= n distinct eigenvalues of an nn unitary matrix A. Vandermode matrix P will transform P-1AP into the diagonal = 11 2 1 1 22 2 2 1 21 111 n n nn n n L MMM L L L P = 121 1000 0100 0010 aaaa nnn L L MMMM L L A = n 0 0 2 1 1AP P 2004-3-16Y. X. Liu, Inner Mongolia University24 Diagonalization of nn matrix 3 ?An nnmatrix A involves m equal eigenvalues (1=m), and other (n-m) distinct eigenvalues. If there are independent real eigenvectors p1,pmwhen solution (iI-A)pi=0 (i=1,m) will transform P-1AP into the diagonal. = + n m 0 0 1 1 1 1 O O APP nmm ppppLL 11+ =p 2004-3-16Y. X. Liu, Inner Mongolia University25 Jordan canonical form 1 ? Matrix A involves m equal eigenvalues 1, others (n-m) distinct eigenvalues. If there is only one real eigenvector p1 when solution (iI-A)pi=0 will transform P-1AP into Jordan only. = + n m 0 1 01 1 1 1 1 1 O O O APP nmm ppppLL 11+ =p mm ppppppL O O L 21 1 1 1 21 1 1 A= p1pm: generalized real eigenvector: 2004-3-16Y. X. Liu, Inner Mongolia University26 Jordan canonical form 2 ? Unitary Matrix A involves m equal eigenvalues 1, and there is only one real eigenvector p1, the P transform P-1AP into Jordan canonical form, and Ps form as follows: = = + 1 1 2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 n nm m m p pp ppp p M LLP 2004-3-16Y. X. Liu, Inner Mongolia University27 Jordan canonical form 3 ? Suppose Matrix A involves 5 equal eigenvalues 1, but there are 2 independent real eigenvector p1, p2, others (n-5) distinct eigenvalues, will transform P-1AP into possible Jordan canonical form as follows: = n 0 1 1 01 6 1 1 1 1 1 1 O APP = n pp p p pp pL 6 2 2 2 2 1 1 2 1 1 1 P 2004-3-16Y. X. Liu, Inner Mongolia University28 Properties of state-transition matrices ( ) ( ) ( )( )( ) ()( ) ()() ( ) ( )()()( ) ( )() ( ) ()() () ( )( ) () () () () ( ) ( ) = = = = = = = = = = = += + + PP xPx Axx BAAB BAAB xx AA AAA A ABBABA ABBABA A t ttttt ttttt k kkt e eeeee eeeee ktt tttttt tttt tttt tttttt ttt I t k ttIet 1 011202 00 11 122121 22 t define ofmatrix n transitiostate is t suppose .9 .8 .7 .6 .5 .4 .3 .2 0.1 ! 1 2 1 & & LL 2004-3-16Y. X. Liu, Inner Mongolia University29 State-transition matrices ? A=diag1,n, and distinct elements ( ) = t t n e e t 0 0 1 O ?mm Jordan matrix A = 0 1 01 O O A ( ) () () = t t t m tt t m ttt e te e m t tee e m t e t tee t L L MMMM L L 000 000 !2 0 !12 2 12 2004-3-16Y. X. Liu, Inner Mongolia University30 Results in vector-matrix analysis ? Cayley-Hamilton theorem ?An nn matrix A and its characteristic equation ?matrix A satisfies its own characteristic equation ( )0 1 1 1 =+= nn nn aaafLAI ( )0 1 1 1 =+= IAAAA nn nn aaafL 2004-3-16Y. X. Liu, Inner Mongolia University31 Minimal polynomial ? The least-minimal polynomial having A as root is called the minimal polynomial. ?The minimal polynomial of an nn matrix A as polynomial () of least degree, ?Such that (A)=0, or ?The minimal polynomial () is given by ?d(), a polynomial of in , is the greatest common divisor of all the elements of adj(I-A). ( )nmaaa mm mm += 1 1 1 L ( )0 1 1 1 =+= Iaaa mm mm AAAAL ( ) ( ) d AI = 2004-3-16Y. X. Liu, Inner Mongolia University32 Computation of eAt (method 1) ? An nn matrix A, If D=P-1AP=diag1, ,n, then eAtcan be given by: 11DA PPPP = t t t tt n e e e ee 0 0 2 1 O ?An nn matrix A, If J=S-1AS is a Jordan matrix, then eAt can be given by: () () 11JA SSSS = t t t m tt t m ttt tt e te e m t tee e m t e t tee ee L L MMMM L L 000 000 !2 0 !12 2 12 2004-3-16Y. X. Liu, Inner Mongolia University33 Computation of eAt (method 2) ? The Laplace transform approach. eAtcan be given as follow: () 1 1 =AI A se t L L 2004-3-16Y. X. Liu, Inner Mongolia University34 Computation of eAt (method 3) ? Case 1: minimal polynomial of A involves only distinct roots. ?By using Sylvesters interpolation formula, it can be shown that eAt can be obtained by solving the following determinant equation: 0 1 1 1 12 12 1 2 2 22 1 1 2 11 2 1 = tm tm mmm tm tm e e e e m A AAAIL L MMMMM L L ?Solving equation above for eAtis the same as writing ( )( )( )( ) 1 1 2 210 += m m t tatatataeAAAI A L and determining the ak(t) (k=0,1,2,m-1)by solving the following set of m equation for ak(t): ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) tm mmmm tm m tm m m etatatata etatatata etatatata =+ =+ =+ 1 1 2 210 1 21 2 22210 1 11 2 12110 2 1 L M L L 2004-3-16Y. X. Liu, Inner Mongolia University35 Computation of eAt (method 3) ? Case 2: minimal polynomial of A involves multiple roots. ?Assume the minimal polynomial of A involves three equal roots (1=2=3) and other roots (4,5,m) that are all distinct. By using Sylvesters interpolation formula, it can be shown that eAtcan be obtained from the following determinant equation: ()() () 0 1 1 1 13210 22 21 3100 132 132 1 4 3 4 2 44 1 1 3 1 2 11 2 1 2 11 2 3 11 4 1 1 1 = tm tm mmmm tm tm tm tm e e e e tem e tmm m A AAAAIL L MMMMMM L L L L 2004-3-16Y. X. Liu, Inner Mongolia University36 Computation of eAt (method 3) ?It is noted that, just as in case 1, solving equation above for eAtis the same as writing ( )( )( )( ) 1 1 2 210 += m m t tatatataeAAAI A L and determining the ak(t)s (k=0,1,2,m-1)from ( )( ) ()() ( ) ( )( )( )()( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) tm mmmm tm m tm m tm m tm m m etatatata etatatata etatatata tetamtatata e t ta mm tata =+ =+ =+ =+ = + 1 1 2 210 1 41 2 42410 1 11 2 12110 1 11 2 13121 2 1 11132 4 1 1 1 132 22 21 3 L M L L L L 2004-3-16Y. X. Liu, Inner Mongolia University37 Linear independence of vectors ? The vectors x1,x2,xnare said to be linearly independent if 0 2211 =+ nn cccxxxL Where c1,c2,cnare constants, implies that 0 21 = n cccL ?If an nnmatrix Ais nonsingular, then ncolumn (or row) vectors are linearly independent. ?If the nnmatrix Ais singular, then ncolumn (or row) vectors are linearly dependent. rnonsingulaor0nrank=AA singularor0nrank=AA 2004-3-16Y. X. Liu, Inner Mongolia University38 State controllability 1 ? The eigenvetors of A are distinct ?The condition for complete state controllability is that the nnr matrix BAABB 1n L be of rank n, or contain nlinearly independent column vectors. 2004-3-16Y. X. Liu, Inner Mongolia University39 State controllability 2 ? If P-1AP= D= diag1, ,n. Let us define x=Pz + = += rnrn r r nn u u u ff ff ff z z z M L MM L L O & 2 1 1 221 111 2 1 2 1 11 0 0 BuPAPzPz If the eigenvectors of A are distinct then the system is completely state controllable if and only if no row of P-1B has all zero elements. 2004-3-16Y. X. Liu, Inner Mongolia University40 State controllability 3 ? If S-1AS=J is Jordan canonical form. Let us define x=Sz + = += r nrn r r r r r r nn u u u ff ff ff ff ff ff ff z z z z z z z M MM L L L L L L O & 2 1 1 661 551 441 331 221 111 6 5 4 3 2 1 6 4 4 1 1 1 11 0 0 1 00 10 001 BuSASzSz 2004-3-16Y. X. Liu, Inner Mongolia University41 For Jordan canonical form ? The system is completely state controllable if and only if ?no two Jordan blocks in J are associated with the same eigenvalues, ?the elements of any row of S-1B that correspond to the last row of each Jordan block are not all zero, ?The elements of each row of S-1B that correspond to distinct eigenvalues are not all zero. 2004-3-16Y. X. Liu, Inner Mongolia University42 Output controllability ? The system is complete output controllability if and only if the m(n+1)r matrix DBCABCACABCB 12n L be of rank m, or contain m linearly independent column vectors. 2004-3-16Y. X. Liu, Inner Mongolia University43 State observability 1 ? The system is complete observability that this requires the rank of the n