Bounding the Distance of a Controllable and Observable

Let (A;B;C) be a triple of matrices representing a time-invariant linear system _ x(t) = Ax(t) +Bu(t) y(t) = Cx(t) under similarity equivalence, corresponding to a realization of a prescribed transfer function matrix. In this paper we measure the distance between a irreducible realization, that is to say a controllable and observable triple of matrices (A;B;C) and the nearest reducible one that is to say uncontrollable or unobservable one. Di erent upper bounds are obtained in terms of singular values of the controllability matrix C(A;B;C) , observability matrix O(A;B;C) and controllability and observability matrix CO(A;B;C) associated to the triple. 2 Introduction We consider triples of matrices (A;B;C) with A 2 Mn(R) , B 2 Mn m(R) and C 2Mp n(R) corresponding to a time-invariant linear systems _ x(t) = Ax(t) +Bu(t) y(t) = Cx(t) . We consider the following action of the general linear group Gl(n;R) , according to the formula (A1; B1; C1) = (P 1AP;P 1B;CP ) The equivalence relation obtained from this action is such that two equivalent triples of matrices have the same transfer-function matrix. We denote the space of these triples of matrices by M and the general linear group Gl(n;R) by G We consider the set Mco = f(A;B;C) 2 M; (A;B;C) controllable and observableg . This is an open set in the space of all triples of matrices M and it is invariant with respect to the G -action. For each (A;B;C) 2 Mco there exists an open neigbourhood of (A;B;C) relatively small, such that all triples of matrices in it are controllable and observable. Then it makes sense to consider the distances to the nearest uncontrollable, unobservable or uncontrollable and unobservable one, and to deduce safety neighbourhoods for controllable and observable triples of matrices. The main goal of this paper is to show that di erent bounds of theese distances can be obtained. The method used for that as this one used in [1] for the case of pairs of matrices, is to explore the singular values of the controllability and observability matrices of the triple (A;B;C) . Several authors [1], [2], [4] analyze bounds on the distance from a given pair of matrices or a given pencil with qualitative di erent structure pair or pencil under di erent equivalent relation for pairs or strictly equivalence for pencils, as well as [5], [6], [7], [9] analyze the structural stability of a pair or a pencil and the hierarchic closure for pencils. In this paper, the norm considered is the 2-norm, and given a triple (A;B;C) 2 M . We denote by Ac the companion matrix for A , that is to say Ac = 0BBBB@ 0 1 0 : : : 0 0 0 1 : : : 0 ... ... 0 0 0 : : : 1 n n 1 n 2 : : : 11CCCCA where i are such that det(tI A) = tn + 1tn 1 + : : :+ n . 3 1. Preliminaries We consider the following action of G on M , : G M !M de ned by (P; (A;B;C)) = (P 1AP;P 1B;CP ) The action de ned by induces the following equivalence relation between triples of matrices: (A1; B1; C1) and (A2; B2; C2) are called equivalent if and only if there exists P 2 G such that (P; (A1; B1; C1)) = (A2; B2; C2) . The controllability matrix of a triple (A;B;C) 2 M is de ned as C(A;B;C) = (B AB : : : An 1B ) : The observability matrix of a triple (A;B;C) 2 M is de ned as O(A;B;C) = 0BB@ C CA ... CAn 11CCA : They are well known the following propositions (see [3] for more details). Proposition (1.1). a) The rank of the controllability matrix is invariant under the equivalence relation considered. b) A triple of matrices (A;B;C) 2 M is controllable if and only if the controllability matrix has full rank, i.e. rankC(A;B;C) = n: Proposition (1.2). a) The rank of the observability matrix is invariant under the equivalence relation considered. b) A triple of matrices (A;B;C) 2 M is observable if and only if the observability matrix has full rank, i.e. rankO(A;B;C) = n: The controllability and observability matrix of a triple (A;B;C) 2 M is de ned asCO(A;B;C) = O(A;B;C) C(A;B;C) =0@ CB CAB : : : CAn 1B ... ... CAn 1B CAnB : : : CA2n 2B1A 4 Proposition (1.3). The rank of the controllability and observability matrix is invariant under the equivalence relation considered. Proof: Let (A1; B1; C1) and (A2; B2; C2) equivalent triples. Then there exist invertible matrix P such that (A2; B2; C2) = (P 1A1P;P 1B1; C1P ) . So C2Ak2B2 = C1PP 1Ak1PP 1B1 = C1Ak1B1: Proposition (1.4). A triple of matrices (A;B;C) is controllable and observable if and only if rankCO(A;B;C) = n: Proof: It follows from Sylvester's inequality (see [8] for details), rankO(A;B;C) + rankC(A;B;C) n rankCO(A;B;C) min (rankO(A;B;C); rankC(A;B;C)): 2. The -Distance. The open character of Mco , allows us to ensure that if (A;B;C) 2 M is a controllable and observable triple of matrices there exists a neigborhood U in M such that for all (A1; B1; C1) 2 U then (A1; B1; C1) is also a controllable and observable. Therefore it makes sense to consider the distance to the nearest uncontrollable or unobservable or uncontrollable and unobservable triple. Definition (2.1): We de ne a norm in the space M in the following manner for all (A;B;C) 2 M; k(A;B;C)k = A B C 0 ; where A B C 0 is any matrix norm. Definition (2.2): For a given controllable and observable triple of matrices (A;B;C) 2 Mco we de ne the distance between (A;B;C) and a nearest uncontrollable triple by c(A;B;C) = min ( A; B; C) k( A; B; C)k where ( A; B; C) 2 M such that (A + A;B + B;C + C) is uncontrollable. Definition (2.3): For a given controllable and observable triple of matrices (A;B;C) 2 Mco we de ne the distance between (A;B;C) and a nearest unobservable triple by o(A;B;C) = min ( A; B; C) k( A; B; C)k 5 where ( A; B; C) 2 M such that (A + A;B + B;C + C) is unobservable. Definition (2.4): For a given controllable and observable triple of matrices (A;B;C) 2 Mco we de ne the distance between (A;B;C) and a nearest uncontrollable and unobservable triple by co(A;B;C) = min ( A; B; C)k( A; B; C)k where ( A; B; C) 2 M such that (A + A;B + B;C + C) is uncontrollable and unobservable. We remark that co maxf c; og as we can see in the following example: Let (A;B;C) with A 2 M1(R) , A = (a) , B 2 M1(R) , B = (1), C 2 M1(R) , C = (1), c = 1, o = 1, co = p2. Then make sense to consider co . The matrix norm considered in the follows is the 2-norm: kAk2 = 1 where 1 is the largest singular value of A . It is evident that if P is an orthogonal matrix and we consider (A1; B1; C1) = (P 1AP;P 1B;CP ) we have (A1; B1; C1) = (A;B;C): for = c; o; or co . 3. -distance and relationship with C(A;B;C) , O(A;B;C) and CO(A;B;C) matrices Now we analyze as a bound of k( A; B; C)k2 can be deduced from the controllability, observability and controllability and observability matrices of a given triple of matrices (A;B;C) . In this case we obtain bounds for , = c; o; or co . Given a triple (A;B;C) 2 Mco , the controllability matrix of (A;B;C) , is independent of the matrix C . Then we can reduce to the pair (A;B) and consider the bound of k( A; B)k2 where ( A; B) is in such a way that (A+ A;B + B) is uncontrollable, obtained by D.L. Boley and W-S Lu in [2], and we deduce the following Theorem. Theorem (3.1). For a given triple (A;B;C) 2 Mco we have c(A;B;C) min 1 + kAck2 c 1 c 2; : : : ; 1 + kAck2 c n 1 c n : where c i , i = 1; : : : ; n are the singular values of the controllability matrix C(A;B;C) . Now, taking into account that the observability matrix O(A;B;C) is independent of the matrix B and O(A;B;C)t = C(At; Ct; Bt) , we have 6 Theorem (3.2). For a given triple (A;B;C) 2 Mco we have o(A;B;C) min 1 + kAck2 o 1 o 2; : : : ; 1 + kAck2 o n 1 o n where o i , i = 1; : : : ; n are the singular values of the observability matrix O(A;B;C) . Now we are interested to obtain a bound related to the CO(A;B;C) matrix. Firstly, we obtain a bound relating the O(A;B;C) and CO(A;B;C) Calling co 0 0 0 the s.v.d. of CO(A;B;C) and co i the singular values we have CO(A;B;C) = Xt co 0 0 0 Y where X and Y are orthogonal matrices. Remark (3.1): If (A1; B1; C1) = (XAX 1;XB;CX 1) with Xt = X 1 , then CO(A1; B1; C1) = O(A1; B1; C1)C(A1; B1; C1) = = O(A;B;C)XtXC(A;B;C) = = O(A;B;C)C(A;B;C) = CO(A;B;C) Lemma (3.1). For a given triple (A;B;C) 2 Mco there exists an orthogonal matrix P such that A1 = P 1AP = A1 A2 A3 A4 ; B1 = P 1B = B1 B2 ; C1 = CP = (C1 C2 ) where A1 2Mr(R) , B1 2Mr m(R) , C1 2Mp r(R) 1 r n 1 , with kA2k2 kAck2 o r+1 o r ; kB1k2 co 1 o r and kC2k2 o r+1 Proof: Let (A;B;C) be a triple in Mco , co 0 0 0 the s.v.d of CO(A;B;C) : CO(A;B;C) = Xt co 0 0 0 Y where X and Y are orthogonal matrices. Xt co 0 0 0 Y = O(A;B;C)C(A;B;C) = O(A;B;C)PP 1C(A;B;C) 7 where P is such that O(A;B;C) = Q o 0 P 1 , P , Q being orthogonals and o 0 the s.v.d of O(A;B;C) . We consider (A1; B1; C1) = (P 1AP;P 1B;CP ) , then O(A1; B1; C1) = Q o 0 Xt co 0 0 0 Y Im 0 = Q o 0 B1 B1 = Q o 0 +Xt co 0 0 0 Y Im 0 = = o 0 +QtXt co 0 0 0 Y Im 0 o 0 = BBBBBBBBBBBBBB@ 0 1 . . . o r 0 r+1 . . . o n 0 0 ... ... 0 0 CCCCCCCCCCCCCCA = 0@ or 0 0 o r 0 0 1A o 0 + = or 1 0 0 0 or 1 0 We denote by Ym the upper left m m submatrix of Y, then Y Im 0 = Ym Ynm m and co 0 0 0 Y Im 0 = co 0 0 0 Ym Ynm m = coY p m 0 where Y p m denote the upper n m submatrix of Ym Ynm m In the other hand, partitioning the matrix QtXt = 0@S1 S2 S31A with S1 2Mr np(R) , S2 2M(n r) np(R) , S3 2M(np n) np(R) o 0 +QtXt = or 1S1 o r 1S2 8 soB1 =or 1S1or 1S2coY pm0Now partitioning the matrix S1 = (Sp1 Ss1 ) with Sp1 2 Mr n(R) , S2 = (Sp2 Ss2 )with Sp2 2M(n r) n(R) , the matrix B1 can be witten asB1 =or 1Sp1 coY pmor 1Sp2 coY pm = B1B2with B1 2Mr m(R)kB1k2 = kor 1Sp1 coY pmk2 kor 1k2kSp1k2k cok2kY pmk2Taking into account that kor 1k2 = or 1; kcok2 = co1kSp1k2 0@S1S2S31A 2 = 1kY pmk2 kY k2 = 1andkB1k2co1orTheorem (3.3). For a given triple (A;B;C) 2 Mco we haveco(A;B;C) (kAck2 or+1 + co1 ) 1or + or+1; 1 r n 1Proof: We consider (A1 + A1; B1 + B1; C1 + C1) withA1 = 0 A20 0 ; B1 = B10 ; C1 = ( 0C2 )The triple (A1 + A1; B1 + B1; C1 + C1) is an uncontrollable and unobservabletriple of matrices for all 1 r n 1.Thenk( A1; B1; C1)k2co(A1; B1; C1) = co(A;B;C)Finally, in this case we havek( A1; B1; C1)k2 k A1k2 + k B1k2 + k C1k2 == kA2k2 + kB1k2 + kC2k2 kAck2 or+1or + co1or + or+1:Now we deduce a bound relating the C(A;B;C) and CO(A;B;C) matrices usingthe duality relation that there exist into C(A;B;C) and O(A;B;C) .9 Theorem (3.4). Let (A;B;C) 2 Mco , thenco(A;B;C) kAck2 cr+1 + co1 1cr + cr+1; 1 r n 1proof: For that it su ces to observe that if (A;B;C) , A 2Mn(R) , B 2Mn m(R) ,C 2 Mp n(R) is a controllable and observable triple then the triple (At; Ct; Bt) isalso a controllable and observable triple. And in the other hand, we observe thatgiven a matrix M 2Mr s(R) , M and M t have the same non-zero singular valuesTheorems (3.3) and (3.4) permit us to deduce the following bound for co .Corollary (3.4). Let (A;B;C) 2 Mco . Thenco(A;B;C)min kAck2 or+1 + co1 1or + or+1; kAck2 cr+1 + co1 1cr + cr+1 ;for 1 r n 1 .Example (3.1): Let (A;B;C) the triple de ned as followsA = 0@ 0:1 0:1 00 0:01 0:010 0 0:011A ; B = 0@ 000:11A ; C = ( 0:1 0 0 )C(A;B;C) = 0@ 00 0:00010 0:001 0:000020:1 0:001 0:000011AO(A;B;C) =0@ 0:1000:01 0:0100:001 0:0011 0:00011ACO(A;B;C) = 0@ 0010 5010 50:12 10 510 5 0:12 10 5 0:123 10 61Ac = 0:01107294359o = 0:01010136582co = 0:01118193072References10 [1] D. Boley; Estimating the Sensitivity of the Algebraic Structure of Pencils withsimple Eigenvalue estimates. SIAM J. Matrix Anal. Appl. 11 (4), 632-643,(1990).[2] D. Boley, Wu-Sheng Lu; Measuring How Far a Controllable System is froman Uncontrollable One. IEEE Trans. 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