Rank Reduction for Matrix Pair and Its Application in Singular Systems

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1 Rank Reduction for Matrix Pair Its Application in Singular Systems Jing Wang Wanquan Liu, Qingling Zhang Xiaodong Liu 1 Institute of System Sciences Northeasten University, PRChina wj--zhang@163com Department of Computing Curtin University of Technology, Australia wanquan@cscurtineduau 1 Deptof Mathematics Physics Dalian Maritime University Dalian, P R China Abstract In this paper, the rank reduction problem for a rectangle matrix pair is investigated First, the rank reduction problem is defined it is solved via an algebraic approach In fact, the proposed method is a procedure for getting the maximal value for the uncertain parameter such that the rank of the perturbed matrix will remain the same Based on the results, the maximal robust stability radius problem of singular systems has been solved completely Finally, three examples are used to illustrate the effectiveness of the proposed approach It is known that the rank reduction problem is a very important issue mathematically with many potential applications in control signal processing [1] In addition, many related issues have been investigated about this rank reduction issue [2, 3] Though many researchers have put much effort on this problem, it has not been thoroughly solved due to its complexity In this paper, we will discuss the rank reduction problem from a new point of view First, the rank reduction problem of a rectangle matrix pencil is transformed to a corresponding problem of square matrix with lower order via a matrix transformation Then a new method is given to completely solve the rank reduction problem via the matrix theory The advantage of the proposed method is its simplicity In the second half of this paper, an important application is investigated It is well known that many control systems are subject to perturbations in terms of uncertain parameters an important quantitative measure of stability robustness for a system to such perturbations is what is called the real or complex stability radius depending on the perturbation nature The problem of stability radius was introduced in [4] from a state space point of view, then it attracted a lot of interests, see [5, 6, 7] All these results are for the normal systems In this paper, we will apply the obtained results to solve the robust stability radius problem for singular systems which have special structure properties have some specific mechanical examples [8] Finally, three examples are used to illustrate the effectiveness of the proposed approachthroughout this paper, the uncertain parameter δ belongs to the real region 1 Introduction 2 Rank reduction The rank reduction problem can be defined as follows in scalar case: Problem Given A, B R n n, compute δ m =sup{ 0 δ R, δ <, rank[a+δb]=rank[a]} This is to say that we need to find the maximum perturbation radius for the matrix pair In order to solve this problem, we need the following important lemma Lemma 1 [9] Given any two matrices A, B R n n, there always exist two nonsingular matrices Q, P such that B = QBP = diag(0,l 1,L 2,,L p,l 1,L 2,,L q,i h,n)(1) Ã = QAP = diag(0,t 1,T 2,,T p,t 1,T 2,,T q,a 1,I g ) (2)

2 where L i =,T i=, L j=,t j= 0 1 0, N ki =, N = diag(n k1,n k2,,n kr ) R g g, L i,t i R m i (m i +1),, 2,,p, L j,t j R (n j+1) n j,j =1, 2,,q, N ki R ki ki, i =1, 2,,r, 0 R m0 m0,a 1 R h h, m 0 + p m i + q (n j +1)+ r k i + h = m, n 0 + q n j + p (n i +1)+ r k i + h = n, r k i = g From lemma 1, let Q, P satisfy the decomposition (1) (2), then the following holds: rank[a + δb] = rank[δdiag(0,l 1,L 2,,L p,l 1,L 2,,L q,i h,n) +diag(0,t 1,T 2,,T p,t 1,T 2,,T q,a 1,I g ] p q = rank[δl i + T i ]+ rank[δl j + T j ] +rank[δi h + A 1 ]+rank[δn + I g ] To discuss each item separately, we have the following: Further, rank[δn + I g ] g = rank[i g ] (3) δl i + T i = δ δ δ δ 1 Therefore, rank[δl i + T i ] m i = rank[t i ], i =1, 2,,p (4) Similarly, rank[δl j + T j ] n j = rank[t j ], j =1, 2,,q (5) In addition, rank[a] = rank[qap ] p = rank[t i ]+ q rank[t j ]+rank[a 1 ]+rank[i g ] Then it follows from (3)-(6) that rank[a + δb] = rank[a] =rank[a 1 ] which implies that the rank reduction problem of a rectangle matrix pencil is transformed to the corresponding problem of the square matrix with lower dimensions Hence (6) δ m = sup{ 0 δ R, δ <, rank[a+δb]=rank[a]} = sup{ 0 δ R, δ <, rank[a 1 +δi h ]= rank[a 1 ]} In the following, we will discuss the range of the parameter δ such that =rank[a 1 ] Let α 1,α 2,,α l be all of the different finite eigenvalues of the matrix A 1, that is det(α i I h +A 1 )=0, i = 1, 2,, l Without generality, one assumes that α 1,α 2,,α s are real, α s+1,α s+2,,α l are not real with 0 α 1 < α 2 < < α s ; If α i is not real for all 1 i l, wesays = 0 Then assume that its Jordan canonical form decomposition is the following: where A 1 = TJT 1, (7) J = diag{j 1,J 2,,J l },J i = diag{j i1,j i2,,j isi }, α i α i 1 0 J ij = C nij nij, 0 0 α i α i s i n ij = h,

3 T is nonsingular The following cases will be discussed respectively according to the value of α 1 Case 1 α 1 > 0 If s = 0, that is none of α 1,α 2,,α l are real, then it is obvious that δ m = If s 0, then we have the following two cases: When δ = α i,, 2,,s, there holds that rank[j i δi] = dim J i m i = rank[j i ] m i ; (8) rank[j j δi] = dim J j =,j i, j s Therefore, And when δ = α 1 =0, = =rank[a 1 ] When δ>0, there holds that rank[ J 1 + δi] = dim J 1 > rank[j 1 ] (9) dim J 1 rank[j 1 ]=m 1, (10) Let the set Ω={i m i = m 1, 2 i s}, = rank[j i δi]+ rank[j j δi] j i Ω ={i m i m 1, 2 i s} = Ω 1 Ω 2, = dim J i m i + j i = rank[j i ] m i + j i = m i < rank[a 1 ] So, in the case δ = α i,i =1, 2,,s, one can obtain the following: rank[a + δb] < rank[a] When δ α i,, 2,,s, the following hold therefore, rank[ J i + δi] = dim J i = rank[j i ], = =rank[a 1 ], which implies the following holds rank[a + δb] =A, From the above, one can obtain that in the case s 0, the value δ m is given by δ m = α 1 Case 2 α 1 =0 From the decomposition (7), we have =rank[ J + δi h ]= rank[ J i + δi] where Ω 1 = {i m i >m 1, 2 i s}, Ω 2 = {i m i <m 1, 2 i s} Then we have the following results When δ = α i,i Ω, we can get the following = rank[j 1 δi]+rank[j i δi]+ rank[j j δi] = dim J 1 + dim J i m i + = rank[j 1 ]+m 1 + rank[j i ] m i + = = rank[a 1 ] When δ = α i,i Ω 1, = rank[j 1 δi]+rank[j i δi]+ rank[j j δi] = dim J 1 + dim J i m i + = rank[j 1 ]+m 1 + rank[j i ] m i + < = rank[a 1 ]

4 When δ = α i,i Ω 2, = rank[j 1 ]+m 1 + rank[j i ] m i + > = rank[a 1 ] If δ α i, i =1, 2,,s, then rank[ J j + δi] = dim J j =, j =2, 3,,l, = rank[j 1 δi]+rank[j i δi]+ = dim J 1 + j=2 = rank[j 1 ]+m 1 + > rank[j i ] = rank[a 1 ] j=2 rank[j j δi] So in this case α 1 = 0, we have the following conclusion: δ m =0 Let M big = {δ rank[a + δb] >A}; M equal = {δ rank[a + δb] =A}; M small = {δ rank[a + δb] <A} Then we will have the following theorem based on previous analysis Theorem 2 With α 1 = 0, the following hold: M big = {α i i Ω 2 } {α α R,α α i,, 2,,s}; M equal = {α i i Ω} {0}; M small = {α i i Ω 1 }; With α 1 > 0, the following hold: if s = 0, then M big =M small =Ø; M equal = {α α R}; if s 0, then M big =Ø; M equal = {α α R,α α i,, 2,,s}; M small = {α i i =1, 2,,s} Further, we have the following important theorem for the perturbation radius: Theorem 3 Given A, B R n n, let δ m =sup{ 0 δ R, δ <, rank[a+δb]=rank[a]} then (i) if A 1 is nonsingular, s = 0, then δ m = ; (ii) if A 1 is nonsingular, s 0, then δ m = α 1 ; (iii) if A 1 is singular, δ m =0 Now the rank reduction problem in the scalar case has been solved Next we will use these results to investigate some control problems 3 Robust radius for singular systems Consider the following singular systems Eẋ(t) =Ax(t)+δBx(t),x(0 ) =x 0, (11) where x(t) R n is the state vector, E R n n, A R n n, B R n n, are constant matrices with E possibly singular δ is a real uncertain parameterassume that the system (11) is regular stable, that is se A = 0 σ(e,a) C, where σ(e,a) ={λ C det(λe A) =0}, C represents the open left half complex plane All the matrices in this part are assumed to have approximate dimensions The maximal robust stability radius problem of system (11) is to find the maximum δ m such that system (11) will be still stable for all 0 < δ <δ m, that is δ m = max { 0 δ R, δ <,σ(e,a+δb) C } In this paper, we will use the results in last section to discuss this problem First, the following theorem converts the maximum stability radius problem into a rank reduction problem Theorem 4 If system (11) is stable regular, then max { 0 δ R, δ <,σ(e,a + δb) C } = max{ 0 δ R, δ <, rank[a + δb] =n} Proof Because the system (E,A) is stable, then rank[λe A] =n, for λ C +, (12)

5 where C + represents the closed right half complex plane And further, from (12), the following holds: rank[jwe A] =n, for w R where j = 1 Since eigenvalues of the matrix pencil (E,A + δb) are continuous with the parameter δ, so the stability only fails in the imaginary axis, ie, max { 0 δ R, δ <,σ(e,a + δb) C } = max{ 0 δ R, δ <, rank[jwe A δb] =n, w R} Notice, since δ R, there holds that rank[a + δb] =n rank[jwe A δb] =n, then we have the following: max{ 0 δ R, δ <, w R rank[jwe A δb] =n, w R} = max{ 0 δ R, δ <, rank[a + δb] =n} so the equality in this theorem is true this completes the proof This result is very interesting, it shows that when system (11) is stable regular, the maximal robust stability radius of system (11) is independent of the matrix E, only depends on the matrix pencil (A, B) Let δm =max{ 0 δ R, δ <, rank[a+δb]=n}, then δ m = δ m Now δ m can be computed by theorem 3 For the matrix pencil (B,A), because A is nonsingular, there holds δb + A 0, therefore there exist nonsingular matrices P m,q m such that Q m BP m = diag(i m,n), (13) Q m AP m = diag(a m,i N ), (14) where N is nilpotent Then from theorem 3, we have the following theorem immediately: Theorem 5 If (E,A) is stable regular, then if the matrix A m have real eigenvalues, then δ m is equal to the minimum module of all of the finite real eigenvalues of the matrix A m, otherwise δ m = Remark 1 In fact, from the above analysis, in order to get the maximal robust stability radius of system (11), we only need to compute the finite eigenvalues of the matrix pencil (B, A) instead of decomposing (13) (14), since the finite eigenvalues of A m are the same with the finite eigenvalues of (B, A) 4 Illustrative examples Example 1 Given matrices E, A B as below: E = , A = , B = We can check that the system (E,A) is stable regular After computing, the finite eigenvalues of (B, A) are 1, 1, so from theorem 5, δ m =1 Example 2 Given matrices E, A B as below: E = B = ,A= , The system (E,A) can be checked to be stable regular After computing, the different finite eigenvalues of (B, A) are 27980, 08947, ± 03084j, so from theorem 5, δ m =08947 Example 3 Given matrices E, A B as below: E = B = ,A= , The system (E,A) is stable regular After computing, the different finite eigenvalues of (B, A) are ± 13584j, ± 06254j, that is there are no real eigenvalues, so from theorem 5, δ m = 5 Conclusions In this paper, the rank reduction problem for a rectangle matrix pair has been investigated Then it is proved that the maximal robust stability radius problem of singular systems can be converted as a rank reduction problem can been solved For the case that the uncertain parameter δ belongs to the complex region, one can get similar conclusions with similar methods And they are omitted here

6 In the future, we will investigate the robust stability radius in general case that is the perturbation is of matrix form for singular systems References [1] Hua Y, Liu W, Generalize KarhunenLoeve Transform, IEEE signal processing Letters, IEEE, New York, USA, Vol 5, No 6, pp , June, 1998 [2] L EI Ghaoui P Gahinet, Rank minimization under LMI constraints: a framework for output feedback problems, in the proceedings of the European Control Conference, July, 1993 [3] M Mesbahi, Solving a class of rank minimization problem as semi-definite programs with applications to fixed order output feedback synthesis, Systems control letters, Vol 33, No1, pp 31 36, 1998 [4] D Hinrichsen A J Pritchard, Stability radius for structured perturbations the algebraic Riccatti equation, Systems Control Letters, Vol 8, pp ,1986 [5] L Qiu, B Bernhardsson, A Rantzer, E Davison, P Young JDoyal, A formula for computation of the real stability radius, Automatica, Vol 31, pp , 1995 [6] R Genesil, A Tesi, Results on the stability robustness of systems with state space perturbations, Systems Control Letters, Vol 22, pp , 1994 [7] W Yan, J Lam, On the computation of the stability radius for nonlinearly structured perturbations, Systems control Letters, Vol 34, No 5, pp , 1998 [8] Recent developments in mechanical systems geometric control theory, a special session in fourth international conference on dynamical systems differential equations, Wilmington, NC,USA, May, pp 24 27, 2002 [9] Gantmacher, FR, The theory of matrices, Chelsea, New York, 1974

ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS

INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5 Number 3-4 Pages 480 487 c 2009 Institute for Scientific Computing and Information ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF