# Robust controllability for linear uncertain descriptor systems

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(2) J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. 633. 1. Introduction In recent years, the controllability problems of linear descriptor systems have attracted some attention in the literature (for example, see [1,5,7–9,11–14] and references therein) due to the significance of descriptor systems in both theory and applications. Sometimes the descriptor system is called singular system, generalized state-space system, implicit system, or semistate system [6]. To the authors’ best knowledge, only Lin et al. [7–9] and Chou et al. [1] studied the robust controllability problems of linear descriptor systems. That is, the research on the robust controllability of linear descriptor systems is considerably rare and almost embryonic. Lin et al. [7] proposed some sufficient conditions for robust controllability of linear descriptor systems with unstructured (norm-bounded) parameter uncertainties, where the unstructured parameter uncertainties are in the system matrix only. Lin et al. [8] studied the robust C-controllability (complete controllability) problem for linear uncertain descriptor systems with all matrices (structure information matrix, system matrix, and input matrix) being interval matrices. Lin et al. [9] also studied the robust controllability problems of linear descriptor systems with structured (elemental) parameter uncertainties in the system matrix and the input matrix. Based on the structured singular value approach, the sufficient conditions proposed by Lin et al. [8,9] are obtained by transforming the robust controllability problems into checking the nonsingularity of a class of uncertain matrices. On the other hand, it is well known that an approximate system model is always used in practice and sometimes the approximation error should be covered by introducing simultaneously both structured (elemental) and unstructured (norm-bounded) parameter uncertainties in control system analysis and design. That is, it is not unusual that at times we have to deal with a system consisting of two parts: one part has only the structured parameter uncertainties, and the other part has the unstructured parameter uncertainties. Therefore, Chou et al. [1] investigated the robust controllability problems of linear descriptor systems with both structured (elemental) and unstructured (norm-bounded) parameter uncertainties simultaneously in the system matrix and the input matrix. Note that only the article of Chou et al. [1] considered both structured and unstructured parameter uncertainties, and the sufficient conditions of Chou et al. [1] are the generalized versions of the results given by Lin et al. [7]. The purpose of this paper is to propose a new approach to investigate the robust controllability problems of linear descriptor systems with both structured (elemental) and unstructured (normbounded) parameter uncertainties in the system matrix and the input matrix. The main results are presented in Section 2. In Section 3, we extend the results given in Section 2 to study the robust controllability problems of a class of linear descriptor systems with structured uncertainties in the structure information matrix as well as with both structured and unstructured parameter uncertainties in the system matrix and the input matrix simultaneously. Three numerical examples are given in Section 4 to illustrate the applications of the proposed sufficient conditions, and to make some comparisons between the proposed sufficient conditions and those of Lin et al. [8,9] and Chou et al. [1]. Finally, Section 5 offers some conclusions. 2. Controllability robustness Consider the linear continuous-time uncertain descriptor system described by m m E x(t) ˙ = Ax(t) + αi Ai x(t) + Ax(t) + Bu(t) + αi Bi u(t) + Bu(t), i=1. i=1. or the linear discrete-time uncertain descriptor system described by. (1).

(3) 634. J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. Ex(k + 1) = Ax(k) +. m i=1. αi Ai x(k) + Ax(k) + Bu(k) +. m . αi Bi u(k) + Bu(k),. (2). i=1. where E ∈ R n×n is the structure information matrix, A ∈ R n×n is the system matrix, and B ∈ R n×q is the input matrix; x(t) and x(k) are the n × 1 state vectors; u(t) and u(k) are the q × 1 input vectors; αi (i = 1, 2, . . . , m) are the uncertain parameters; Ai and Bi (i = 1, 2, . . . , m) are the given n × n and n × q, respectively, constant matrices which are prescribed a prior to denote the linearly dependent information on uncertain parameters αi ; the unstructured uncertain and B are assumed to be bounded, i.e., matrices A β1 (3) A and β2 , B. (4). where β1 and β2 are nonnegative real constant numbers, and · denotes any matrix norm. Here the matrix E may be a singular matrix with rank(E) n. In many applications, the matrix E is a structure information matrix rather than a parameter matrix, i.e., the elements of E contain only structure information regarding the problem considered. In this paper, the linear nominal system (E, A, B) is assumed to be regular and controllable. Due to inevitable uncertainties, the linear nominal system (E, mA, B) is perturbed into the linear uncertain system (E, A + A, B + B), where A = i=1 αi Ai + A and B = m α B + B. Our problem is to determine the conditions such that the linear uncertain system i=1 i i (E, A + A, B + B) is still regular and controllable. Although only the controllability problems are considered, the corresponding results for the dual observability robustness problems are straightforward extensions and are omitted. Before we investigate the robust properties of regularity and controllability of the linear uncertain system (E, A + A, B + B), the following definitions and lemmas need to be introduced first. Definition 1 [15]. The system (E, A, B) is called completely controllable (C-controllable), if for any t1 > 0 (or k1 > 0), x(0) ∈ R n and w ∈ R n , there exists a control input u(t) (or u(k)) such that x(t1 ) = w (or x(k1 ) = w). Definition 2 [15]. The system (E, A, B) is called R-controllable, if it is controllable in the reachable set. Definition 3 [10]. The system (E, A, B) is called impulse controllable (I-controllable), if there is a state feedback u(t) = Kx(t) (or u(k) = Kx(k)) such that the closed-loop system (E, A + BK) is impulse-free. Definition 4 [10]. The system (E, A, B) is called strongly controllable (S-controllable), if it is both R-controllable and I-controllable. Definition 5 [4]. The measure of a matrix W ∈ C n×n is defined as (I + θ W − 1) , µ(W ) ≡ lim θ→0 θ where · is the induced matrix norm on C n×n ..

(4) J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. 635. Lemma 1 [15]. The system (E, A, B) is regular if and only if rank[En Ed ] = n2 , where En ∈ 2 2 2 R n ×n and Ed ∈ R n ×n are given by E A 0 E A · · · and Ed = . En = (5) · · · · · · 0 E A Lemma 2 [7]. Suppose that the system (E, A, B) is regular. The system (E, A, B) is I-controllable if and only if rank[ASE. E. B] = n,. (6). where SE ∈ R n×(n−r) is the maximum right annihilator matrix of E, in which r = rank[E]. Lemma 3 [3]. Suppose that the system (E, A, B) is regular. The system (E, A, B) is R-con2 2 trollable if and only if rank[Ed Eb ] = n2 , where Ed ∈ R n ×n is given in Eq. (5) and Eb = 2 diag{B, B, . . . , B} ∈ R n ×nq . Lemma 4 [2]. Suppose that the system (E, A, B) is regular. The system (E, A, B) is C-controllable if and only if it is R-controllable and rank[E B] = n. Lemma 5 [4]. The matrix measures of the matrices W and V , µ(W ) and µ(V ), are well defined for any norm and have the following properties: (i) µ(±I ) = ±1, for the identity matrix I ; (ii) −W −µ(−W ) Re(λ(W )) µ(W ) W , for any norm · and any matrix W ∈ C n×n ; (iii) µ(W + V ) µ(W ) + µ(V ), for any two matrices W, V ∈ C n×n ; (iv) µ(γ W ) = γ µ(W ), for any matrix W ∈ C n×n and any nonnegative real number γ ; where λ(W ) denotes any eigenvalue of W, and Re(λ(W )) denotes the real part of λ(W ). Lemma 6. For any γ < 0 and any matrix W ∈ C n×n , µ(γ W ) = −γ µ(−W ). Proof. From the property (iv) in Lemma 5, this lemma can be immediately obtained.. . Lemma 7. Let W ∈ C n×n . If µ(−W ) < 1, then det(I + W ) = / 0. Proof. Since µ(−W ) < 1, then, from the property (ii) in Lemma 5, we have Re(λ(W )) −µ(−W ) > −1. This implies that λ(W ) = / −1. Thus, we can get that det(I + W ) = / 0. Now, let the singular value decompositions of R0 = [En Ed ], N0 = [ASE E B], Q0 = [Ed Eb ] and M0 = [E B] be, respectively, R0 = U [S. On2 ×n ]V H ,. (7).

(5) 636. J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. N0 = UI [SI. On×(n−r+q) ]VIH ,. (8). Q0 = UR [SR. On2 ×nq ]VRH ,. (9). M0 = UC [SC. On×q ]VCH ,. and (10). where U ∈ R n ×n and V ∈ R (n +n)×(n +n) are the unitary matrices, S = diag{σ1 , σ2 , . . . , σn2 }, and σ1 σ2 · · · σn2 > 0 are the singular values of R0 ; UI ∈ R n×n and VI ∈ R (2n−r+q)×(2n−r+q) are the unitary matrices, r = rank(E), SI = diag{σ1 , σ2 , . . . , σn }, and σ1 2 2 2 2 σ2 · · · σn > 0 are the singular values of N0 ; UR ∈ R n ×n and VR ∈ R (n +nq)×(n +nq) are the unitary matrices, SR = diag{σ1 , σ2 , . . . , σn2 }, and σ1 σ2 · · · σn2 > 0 are the singular values of Q0 ; UC ∈ R n×n and VC ∈ R (n+q)×(n+q) are the unitary matrices, SC = diag{σ1 , σ2 , . . . , σn }, and σ1 σ2 · · · σn > 0 are the singular values of M0 ; V H , VIH , VRH and VCH denote, respectively, the complex-conjugate transposes of the matrices V , VI , VR and VC . In what follows, with the preceding definitions and lemmas, we present some sufficient conditions for ensuring that the linear uncertain descriptor system (E, A + A, B + B) remains regular and controllable. 2. 2. 2. 2. Theorem 1. Suppose that the linear nominal descriptor system (E, A, B) is regular and Icontrollable. The linear uncertain descriptor system (E, A + A, B + B) is still regular and I-controllable, if the following inequalities simultaneously hold m . αi ϕi + β1 S −1 U H V [In2 , On2 ×n ]T < 1. (11a). i=1. and m . αi φi + β1 SI−1 UIH SE VI [In , On×(n−r+q) ]T. i=1. + β2 SI−1 UIH VI [In , On×(n−r+q) ]T < 1,. (11b). where In2 and In denote, respectively, the n2 × n2 and n × n identity matrices;.

(6) −1 H T for αi 0, µ −

(7) S−1 UH Ri V [In2 , On2 ×nT] ϕi = i V [In2 , On2 ×n ] for αi < 0; −µ S U R. −1 H T µ(−SI UI Ni VI [In , On×(n−r+q) ] ) for αi 0,.

(8) φi = for αi < 0; −µ SI−1 UIH Ni VI [In , On×(n−r+q) ]T i = [On2 ×n R. 2 ×(n2 +n). Ri ] ∈ R n. ;. n2 ×n2. ; Ri = diag{Ai , . . . , Ai } ∈ R Ni = [Ai SE On×n Bi ] ∈ R n×(2n−r+q) ; the matrices S, U, V , SI , UI and VI are defined in Eqs. (7) and (8), respectively. Proof. Firstly, we show the regularity. Since the nominal system (E, A, B) is regular, then, 2 2 from Lemma 1, we can get that the matrix R0 = [En Ed ] ∈ R n ×(n +n) has full row rank (i.e.,.

(9) J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. 637. rank(R0 ) = n2 ). With the uncertain matrices A + A and B + B, the uncertain descriptor system (E, A + A, B + B) is regular if and only if = R0 + R. m . , i + F αi R. (12). i=1. = [On2 ×n F ] ∈ R n i = [On2 ×n Ri ] ∈ R n ×(n +n) , F has full row rank, where R 2 2 . . . , A} ∈ R n2 ×n2 . diag{Ai , . . . , Ai } ∈ R n ×n , and F = diag{A, It is known that = rank(S −1 U H RV ). rank(R) 2. 2 ×(n2 +n). 2. , Ri =. (13). we can discuss the rank of Thus, instead of rank(R), m . [In2 , On2 ×n ] +. i + F , αi R. (14). i=1. i V and F V , for i = 1, 2, . . . , m. Since a matrix has at least = S −1 U H F i = S −1 U H R where R 2 rank n if it has at least one nonsingular n2 × n2 submatrix, a sufficient condition for the matrix in Eq. (14) to have rank n2 is the nonsingularity of L = In2 +. m . αi R i + F ,. (15). i=1. i V [In2 , On2 ×n ]T (for i = 1, 2, . . . , m), and F = S −1 U H F V [In2 , On2 ×n ]T . where R i = S −1 U H R Using the properties in Lemmas 5 and 6, and from (3) and (11a), we get m µ − αi R i − F i=1. . =µ − µ −. m i=1 m . αi S. −1. i V [In2 , On2 ×n ]T U HR. −1. i V [In2 , On2 ×n ]T U HR. −S . αi S. −1. H. U F V [In2 , On2 ×n ]. T. V [In2 , On2 ×n ]T. + S −1 U H F. i=1 m .

(10) V [In2 , On2 ×n ]T. i V [In2 , On2 ×n ]T + S −1 U H F µ − αi S −1 U H R i=1. = . m i=1 m . V [In2 , On2 ×n ]T. αi ϕi + S −1 U H F. αi ϕi + β1 S −1 U H V [In2 , On2 ×n ]T. i=1. < 1.. (16). From Lemma 7, we have that m det In2 + αi R i + F = / 0. i=1. (17).

(11) 638. J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. has full row rank. Thus, from Lemma 1, the regularity of the This implies that the matrix R uncertain descriptor system (E, A + A, B + B) is ensured. Next, we show the I-controllable. Since the nominal system (E, A, B) is I-controllable, then, from Lemma 2, we can get that the matrix N0 = [ASE E B] has full row rank (i.e., rank(N0 ) = n). With the uncertain matrices A + A and B + B, the uncertain descriptor system (E, A + A, B + B) is I-controllable if and only if N = N0 +. m . αi Ni + H 1 + H 2 ,. (18). i=1. has full row rank, where r = rank(E), Ni = [Ai SE On×n Bi ] ∈ R n×(2n−r+q) , H1 = E On×n On×q ] ∈ R n×(2n−r+q) , and H2 = [On×(n−r) On×n B] ∈ R n×(2n−r+q) . [AS It is known that rank(N ) = rank(SI−1 UIH N VI ).. (19). Thus, instead of rank(N ), we can discuss the rank of [In , On×(n−r+q) ] +. m . 1 + H 2 , i + H αi N. (20). i=1. i = S −1 U H Ni VI , H 1 = S −1 U H H1 VI and H 2 = S −1 U H H2 VI , for i = 1, 2, . . . , m. Since where N I I I I I I a matrix has at least rank n if it has at least one nonsingular n × n submatrix, a sufficient condition for the matrix in Eq. (20) to have rank n is the nonsingularity of LI = In +. m . αi N i + H 1 + H 2 ,. (21). i=1. where N i = SI−1 UIH Ni VI [In , On×(n−r+q) ]T (for i = 1, 2, . . . , m), H 1 = SI−1 UIH H1 VI [In , On×(n−r+q) ]T and H 2 = SI−1 UIH H2 VI [In , On×(n−r+q) ]T . Using the properties in Lemmas 5 and 6, and from (3), (4) and (11b), we have m µ − αi N i − H 1 − H 2 i=1. . =µ −. m . αi SI−1 UIH Ni VI [In , On×(n−r+q) ]T − SI−1 UIH H1 VI [In , On×(n−r+q) ]T. i=1. . − SI−1 UIH H2 VI [In , On×(n−r+q) ]T µ −. m . αi SI−1 UIH Ni VI [In , On×(n−r+q) ]T. i=1. −1 H + SI UI H1 VI [In , On×(n−r+q) ]T + SI−1 UIH H2 VI [In , On×(n−r+q) ]T. m

(12). µ − αi SI−1 UIH Ni VI [In , On×(n−r+q) ]T i=1. + SI−1 UIH H1 VI [In , On×(n−r+q) ]T + SI−1 UIH H2 VI [In , On×(n−r+q) ]T.

(13) J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. 639. m

(14). µ − αi SI−1 UIH Ni VI [In , On×(n−r+q) ]T i=1. + β1 SI−1 UIH SE VI [In , On×(n−r+q) ]T + β2 SI−1 UIH VI [In , On×(n−r+q) ]T. =. m . αi φi + β1 SI−1 UIH SE VI [In , On×(n−r+q) ]T. i=1. + β2 SI−1 UIH VI [In , On×(n−r+q) ]T. < 1.. (22). From Lemma 7, we have that m det In + / 0. αi N i + H 1 + H 2 =. (23). i=1. This implies that the matrix N has full row rank. Hence, from the results mentioned above and Lemma 2, the I-controllability of the uncertain descriptor system (E, A + A, B + B) is ensured. Theorem 2. Suppose that the linear nominal descriptor system (E, A, B) is regular and R-controllable. The linear uncertain descriptor system (E, A + A, B + B) is still regular and R-controllable, if the following inequalities simultaneously hold m . αi ϕi + β1 S −1 U H V [In2 , On2 ×n ]T < 1 (24a) i=1. and. m . αi θi + (β1 + β2 ) SR−1 URH. VR [In2 , On2 ×nq ]T < 1,. (24b). i=1. where.

(15). µ − SR−1 URH Qi VR [In2 , On2 ×nq ]T for αi 0,

(16). θi = −µ SR−1 URH Qi VR [In2 , On2 ×nq ]T for αi < 0; Bi Ai A Bi i · · Qi = · · · · Ai Bi. ∈ R n2 ×(n2 +nq) ; . ϕi (i = 1, 2, . . . , m) are given in Theorem 1; the matrices S, U, V , SR , UR and VR are defined in Eqs. (7) and (9), respectively. Proof. Firstly, following the same proof procedure given in Theorem 1, we can prove that the sufficient condition (24a) ensures the uncertain descriptor system (E, A + A, B + B) to be regular. Next, we show the R-controllability. Since the nominal system (E, A, B) is R-controllable, then, from Lemma 3, we have that the matrix Q0 = [Ed Eb ] has full row rank (i.e., rank(Q0 ) = n2 ). With the uncertain matrices A +.

(17) 650. J.-H. Chou et al. / Linear Algebra and its Applications 414 (2006) 632–651. between the proposed sufficient conditions and those of Lin et al. [8,9] and Chou et al. [1] under the assumption that the matrix R0 = [En Ed ] has full row rank. From the above numerical examples, under the assumption that the matrix R0 = [En Ed ] has full row rank, we can see that the proposed sufficient conditions may obtain less conservative results than those of Lin et al. [8,9] and Chou et al. [1]. The reasons why the proposed sufficient conditions are less conservative are: (i) The proposed sufficient conditions take the directional information into consideration. This can be explained by the fact that as a parameter varies in different directions, it affects the system’s properties differently. That is, the effect of a single parameter α on the system’s properties can be completely different for the same |α| and opposite sign. Therefore, any sufficient conditions, that ignore the signs, may obtain more conservative results. (ii) The singular value decomposition used to derive the proposed sufficient conditions can be exploited to simplify the analysis and to gain insight into the underlying important factors of the matrices R0 , N0 , Q0 and M0 . Therefore, the proposed sufficient conditions may give less conservative results under the assumption that the matrix R0 = [En Ed ] has full row rank. On the other hand, it may be believed that the sufficient conditions proposed in this paper and the sufficient conditions of Lin et al. [8,9] and Chou et al. [1] can be complemented by each other such that the tools of controllability robustness analysis of linear uncertain descriptor systems are more complete.. 5. Conclusions In this paper, some sufficient conditions have been established for the robust regularity, the robust I-controllability, the robust R-controllability and the robust C-controllability of the linear descriptor systems with both structured and unstructured parameter uncertainties. The corresponding results for the dual observability robustness problems are straightforward extensions. The results in the proposed theorems have also been extended to obtain another sufficient conditions on the robust regularity, the robust I-controllability, the robust R-controllability and the robust C-controllability for a class of linear descriptor systems with both structure information uncertainties and parameter uncertainties. Three numerical examples have been given to illustrate the applications of the proposed sufficient conditions, and it has been shown that the proposed sufficient conditions could be less conservative than the existing conditions given by Lin et al. [8,9] and Chou et al. [1] under the assumption that the matrix R0 = En Ed has full row rank. The reasons why the proposed sufficient conditions could be less conservative are also given in Remark 4. Acknowledgments This work was supported by the National Science Council, Taiwan, Republic of China, under grant numbers NSC-92-2213-E-327-001 and NSC-92-2213-E-151-006. The authors thank the referees for their constructive and helpful comments and suggestions. References [1] J.H. Chou, S.H. Chen, R.F. Fung, Sufficient conditions for the controllability of linear descriptor systems with both time-varying structured and unstructured parameter uncertainties, IMA J. Math. Control Inform 18 (2001) 469–477. [2] J.C. Cobb, Controllability, observability and duality in singular systems, IEEE Trans. Automat. Control 29 (1984) 1076–1082..

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