Available online at www.sciencedirect.com Systems & Control Letters 54 (2005) 33 – 41 www.elsevier.com/locate/sysconle Implicit-system approach to the robust stability for a class of singularly perturbed linear systems Nguyen Huu Dua , Vu Hoang Linha;∗ a Faculty of Mathematics, Mechanics, and Informatics, University of Natural Sciences, Vietnam National University 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Received 15 January 2003; received in revised form June 2004; accepted June 2004 Available online 17 July 2004 Abstract Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear di erential-algebraic equations (DAEs) are studied The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only Our results can be extended to other singular perturbation problems for DAEs of more general form c 2004 Elsevier B.V All rights reserved Keywords: Asymptotic stability; Stability radius; Singular perturbation; Implicit systems; Index of matrix pencil Introduction by linear di erential equations In the past decade, considerable attention has been devoted to some robustness measures of continuous and discrete time systems The concept of stability radii introduced by Hinrichsen and Pritchard has been investigated for di erent classes of systems in a large amount of research papers, e.g., for the ÿnite dimensional case, see [2,3,5,6,9–14] A number of models arising in applications are di erential-algebraic equations (DAEs) which may contain some small parameters as well, for example, see [1, Section 1.3] We consider the coupled implicit system described E11 y1 (x) = A11 y1 (x) + A12 y2 (x); ∗ Corresponding author E-mail address: vhlinh@hn.vnn.vn (V.H Linh) E22 y2 (x) = A21 y1 (x) + A22 y2 (x); (1.1) where the solutions yi (x) are vector functions, yi (:) : R → Cni ; i = 1; 2; the coe cients Eii , Aij are constant matrices in Cni ×ni and Cni ×nj , respectively, with i; j = 1; 2; is a small positive parameter In addition, we suppose Assumption A1 E11 and A22 are nonsingular The matrix E22 may be singular but, excluding the trivial case, we assume E22 = When no confusion arises, the argument x will be omitted for brevity We 0167-6911/$ - see front matter c 2004 Elsevier B.V All rights reserved doi:10.1016/j.sysconle.2004.06.003 34 N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 can rewrite (1.1) in the block form as follows: E y = Ay; (1.2) where E = y= E11 0 E22 y1 ; A= A11 A12 A21 A22 ; : y2 We note that in the case of = 0; the system is led to a semi-explicit index-1 system of DAEs, e.g., see [1,8] The system (1.1) can be considered as a generalization of the classical singular perturbation problem investigated in [3] and a number of papers where Eii ; i = 1; 2; are set identity matrices Assuming that there exists ¿ such that (1.1) is asymptotically stable (see Deÿnition in Section 2) for all ∈ (0; ], we consider the perturbed system E11 y1 = (A11 + B1 C1 )y1 + (A12 + B1 C2 )y2 ; E22 y2 = (A21 + B2 C1 )y1 + (A22 + B2 C2 )y2 (1.3) or in the block form E y = (A + B C)y (1.4) with B= B1 B2 ; C = (C1 C2 ): Here ∈ Cp×q is an uncertain disturbance, Bi , Ci ; i = 1; 2; are given matrices in Cni ×p and Cq×ni , respectively, and they specify the structure of the perturbation Following the concept due to Hinrichsen and Pritchard [9,10], we want to determine the value of the so-called structured complex stability radius for (1.1) deÿned by r(E ; A; B; C) = inf { ; ∈ Cp×q and (1:4) is not asymptotically stable} (1.5) for each ∈ (0; ] We note that the norm used here is an arbitrary matrix norm induced by a vector norm in Cp×q When E22 is invertible, multiplying both sides of the ÿrst and the second equations of (1.1), (1.3) −1 −1 and −1 E22 , respectively, one obtains an by E11 explicit system of ordinary di erential equations (ODEs) which has been well studied in the literature, e.g., see [9,10,12,14] However, even in this case, one would prefer avoiding the inconvenient computation of the inverse matrices In addition, because of the appearance of the term −1 on the right-hand side of the system, the computation of the stability radius may become an ill-posed problem Therefore, a direct investigation of the asymptotic stability and the asymptotic behavior of the stability radius r(E ; A; B; C) as tends to are of interest In [3], Dragan considered the problem with identity matrices E11 , E22 Based on the generalized Popov–Yakubovich theory and the asymptotic theory of singularly perturbed di erential equations, he showed that when the small parameter in the leading term tends to zero, the stability radius for the singularly perturbed system tends to the smallest value of the stability radius for the “reduced slow system” and that for the “fast boundary layer system” This means it may happen that r(E ; A; B; C) does not tend to r(E0 ; A; B; C) as tends to zero Another result closely related to that of [3] was obtained by Tuan and Hosoe [15] In their paper a new version of the Tikhonov Theorem was developed and applied to show the asymptotic behavior of the H∞ norm of the transfer function for the classical singular perturbation problem The paper is organized as follows In Section we summarize some preliminary results of theory for general implicit systems In Section 3, ÿrst, we analyze the algebraic structure of the coe cient matrix pair and give su cient conditions ensuring the asymptotic stability of the general system (1.1) for all su ciently small Then, by an approach di erent from those used in [3,15], we give a short and easy-to-follow proof to obtain an asymptotic formula of the stability radius for the implicit system (1.1) Our result is based on a formula of the stability radius for DAEs proposed by Du and Lien in a recent paper [5] In the particular case, for index-1 DAEs this formula of the stability radius was in fact obtained previously by Byers and Nichols [2] and independently, by Qiu and Davison [13] Finally, we discuss some related and more general problems in the robust stability of implicit systems with respect to (singular) perturbations in the leading coe cient matrix N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 Preliminaries Consider a general implicit system of linear di erential equations Ey (x) = Ay(x); (2.1) where E and A are given constant matrices in Cn×n The leading coe cient matrix E may be nonsingular or singular Deÿnition The matrix pencil {E; A} is said to be regular if there exists ∈ C such that the determinant of ( E − A), denoted by det( E − A), is di erent from zero Otherwise, if det( E − A) = ∀ ∈ C, we say that {E; A} is irregular If {E; A} is regular, then a complex number is called a (generalized ÿnite) eigenvalue of {E; A} if det( E −A)=0 The set of all eigenvalues is called the spectrum of the pencil {E; A} and denoted by {E; A} If the matrix E is singular and the pencil {E; A} is regular, then there exist nonsingular matrices W; T ∈ Cn×n such that E=W A=W Ir 0 N T −1 ; H 0 In−r T −1 ; (2.2) where Ir ; In−r are identity matrices of indicated size, H ∈ Cr×r , and N ∈ C(n−r)×(n−r) is a matrix of nilpotency index k; k ∈ N = {1; 2; : : :}, i.e., N k = 0, N i = for i = 1; 2; : : : ; k − Eq (2.2) is well known to be the canonical Weierstrass–Kronecker form of pencil {E; A}, see [1,8] If N is a zero matrix, then k = holds Deÿnition The nilpotency index of N in the Weierstrass–Kronecker form (2.2) is called the index of matrix pencil {E; A} and we write index{E; A} = k If E is nonsingular, we set index{E; A} = From (2.2), it is easy to verify that for a regular matrix pencil {E; A} deg{ → det( E − A)} = rank E if index{E; A} 1: (2.3) 35 Deÿnition We say that the trivial zero solution of (2.1) is asymptotically (and also exponentially) stable if, for an arbitrary vector y0 ∈ Cn , there are positive constants c; such that the solution of the initial value problem Ey (x) = Ay(x); x ∈ [0; ∞); P(y(0) − y0 ) = exists uniquely and the estimate y(x) c Py0 e− x holds for all x ¿ Here, P is an appropriately chosen projector in Cn×n If the zero solution of (2.1) is asymptotically stable, we then also say that the system (2.1) is asymptotically stable For instance, in case index{E; A} = one may choose P =I −Q where Q is the projector onto ker(E) along S={z ∈ Cn ; Az ∈ im E}, see [8] A di erence between ODEs and DAEs is that the equality y(x0 ) = y0 is not expected here We also remark that, for linear time-invariant systems, the concepts of asymptotic stability and exponential stability are equivalent One may easily verify the following: Proposition The system (2.1) is asymptotically stable, if and only if, the matrix pencil {E; A} is (asymptotically) stable, i.e., (E; A) ⊂ C− ; where C− denotes the open left-half complex plane We refer to [8, Section 1.2.5] for some more details on stability of DAEs Now, let us suppose that system (2.1) is asymptotically stable and consider the disturbed system Ey = (A + B C)y; where B ∈ Cn×p ; C ∈ Cq×n are given matrices and ∈ Cp×q is an uncertain disturbance The matrix B C is called a structured perturbation We deÿne V(E; A; B; C) = { ∈ Cp×q ; pencil {E; A + B C} is either unstable or irregular}: In [5] the structured complex stability radius of (2.1) is deÿned as r(E; A; B; C) = inf { ; ∈ V(E; A; B; C)}; 36 N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 where · is an arbitrary matrix norm induced by a vector norm The following result is analogous to that for explicit linear systems [9,10] Proposition (Du and Lien [5], see also Byers and Nichols [2] and Qiu and Davison [13]) Suppose that the matrix pencil {E; A} is regular and asymptotically stable Then the complex stability radius of (2.1) has the representation r(E; A; B; C) = sup C(sE − A)−1 B : s∈iR (2.4) Remark The matrix function G(s)=C(sE −A)−1 B is called the associated transfer matrix [10] For a nonsingular matrix E, it is easy to verify that |s|→∞ G(s) = lim |s|→∞ C(sE − A)−1 B = 0: In the case of a singular E, by using the canonical Kronecker form (2.2), we write G(s) = C(sE − A)−1 B  (sIr − H )−1  = CT   Asymptotic behavior of the stability radius for the singularly perturbed system k−1 (sN ) i Throughout this section, in addition to the general assumptions on the coe cient matrices of (1.1) and Assumption A1 formulated in Section 1, we will use Assumption A2 index{E22 ; A22 } = k and one of the following conditions Assumption A3 (A11 − A12 A−1 22 A21 ) is nonsingular Assumption A3# {E22 ; A22 } ⊂ C− and {E11 ; A11 − − A12 A−1 22 A21 } ⊂ C Remark Consider (1.1) and assume Assumption A1 Then the following holds true:  − In the next section we also need a factorization formula of a block matrix, see e.g [7, Section 2.5] −1 Here, iR denotes the imaginary axis of the complex plane lim degree of the generalized characteristic polynomial not change  −1 W B  (i) For k = 0; and ¿ we have index{E ; A} = k ⇔ index {E22 ; A22 } = k: i=0 (2.5) and easily deduce that G(s) tends to either a ÿnite number (for example, in case k = 1) or inÿnity when |s| → +∞ It is clear that the stability radius for a system of index less than or equal to is strictly positive This fact does not hold for a higher-index system with respect to an arbitrary perturbation structure For example, if k ¿ and B = C = I one may ÿnd that (ii) Assumption A3 implies nonsingularity of A This follows from det(A) = det(A22 ) det(A11 − A12 A−1 22 A21 ): These facts will be essential in the proof of Theorem Furthermore, we note that which implies r(E; A; B; C) = (iii) If index{E22 ; A22 } = 0, then index{E0 ; A} = ¿ index{E ; A} = (iv) If index{E22 ; A22 } = 1, then index{E0 ; A} = index{E ; A} = but rank E0 = n1 ¡ rank E = n1 + r for ¿ Remark There are di erent ways to show that for (semi-explicit) index-1 systems, the stability radius introduced here has also the structure preserving property, e.g., see [2,13] That is, under a perturbation with the norm less than the value of the stability radius, the index of the perturbed matrix pencil as well as the Therefore, the problem we deal with here is a singular perturbation problem in a generalized sense If is zero, the system (1.1) is called the reduced system Analogously to (1.5), we deÿne the structured complex stability radius of the reduced system and denote it by r(E0 ; A; B; C) lim |s|→∞ G(s) = +∞ N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 First, we investigate the asymptotic stability of the system (1.1) via a generalized eigenvalue problem Let us denote B (a) := {z ∈ C| |a − z| ¡ } for all a ∈ C; ¿ 0: Theorem Consider (1.1) and assume A1–A3 hold true Let r := rank E22 ; ¡ r n2 Denote the eigenvalues (counted with multiplicity) of the matrix pencils for ¿ 0, resp., {E22 ; A22 } = { ; : : : ; r }; {E11 ; A11 − A12 A−1 22 A21 } = { ; : : : ; {E ; A} = {t1 ( ); t2 ( ); : : : ; tr+n1 ( )}: Then for any ¿ there exists ¿ such that, for an appropriate order of the eigenvalues, ti ( ) ∈ B ( i ); Since the roots of a polynomial depend continuously on its coe cients (see [16, Section 2]) and since i ; i = 1; 2; : : : ; r, denote the nonzero roots of polynomial q0 (:), there exists ¿ such that ti ( ) ∈ B ( i ); i = 1; 2; : : : ; r; holds for all ∈ (0; ] To verify the asymptotic behavior of the other n1 eigenvalues of {E ; A}, note that t = is not a root of det(tE − A), hence we can consider the equation det n1 }; 37 E11 − t −1 A11 −t −1 A12 −t −1 A21 E22 − t −1 A22 = 0: Set qˆ (s) := det(E − sA); ¿ 0; s ∈ C: (3.2) Then ∀ ∈ (0; ]; i = 1; 2; : : : ; r and p (t) = if and only if qˆ (t −1 ) = ∀ ¿ 0; t = 0: tr+j ( ) ∈ B ( j ); ∀ ∈ (0; ]; j = 1; 2; : : : ; n1 : Matrix A is nonsingular (see Remark 3(ii)), therefore, Proof By Remark 3(i), (ii) and (2.3), the statement on the numbers of eigenvalues of the three matrix pencils holds Furthermore, all the eigenvalues are nonzero We set p (t) := det tE11 − A11 −A12 −A21 tE22 − A22 sE11 − A11 − A12 −A21 sE22 − A22 × det(s(A11 − A12 A−1 22 A21 ) − E11 ): ; sj ( ) = (3.1) Note that ∀ ¿ 0: When tends to 0, the coe cients of q (:) tend one by one to those of polynomial s → q0 (s) = det(sE11 ) det(sE22 − A22 ): Furthermore, deg q (·) = deg q0 (·) = n1 + r ∀ ¿ 0: s → qˆ0 (s) = (−1)n1 +n2 det(sA22 ) Let ˆ be an arbitrary small positive number Referring once again to the continuity of roots of a polynomial, there exists a su ciently small ˆ2 such that ¿ 0; s ∈ C: p (t) = if and only if q ( t) = ∀ ¿ 0: As tends to zero, the coe cients of qˆ (:) tend one by one to those of the polynomial ; ¿ 0; t ∈ C; q (s) := det deg qˆ (:) = rank A = n1 + n2 ; and sj ( ) ∈ B ˆ ( −1 j ); j = 1; 2; : : : ; n1 ; ∀ ∈ (0; ˆ2 ]: Here sj ( ); j =1; 2; : : : ; n1 ; are n1 appropriately chosen roots of qˆ (·) By invoking continuity of the inverse mapping, we may choose ¿ such that tr+j ( ) = sj−1 ( ) ∈ B ( j ); j = 1; 2; : : : ; n1 ; ∀ ∈ (0; ]; where tr+j ( ); j = 1; 2; : : : ; n1 are the other n1 roots (ordered appropriately) of p (·) Taking := max{ ; }, the proof of Theorem is complete 38 N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 Using Assumption A3# instead of A3, we obtain Corollary Let Assumptions A1, A2 and A3# hold true Then there exists a number ¿ such that the system (1.1) is asymptotically stable for all ∈ [0; ] Proof The case of = is trivial since it is easy to check that (E0 ; A) = (E11 ; A11 − A12 A−1 22 A21 ): For = min{−R( 1); : : : ; −R( r); −R( 1); : : : ; −R( n1)} ¿ 0; it follows that B ( i ) ⊂ C− ∀i = 1; 2; : : : ; r; and B ( j ) ⊂ C− ∀j = 1; 2; : : : ; n1 : Applying Theorem 1, there exists a number ¿ such that for all ∈ [0; ] all eigenvalues of the pencil {E ; A} are inside the open left-half complex plane C− Invoking Proposition 1, the proof is complete Remark The classical singular perturbation problem (the case when the leading terms Eii are identity matrices) is a special case of (1.1) Its asymptotic stability has been investigated in a number of papers and books, e.g., [4, Proposition 3.1.1] and the references cited in [3] Here, we have succeeded in giving a direct proof to a more general problem Furthermore, due to Remark 1, the spectrum of a matrix pencil of index less than or equal to is stable with respect to small perturbations occurring in the second term This fact gives us another explanation to the asymptotic behavior of the spectrum of the matrix pencils in (3.1) and (3.2) Remark It is easy to see that the index Assumption A2 (and the nonsingularity condition of E11 ) may be relaxed in some special cases, for example, when either A21 or A12 = The stability Assumption A3# is su cient and “nearly” necessary, since the condition − {E11 ; A11 − A12 A−1 22 A21 } ⊂ C is the necessary and su cient condition for the asymptotic stability of the reduced system ( =0) Furthermore, if one of the matrix pairs in Assumption A3# has an eigenvalue in the open right half C+ of the complex plane, the result of Theorem ensures that system (1.1) should become unstable for all su ciently small If all eigenvalues of {E22 ; A22 } have nonpositive real parts, but at least one eigenvalue is located on iR, both the stability and unstability may occur for the singularly perturbed systems Now, using Proposition 2, we obtain formulae of the stability radius for the singularly perturbed system (1.1) and the reduced one First, for brevity, let us introduce some auxiliary notations Consider (1.3) and deÿne, for all ¿ and t ∈ iR, A (t) = A11 + A12 (t E22 − A22 )−1 A21 ; B (t) = B1 + A12 (t E22 − A22 )−1 B2 ; C (t) = C1 + C2 (t E22 − A22 )−1 A21 ; D (t) = C2 (t E22 − A22 )−1 B2 ; G (t) = D (t) + C (t)(tE11 − A (t))−1 B (t): (3.3) In addition, we use the following notations for the case of = (the “reduced slow system”): A = A11 − A12 A−1 22 A21 ; B = B1 − A12 A−1 22 B2 ; C = C1 − C2 A−1 22 A21 ; D = −C2 A−1 22 B2 ; GS (t) = D + C(tE11 − A)−1 B; ∀t ∈ iR: (3.4) Finally, we deÿne GF (t) = C2 (tE22 − A22 )−1 B2 ; ∀t ∈ iR: (3.5) Lemma Let Assumptions A1, A2, and A3# hold true Then, for all ∈ (0; ], where is the bound given in Corollary 1, the stability radius of the singularly perturbed system (1.1) can be given by −1 r(E ; A; B; C) = sup G (t) t∈iR : N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 Furthermore, the stability radius of the reduced system is formulated as follows: −1 r(E0 ; A; B; C) = sup GS (t) : t∈iR Proof Using (2.4) and the Frobenius formula for computing the inverse of a block matrix [7, Section 2.5], the formulae in Lemma follow from some elementary matrix calculations We note that, as a consequence of Corollary 1, for ∈ (0; ], all formulae in (3.3)–(3.5) are well deÿned on the line iR Consider the “fast boundary layer system” with respect to the perturbation structure {B2 ; C2 } Using (2.4) once again, the formula of the stability radius of the “fast boundary layer system” can be given by −1 sup GF (s) : s∈iR Note that the stability radius of the fast boundary layer system does not depend on ¿ since sup C2 ( tE22 − A22 )−1 B2 t∈iR =sup C2 (sE22 − A22 )−1 B2 ; ∀ ¿0 s∈iR lim r(E ; A; B; C) →+0 Proof Based on the explicit formulae for the stability radii given in Lemma 1, we give a direct proof The main idea is that for su ciently large |t|, the second part of the expression of G (t) is arbitrarily small Therefore, the function G (t) is close to GF (s) with s = t On the other hand, in a given bounded domain, the function G (t) is close to GS (t) for su ciently small because of continuity Now, let us choose an (arbitrarily small) ¿ We show that the inequalities t∈iR r( E22 ; A22 ; B2 ; C2 ) = r(E22 ; A22 ; B2 ; C2 ); ∀ ¿ 0: # Remark Suppose A1, A2, A3 hold and ¿ as in Corollary ensures that (1.1) is asymptotically stable for all ∈ [0; ] Since index{E22 ; A22 } 1, it follows from Remark and A3# that s → (sE22 − A22 )−1 is bounded on iR Scaling the imaginary axis by ¿ yields that sups∈iR (sE22 −A22 )−1 =supt∈iR ( tE22 − A22 )−1 Therefore, the functions B (·) ; Theorem Assume again the same conditions as in Lemma Then, as tends to zero, the stability radius of the singularly perturbed system converges to the minimum of the stability radius of the “reduced slow system” and that of the “fast boundary layer system”, i.e., max sup GS (t) ; sup GF (s) and it follows that A (·) ; However, maybe, their supremums over iR are not close to each other as tends to zero Now, we state the main result which is an analogue of Theorem 3.7 in [3] =min{r(E0 ; A; B; C); r(E22 ; A22 ; B2 ; C2 )}: E22 y2 = A22 y2 ; r( E22 ; A22 ; B2 ; C2 ) = 39 C (·) ; and D (·) are bounded on iR and the bounds are independent of ¿ The latter statement is especially important in the proof of Theorem below At each ÿxed point t ∈ iR; the function G (t) tends to GS (t) as tends to zero because of the continuity s∈iR −2 sup G (t) t∈iR max sup GS (t) ; sup GF (s) t∈iR + s∈iR (3.6) hold for all su ciently small From now on, we consider variables t and s on the axis iR, only (a) First, we prove the latter inequality in (3.6) Since E11 is nonsingular, by Remark 6, it is trivial to verify that the function C (t)(tE11 − A (t))−1 B (t) converges uniformly to zero w.r.t ∈ [0; ] as |t| → ∞ Thus, there exists a su ciently large number T = T ( ) (T is independent of ) such that C (t)(tE11 − A (t))−1 B (t) ; |t| ¿ T: Therefore, for t with |t| ¿ T , we have G (t) C2 (t E22 − A22 )−1 B2 + : 40 N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 Hence, we obtain Therefore, for min{ ; }, the inequality sup G (t) sup G (t) ¿ max sup GS (t) ; sup GF (s) |t|¿T t∈iR sup C2 (t E22 − A22 )−1 B2 + |t|¿T = sup GF (s) + sup GF (s) + : s∈iR |s|¿ T (3.7) On the other hand, on the compact domain {(t; ); |t| T; 6 }, G (t) is continuous, hence, uniformly continuous Therefore, there exists a su ciently small = ( ) such that for , we have sup G (t) sup GS (t) + sup GS (t) + : |t|6T t∈iR |t|6T Thus, for , we obtain sup G (t) max sup GS (t) ; sup GF (s) t∈iR t∈iR s∈iR + : (b) Now, we prove the ÿrst inequality in (3.6) Analogously to (3.7), we have sup G (t) ¿ sup GF (s) − : |t|¿T |s|¿ T Since GF (:) is continuous on iR, there exists a su ciently small = ( ) such that for , the inequality sup GF (s) ¿ sup GF (s) − s∈iR |s|¿ T holds Hence, we obtain sup G (t) ¿ sup GF (s) − : |t|¿T s∈iR On the other hand, since supt∈iR GS (t) is ÿnite, there exists a number t0 = t0 ( ) ∈ iR such that GS (t0 ) ¿ sup GS (t) − : t∈iR In addition, because of continuity of → G (t0 ) , there exists a su ciently small = ( ) such that for , we obtain sup G (t) ¿ G (t0 ) ¿ GS (t0 ) − t∈iR ¿ sup GS (t) − : t∈iR t∈iR s∈iR −2 holds Then, for min{ ; ; }, the inequalities in (3.6) hold The proof of Theorem is complete Remark A natural and still open question arises here when the stability radius r(E ; A; B; C) tends to r(E0 ; A; B; C) and when to r(E22 ; A22 ; B2 ; C2 ) For example, it is obvious that lim GS (t) = GF (0) sup GF (s) : |t|→∞ s∈iR Therefore, if supt∈iR GS (t) is attained at inÿnity, the stability radius r(E ; A; B; C) tends to r(E22 ; A22 ; B2 ; C2 ) as → +0 However, this condition is su cient, only and checking it is not an easy task Final remarks In the present paper a characterization of the robust stability for a class of singularly perturbed systems of di erential-algebraic equations (DAEs) has been given Our results are based upon a study of the algebraic structure of the coe cient matrix pair and the formula of the complex stability radius for an implicit system of di erential equations This approach, which is quite di erent from that of [3], has resulted in a short proof for the asymptotic behavior of the complex stability radius as the parameter tends to zero The result of the paper seems to be useful when one is to face the robust stability analysis of a singularly perturbed system for second-order DAEs equations which arises, for example, when modeling an electrical network by the implicit equation Ey (x) = Ay (x) + Dy(x); (4.1) where E; A; D are square complex matrices, E may be singular, and is a small positive parameter With a new variable z = y , we rewrite (4.1) as I y E z = I y D A z and obtain a system of the form (1.1) Then, it is easy to formulate (su cient) conditions providing the N.H Du, V.H Linh / Systems & Control Letters 54 (2005) 33 – 41 asymptotic stability of the system (4.1) for all sufÿciently small parameter values as well as to analyze the stability radius due to appropriately structured perturbations The approach used here may also be applied to the robust stability analysis of more general problems with a small parameter in the leading term The results in Section may be extended to the (singular) perturbation problem (E + F)y (x) = Ay(x); (4.2) where E; F; A are square complex matrices, is a small positive parameter The matrix F speciÿes the structure of perturbation appearing in the leading term, while the matrix E may be singular The matrix pencil {E; A} is supposed to be stable, i.e., (E; A) ⊂ C− By using the approach introduced here, it is possible to give conditions to ensure the asymptotic stability of the system for all su ciently small Furthermore, asymptotic behavior of the complex stability radius of (4.2) can be characterized as tends to zero, as well Acknowledgements The authors of the paper would like to thank a referee for numerous useful suggestions improving the results and their presentation, especially those concerning Theorem The authors are also grateful to Professor Katalin Balla for lots of useful hints and Professor Vasile Dragan for providing some interesting information concerning the subject of the paper References [1] K.E Brenan, S.L Campbell, L.R Petzold, Numerical Solution of Initial Value Problems in Di erential-Algebraic Equations, North-Holland, New York, 1989 41 [2] R Byers, N.K Nichols, On the stability radius of a generalized state-space system, Linear Algebra Appl 188,189 (1993) 113–134 [3] V Dragan, The asymptotic behavior of the stability radius for a singularly perturbed linear system, Int J Robust Nonlinear Control (1998) 817–829 [4] V Dragan, A Halanay, Stabilization of Linear Systems, Birkhauser, Boston, 1999 [5] N.H Du, D.T Lien, Stability radii for di erential-algebraic equations, submitted for publication [6] N.H Du, D.T Lien, V.H Linh, On complex stability radii for implicit discrete time systems, Vietnam J Math 31 (4) (2003) 475–488 [7] F.R Gantmacher, Theory of Matrices, 4th Edition, Nauka, Moscow, 1988 (in Russian), In English translation: The Theory of Matrices, Vol 1, AMS Chelsea Publishing, Providence, RI, 1998 [8] E Griepentrog, R Marz, Di erential Algebraic Equations and their Numerical Treatment, Teubner Texte zur Mathematik, Vol 88, Teubner, Leipzig, 1986 [9] D Hinrichsen, A.J Pritchard, Stability radii of linear systems, Systems Control Lett (1986) 4–10 [10] D Hinrichsen, A.J Pritchard, Stability radii for structured perturbations and the algebraic Riccati equations, Systems Control Lett (1986) 105–113 [11] D Hinrichsen, N.K Son, Stability radii of positive discrete time systems under a ne parameter perturbations, Int J Robust Nonlinear Control (1998) 1169–1188 [12] L Qiu, B Bernhardsson, A Rantzer, E.J Davison, P.M Young, J.C Doyle, A formula for computation of the real stability radius, Automatica 31 (1995) 879–890 [13] L Qiu, E.J Davison, The stability robustness of generalized eigenvalues, IEEE Trans Automat Control 37 (6) (1992) 886–891 [14] N.K Son, D Hinrichsen, Robust stability of positive continuous time systems, Numer Funct Anal Optim 17 (1996) 649–659 [15] H.D Tuan, S Hosoe, On a state-space approach in robust control for singularly perturbed systems, Int J Control 66 (3) (1997) 435–462 [16] J.H Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1967  ... parameter values as well as to analyze the stability radius due to appropriately structured perturbations The approach used here may also be applied to the robust stability analysis of more general problems... proof for the asymptotic behavior of the complex stability radius as the parameter tends to zero The result of the paper seems to be useful when one is to face the robust stability analysis of a. .. computation of the stability radius may become an ill-posed problem Therefore, a direct investigation of the asymptotic stability and the asymptotic behavior of the stability radius r(E ; A; B; C) as