DSpace at VNU: On the robust stability of implicit linear systems containing a small parameter in the leading term

IMA Journal of Mathematical Control and Information (2006) 23, 67–84 doi:10.1093/imamci/dni044 Advance Access publication on July 27, 2005 On the robust stability of implicit linear systems containing a small parameter in the leading term N GUYEN H UU D U AND V U H OANG L INH† Faculty of Mathematics, Mechanics, and Informatics, University of Natural Sciences, Vietnam National University, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam This paper deals with the robust stability of implicit linear systems containing a small parameter in the leading term Based on possible changes in the algebraic structure of the matrix pencils, a classification of such systems is given The main attention is paid to the cases when the appearance of the small parameter causes some structure change in the matrix pencil First, we give sufficient conditions providing the asymptotic stability of the parameterized system Then, we give a formula for the complex stability radius and characterize its asymptotic behaviour as the parameter tends to zero The structure-invariant cases are discussed, too A conclusion concerning the parameter dependence of the robust stability is obtained Keywords: robust stability; stability radius; singular perturbation; implicit systems; dependence on parameter Introduction Recently, the system robustness analysis occurring in a wide range of applications has attracted considerable attention of researchers from mathematical and engineering communities The concept of stability radii introduced by Hinrichsen & Pritchard (1986a,b) has been applied and analysed for different classes of systems in a large number of research papers, e.g see a fairly up-to-date list of references in Bracke (2000) A lot of problems arising in applied fields, such as modelling electrical circuits, multi-body mechanics, optimal control, etc., can be described by differential–algebraic equations (for short, we write DAEs; other frequently used names are implicit systems, singular systems, generalized state-space systems and descriptor systems) which may contain one or several small parameters as well, see Brenan et al (1989), Section 1.3 and Kurina (1993) In this paper, the robust stability and the sensitivity depending on data for implicit linear time-invariant systems containing a small parameter will be analysed E XAMPLE Consider the classical singular perturbation problem y1 (x) = A11 y1 (x) + A12 y2 (x), εy2 (x) = A21 y1 (x) + A22 y2 (x), where yi (x) ∈ Cni, Ai j ∈ Cni ×n j, i, j ∈ {1, 2}, and ε is a parameter (0 < ε 1) This is the simplest example for implicit systems containing a small parameter in the leading term and it has been discussed widely in the literature of control theory, e.g see Kokotovic et al (1986), Dragan & Halanay (1999) Note that the leading matrix is non-singular for ε > and singular for ε = A concrete numerical example modelling a voltage regulator controlled by a so-called corrected near-optimal state feedback † Email: vhlinh@hn.vnn.vn c The author 2005 Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications All rights reserved Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 [Received on 27 July 2004; revised on January 2005; accepted on 12 January 2005] 68 N H DU AND V H LINH law can be found in Kokotovic et al (1986) and Qiu & Davison (1992) For examples in optimal control problems described by implicit systems of more general form, see Kurina (1993) and references cited therein z = (0 0)y Here, G 0, g > 0, C > are assumed In general, there are five small parasitic parameters G , G , G , C1 , C2 We may be interested in the situation when either C1 or C2 is varying and very close to zero and all the others are fixed for simplicity Note that in this problem the leading matrix is always singular Furthermore, its rank is equal to if C2 = and equal to otherwise We consider the implicit parameterized system of differential equations (E + εF)y (x) = Ay(x), (1.1) where y(·): R → Cn is a vector function, E, F, A are constant matrices in Cn×n and ε is a small positive parameter The matrix E may be singular, but the pencil {E, A} is assumed to be regular (see Section of this paper) The matrix F describes the parameter-perturbation direction in the leading term In the main part of this paper, we suppose the following assumption A SSUMPTION A1 Pencil {E, A} is (asymptotically) stable and index{E, A} = The asymptotic stability of {E, A} means exactly that all finite generalized eigenvalues of the pencil lie in the open left half C− of the complex plane The almost trivial subcase when E is non-singular is treated separately in Section 5.1 It is well known that in the case of a singular E, the asymptotic stability of the system may be lost under an arbitrarily small perturbation affecting the leading term, see Qiu & Davison (1992), Byers & Nichols (1993) and Du et al (2003) The first question is how the direction F should be to ensure the asymptotic stability of (1.1) for all sufficiently small ε? In lots of applications, the coefficient matrix A is under the effect of an uncertain perturbation Let an appropriate matrix F be chosen, we consider the perturbed system (E + ε F)y (x) = (A + B∆C)y(x), (1.2) and determine the complex structured stability radius defined by r (E + εF, A; B, C) = inf{ ∆ , ∆ ∈ C p×q and (1.2) is not asymptotically stable} (1.3) Here, matrix ∆ is an uncertain perturbation, matrices B ∈ Cn× p , C ∈ Cq×n specify the structure of perturbation The norm used here is an arbitrary matrix norm induced by vector norms A formula of the complex stability radius for index-1 systems was proposed in Qiu & Davison (1992), Byers & Nichols (1993), and for general implicit systems in Du (1999), Du et al (2003) It is worth mentioning that the Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 E XAMPLE In general, engineering applications may contain several small parameters and the original equations may be DAEs as well (i.e the leading matrix is singular) Consider the circuit known as a loaded degree-one Hazony section under small loading, see Brenan et al (1989) This circuit has timeinvariant linear resistors, capacitors, a current source and a gyrator The resistances are large and the capacitances are small The problem is described by the linear time-invariant DAE ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ 0 g G1 C1 ⎝0 C2 0⎠ y + ⎝ g G + G −1⎠ y = ⎝0⎠ u, C −C2 G3 −G −1 ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS 69 Preliminary Consider a general implicit system of linear differential equations E y (x) = Ay(x), where E and A are given constant matrices in may be non-singular or singular Kn×n , (2.1) K = C or R The leading coefficient matrix E D EFINITION The matrix pencil {E, A} is said to be regular if there exists λ ∈ C such that the determinant of (λE − A), denoted by det(λE − A), is different from zero Otherwise, if det(λE − A) = 0, ∀ λ ∈ C, we say that {E, A} is irregular Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 stability radius with respect to the so-called admissible perturbations occurring in both the coefficient matrices was examined first by Byers & Nichols (1993) In Hinrichsen & Pritchard (1990), it is shown that the complex structured stability radius of an explicit system, i.e when E = I is set, depends continuously on the data triplet {A, B, C} The second question arising here is how the stability radius of an implicit system depends on the leading term Does r (E + εF, A; B, C) tend to r (E, A; B, C) as ε tends to zero? If it is not true, what is the asymptotic behaviour of r (E + εF, A; B, C) for small ε? Since ε is small, the computation of r (E + εF, A; B, C) may lead to an ill-posed problem, in general Therefore, an answer of the latter question seems to be of interest from both the theoretical and numerical point of view Dragan (1998) considered the classical singular perturbation problem, i.e a special case of (1.1) with E = diag(In , 0), F = diag(0, In ), where In , In are identity matrices of indicated sizes Based on some results in the control theory and the asymptotic theory for singularly perturbed differential equations, he has shown it may happen that the stability radius r (E + εF, A; B, C) does not tend to that of the reduced system as ε tends to zero Moreover, the exact limit was obtained A result closely related to Dragan (1998) was obtained in Tuan & Hosoe (1997), where the asymptotic behaviour of the H∞ norm of the transfer function for the classical singularly perturbed control problem was characterized We should also mention the reference lists in these papers, Kokotovic et al (1986), and Dragan & Halanay (1999) for more results on the classical singular perturbation problem Recently, by developing a new approach which is direct and completely different from that in Dragan (1998), the authors of this paper have extended Dragan’s result to a more general class of singularly perturbed systems, see Du & Linh (2005) Based on the idea presented there, here we aim to give a more complete analysis for the robust stability of the system of quite general form (1.1) Our approach is based rather on some basic results in linear algebra and classical analysis We will show that the uniform convergence of associated artificial transfer functions on the imaginary axis as the parameter tends to zero plays an important role in the question whether the operations of taking limit and supremum commute The paper is organized as follows In Section 2, we recall some basic notions and introduce a computable formula of the complex stability radius for implicit systems In Section 3, based on the structure properties of the matrix pencil related to (1.1), we classify the parameter perturbations and for the sake of a convenient treatment, the system will be transformed into a block form Some auxiliary statements are given, too Section focuses on the main case when the perturbation in the leading term causes structure changes in the matrix pencil Sufficient conditions are given for providing the asymptotic stability of (1.1) and the asymptotic behaviour of r (E + εF, A; B, C) is characterized as ε tends to zero Section discusses the problem with regular perturbation, i.e the case when the appearance of the small parameter ε does not cause changes in the structure of the matrix pencil The paper is closed with some conclusions 70 N H DU AND V H LINH If {E, A} is regular, then a complex number λ is called a (generalized finite) eigenvalue of {E, A} if det(λE − A) = The set of all eigenvalues is called the spectrum of the pencil {E, A} and denoted by σ {E, A} Suppose that the pencil {E, A} is regular If the matrix E is singular, then there exist non-singular matrices W, T ∈ Cn×n such that E=W Ir 0 T −1 , N A=W H 0 T −1 , In−r (2.2) D EFINITION Suppose that {E, A} is regular 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{λ → det(λE − A)} = rank E if and only if index {E, A} (2.3) D EFINITION Suppose that {E, A} is regular We say that the trivial zero solution of (2.1) is asymptotically (and also exponentially) stable if, for an arbitrary vector y0 ∈ Kn , there are positive constants c, α such that the solution of the initial-value problem E y (x) = Ay(x), P(y(0) − y0 ) = x ∈ [0, ∞), exists uniquely and the estimate y(x) c P y0 e−αx holds for all x Here, P is an appropriately n×n chosen projector in K 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} = 1, one may choose P = I − Q where Q is the projector onto ker (E) along S = {z ∈ Cn , Az im E}, see Griepentrog & Măarz (1986) A difference between ordinary differential equations and DAEs is that the equality y(0) = y0 is not expected here, in general 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 statement P ROPOSITION The system (2.1) is asymptotically stable if and only if the matrix pencil {E, A} is (asymptotically) stable, i.e σ (E, A) ⊂ C− , where C− denotes the open left half complex plane We refer to Brenan et al (1989) and Griepentrog & Măarz (1986) for more details on the theory of DAEs Now, let us suppose that system (2.1) is asymptotically stable and consider the disturbed system E y = (A + B∆C)y, Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 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 − Formula (2.2) is well known to be the canonical Weierstrass–Kronecker form of pencil {E, A}, see Brenan et al (1989) and Griepentrog & Măarz (1986) If N is a zero matrix, then k = holds 71 ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS where B ∈ Kn× p , C ∈ Kq×n are given matrices and ∆ ∈ K p×q is an uncertain disturbance The matrix B∆C is called a structured perturbation We define V K (E, A; B, C) = {∆ ∈ K p×q , {E, A + B∆C} is either unstable or irregular} In Du (1999) (see also Du et al., 2003), the structured stability radii of (2.1) are defined as rK (E, A; B, C) = inf{ ∆ , ∆ ∈ VK (E, A; B, C)}, P ROPOSITION (Du, 1999, see also Qiu & Davison, 1992; Byers & Nichols, 1993) Suppose that the matrix pencil {E, A} is regular and asymptotically stable Then the complex stability radius of (2.1) has the representation sup C(s E − A)−1 B r (E, A; B, C) = −1 s∈iR (2.4) Here, iR denotes the imaginary axis of the complex plane Note that the above formula of the stability radius holds not only for the Frobenius norm (see Byers & Nichols, 1993) and the Euclidean norm (see Qiu & Davison, 1992) but also for any matrix norm induced by vector norms R EMARK The matrix function G(s) = C(s E − A)−1 B is called the associated transfer matrix, see Hinrichsen & Pritchard (1986b) For a non-singular matrix E, it is easy to verify that lim |s|→∞ G(s) = lim |s|→∞ C(s E − A)−1 B = In the case of a singular E, by using the canonical Weierstrass–Kronecker form (2.2), we write G(s) = C(s E − A)−1 B = C T (s Ir − H )−1 0 − k−1 i i=0 (s N ) W −1 B (2.5) and easily deduce that G(s) tends to either a finite number (e.g in case k = 1) or infinity when |s| → +∞ Hence, 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 find that lim |s|→∞ G(s) = +∞, which implies r (E, A; B, C) = R EMARK It is not difficult to show that in the case of index-1 systems, the stability radius introduced here has also the structure-preserving property, see Qiu & Davison (1992) and Byers & Nichols (1993) That is, under any perturbation of the norm less than the value of the stability radius, not only the asymptotic stability is preserved, but also the index of the perturbed matrix pencil The latter property also implies the degree invariance for the generalized characteristic polynomial Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 where · is an arbitrary matrix norm induced by vector norms Depending on K = C or K = R, we talk about the complex or real stability radius, respectively In this paper, we focus on the complex one, only Thus, if we not mention explicitly, the stability radius will mean the structured complex one Furthermore, the subscript C in the notation of the stability radius will be omitted for brevity The following result is analogous to that for explicit linear systems, see Hinrichsen & Pritchard (1986b) 72 N H DU AND V H LINH M= M11 M21 M12 M22 be given with appropriate block sizes Suppose that M22 is non-singular Then, the decomposition M= In −1 M12 M22 In −1 M21 M11 − M12 M22 0 M22 In −1 M21 M22 In (2.6) holds A similar decomposition holds true in the case of a non-singular M11 , as well Classification and the simplified block form of the problem Now, we return to the parameterized system (1.1) In the theory of DAEs, relevant structure properties of matrix pencils play important roles In addition to the index assumption on the matrix pencil {E, A}, we require, and later give sufficient conditions, that the index of the parameterized matrix pencil be a fixed constant less than or equal to and the rank of the leading term be invariant for all sufficiently small positive ε We are interested in the following cases: C1 Index-change case: The index of {E + εF, A} changes from to when ε becomes 0, i.e there exists ε0 > such that index{E + εF, A} = for all ε, < ε ε0 , but index{E, A} = Of course, the index change implies the rank change of the leading term C2 Rank-change case: The index of {E +εF, A} is equal to for all sufficiently small ε and ε = 0; the rank of (E + εF) is constant for all sufficiently small ε but changes when ε becomes This means exactly that some generalized finite eigenvalues may be lost as ε reaches C3 Structure-invariance case: The index of the parameterized matrix pencil {E + εF, A} as well as the rank of the leading term not change for all sufficiently small ε, i.e there exists ε0 > such that index {E + ε F, A} = index{E, A} and rank(E + εF) = rankE for all ε, < ε ε0 This parameter perturbation ε F belongs to the class of admissible perturbations defined in Byers & Nichols (1993) If Case C3 occurs, we refer to (1.1) as a regular perturbation problem since the appearance of ε does not cause changes of the relevant structure properties of the matrix pencil, while Cases C1–C2 are called singularly perturbed problems In general, degenerate cases also occur, e.g when—for ε in an arbitrary small right neighbourhood of 0—matrix pencil {E + εF, A} has a varying index, has a constant higher index or is irregular These cases are excluded from our consideration For the sake of simplicity, from now on the problem (1.1) is set in a block form Without loss of generality, we suppose that the triplet {E, F, A} has the form E= E 11 0 , F= F11 F21 F12 , F22 A= A11 A21 A12 , A22 (3.1) Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 R EMARK The concept of the stability radius can be extended in a more general sense as follows Suppose that all the eigenvalues of the undisturbed matrix pencil lie in a prescribed open subset Cg of the complex plane We want to determine how large perturbations the system can tolerate without losing the property that its spectrum remains in Cg In the asymptotic stability analysis of differential equations, the open subset Cg is chosen to be C− It is trivial to obtain a formula of a Cg -stability radius analogously to that in Proposition In fact, we should simply replace iR by the boundary set of Cb = C\Cg As a consequence of the definition, positivity of a Cg -stability radius with an appropriately chosen subset Cg implies the continuity of the pencil spectrum with respect to data In the next sections, we need a well-known decomposition formula of block matrices, see, e.g Gantmacher (1960), Section 2.5 Let an arbitrary matrix ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS 73 where E 11 is non-singular The condition index{E, A} = is equivalent to the non-singularity of A22 Furthermore, the matrices B, C are decomposed into block forms as B= B1 , B2 C2 ) C = (C1 (3.2) Here, all the submatrices are supposed to have appropriate sizes, e.g Ai j has the size n i ×n j , i, j = 1, 2, where n + n = n If the original matrix E is not of the sparse and block form as in (3.1), one may use, e.g a singular-value decomposition E = U ∗ Σ V, E new = Σ, Fnew = U F V ∗ , Anew = U AV ∗ (3.3) Accordingly, the perturbation structure is determined by Bnew = U B, Cnew = C V ∗ Finally, the new matrices should be decomposed appropriately to have the form (3.1) It is obvious that the new system and the original one are equivalent in the sense that they possess the same robust-stability properties (asymptotic stability, the complex structured stability radius) R EMARK As a matter of fact, it is sufficient to use any decomposition of the form E=P E 11 0 Q, where matrices E 11 , P, Q are non-singular, in order to transform the problem (1.1) into the form (3.1) Now, we present some auxiliary statements which will be useful in the next sections The first one can be checked easily Hence, it is stated without proof L EMMA Suppose that P, Q, R are arbitrary matrices in Cn×n , ε is a real parameter and s is a complex variable If P is a non-singular matrix, then there exist a sufficiently small ε0 > such that (a) Matrix (P + εQ) is invertible for all ε, ε ε0 Furthermore, the norm of the inverse matrix (P + εQ)−1 is bounded on [0, ε0 ] (b) The norm of [s(P + εQ) − R]−1 converges uniformly to w.r.t ε on [0, ε0 ] when |s| tends to +∞ The latter statement remains valid in the case when R is a bounded matrix function of variables ε, s The next statement deals with the possible order change of operations of taking limit and supremum for a two-variable continuous function in a non-compact domain L EMMA Suppose that f (x, t) is a continuous real function in domain D = [a, b] × [0, ∞), where a, b are finite numbers Furthermore, f (x, t) converges to a continuous function g(x) uniformly with respect to x ∈ [a, b] as t tends to infinity Then, (a) f (x, t) is uniformly continuous on D (b) limx→x0 supt∈[0,∞) f (x, t) = supt∈[0,∞) f (x0 , t), for all x0 ∈ [a, b] Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 n where U, V are unitary matrices, Σ = diag(σ1 , σ2 , , σn ), {σi }i=1 are the singular values of E in a non-increasing order Then, one obtains an equivalent problem with the new data set 74 N H DU AND V H LINH Proof Let be an arbitrarily small positive number (a) Due to the uniform convergence, there exists a number T such that | f (x, t) − g(x)| /3, ∀ x ∈ [a, b], t T Let us fix such a number T Since g(x) is continuous on [a, b], thus uniformly continuous, there exists a number δ1 > such that |g(x1 ) − g(x2 )| /3, ∀ |x1 − x2 | δ1 | f (x1 , t1 ) − f (x2 , t2 )| | f (x1 , t1 ) − g(x1 )| + | f (x2 , t2 ) − g(x2 )| + |g(x1 ) − g(x2 )| holds for all t1 , t2 T, |x1 − x2 | δ1 On the other hand, since f (x, t) is uniformly continuous on the compact subdomain D1 = [a, b] × [0, T ], there exists a number δ2 > such that | f (x1 , t1 ) − f (x2 , t2 )| , ∀ (x1 , t1 ), (x2 , t2 ) ∈ D1 , (x1 , t1 ) − (x2 , t2 ) δ2 Since f (x, t) is continuous and T is fixed, it is also clear that there exists a number δ3 > such that | f (x1 , t1 ) − f (x2 , t2 )| , ∀ x1 , x2 ∈ [a, b], t1 < T, t2 > T, (x1 , t1 ) − (x2 , t2 ) δ3 Set δ = min{δ1 , δ2 , δ3 } and obtain | f (x1 , t1 ) − f (x2 , t2 )| , ∀ (x1 , t1 ), (x2 , t2 ) ∈ D, (x1 , t1 ) − (x2 , t2 ) δ This proves the uniform continuity of f (x, t) on D (b) As a consequence of the first statement, there exist a number κ (independent of x0 , t) such that f (x0 , t) − f (x, t) f (x0 , t) + , ∀ x, t : |x − x0 | κ, t Taking the supremum with t varying on [0, ∞), one obtains sup f (x0 , t) − t∈[0,∞) f (x, t) sup t∈[0,∞) sup f (x0 , t) + , ∀ x : |x − x0 | κ t∈[0,∞) The proof of the second statement is complete P ROPOSITION Suppose that E ∈ Cn×n is a constant matrix, A(ε): [0, ε0 ] → Cn×n , B(ε): [0, ε0 ] → Cn× p , C(ε): [0, ε0 ] → Cq×n are continuous matrix functions, the pencil {E, A(0)} is of index-1 and stable Then, the relation lim sup G(ε, t) = sup G(0, t) , (3.4) ε→+0 t∈iR where G(ε, t) := C(ε)(tE − A(ε))−1 B(ε), t∈iR holds true Proof First, we note that there obviously exists a number ε1 ε0 such that the pencil {E, A(ε)} is of index-1 and stable for all ε ε1 , too Therefore, the function G(ε, t) is well defined (and continuous) when ε ∈ [0, ε1 ], t ∈ iR Without loss of generality, we assume that the matrices E, A(ε), B(ε), C(ε) Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 Hence, the inequality ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS 75 are in block form (3.1)–(3.2) The index of {E, A(ε)} implies that A22 (ε) is invertible for all ε ∈ [0, ε1 ] Using formula (2.6) for computing the inverse of block matrices, after some matrix calculations, we arrive at G(ε, t) = D(ε) + C(ε)(tE11 − A(ε))−1 B(ε), with A(ε) = A11 (ε) − A12 (ε)A22 (ε)−1 A21 (ε), C(ε) = C1 (ε) − C2 (ε)A22 (ε)−1 A21 (ε), B(ε) = B1 (ε) − A12 (ε)A22 (ε)−1 B2 (ε), (3.5) D(ε) = −C2 (ε)A22 (ε)−1 B2 (ε) C OROLLARY Under the same assumptions as in Proposition 3, we have lim r (E, A(ε); B(ε), C(ε)) = r (E, A(0); B(0), C(0)) ε→+0 We conclude that the complex stability radius of an index-1 system depends continuously on the second term as well as the perturbation structure The singular perturbation problem 4.1 The index-change case In this section and the next one, we suppose that the problem is set of form (3.1), (3.2) First, we characterize the class of direction matrices F inducing the index change (Case C1) P ROPOSITION The parameterized leading term (E + εF) is non-singular for all sufficiently small ε if and only if the matrix pencil {E, F} is regular From now on, we are interested in an extension of the robust-stability result on the classical singular perturbation problem (Dragan, 1998) Techniques used in Du & Linh (2005) can be applied with a slight modification In this subsection, in addition to the assumptions in Section 1, we suppose the following conditions A SSUMPTION A2 Matrix F22 is non-singular A SSUMPTION A3 Matrix pencils {F22 , A22 } and {E 11 , A11 − A12 A−1 22 A21 } are (asymptotically) stable, i.e their spectra belong to the open left half plane C− It is trivial to check the equality σ {E, A} = σ {E 11 , A11 − A12 A−1 22 A21 }, i.e the second stability condition in Assumption A3 has already been provided in fact by the stability of pencil {E, A}, see Assumption A1 First, we establish a statement on the asymptotic stability of (1.1) P ROPOSITION Assume that Assumptions A1–A3 hold true Then the system (1.1), (3.1) is asymptotically stable for all sufficiently small ε, i.e there exists a positive number ε such that σ {E + εF, A} ⊂ C− for each parameter ε ∈ [0, ε] Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 The matrix functions A(ε), B(ε), C(ε), D(ε) are continuous and since E11 is non-singular, the expression C(ε)(tE11 − A(ε))−1 B(ε) converges to zero uniformly with respect to ε ∈ [0, ε1 ] as |t| tends to infinity Thus, the function G(ε, t) fulfils the conditions of Lemma Therefore, the equality (3.4) holds We obtain an extension of Hinrichsen & Pritchard (1990), Proposition 2.2, to index-1 DAEs 76 N H DU AND V H LINH Proof The proposition means exactly that the parameterized system is asymptotically stable for sufficiently small ε if the reduced slow system with pencil {E, A} and the so-called boundary-layer fast system with pencil {εF22 , A22 } are simultaneously asymptotically stable It is clear that the pencil {E + εF, A} has index-0 for sufficiently small ε and consequently, it has exactly n finite (non-zero) eigenvalues Denote the eigenvalues (counted with multiplicity) of the following matrix pencils for ε > 0, respectively, σ {E 11 , A11 − A12 A−1 22 A21 } = µ1 , , µn , σ {F22 , A22 } = ν1 , , νn , We will show that, by ordering the eigenvalues of {E + ε F, A} appropriately, we have ελ j (ε) = ν j + o(1), λn +i (ε) = µi + o(1), j = 1, 2, , n , i = 1, 2, , n , ε → +0, i.e the first n eigenvalues of {E + εF, A} are asymptotically equal to those of the reduced pencil {E, A}, while the other n eigenvalues have asymptotic behaviour like those of pencil {ε F22 , A22 } as ε tends to zero On one hand, we observe that λ is an eigenvalue of {E + εF, A} if and only if ελ is an eigenvalue of the pencil ε A11 ε A12 E 11 + εF11 εF12 , (4.1) F21 F22 A21 A22 The matrix pencil (4.1) contains small parameter perturbations in both terms Since the reduced leading term (when ε = is set) is non-singular, so is the parameterized leading term for sufficiently small ε, see Lemma 1, part (a) Then, the eigenvalue problem of the matrix pair (4.1) may be reduced to the classical eigenvalue problem of a perturbed matrix Invoking the continuity of the spectrum of matrices, there are n eigenvalues of (4.1) (ordered appropriately) such that ελ j (ε) = ν j + o(1), j = 1, 2, , n , ε → +0 Consequently, there exists a number ε1 such that the pencil {E + ε F, A} has n eigenvalues in C− for ε ε Furthermore, those tend to infinity as ε tends to zero On the other hand, if λ is a non-zero eigenvalue of pencil {E + ε F, A}, then the value λ−1 is a non-zero eigenvalue of pencil {A, E + εF} and vice versa Since A is non-singular (otherwise would be an eigenvalue of {E, A} which contradicts the stability assumption), by a similar argument as above, we state that the spectrum of {A, E + εF} tends to that of {A, E} as ε tends to zero Taking into account that µi−1 , i = 1, 2, , n , are non-zero eigenvalues of {A, E}, there are (other) n eigenvalues of {E + ε F, A} (ordered appropriately) such that λn +i (ε)−1 = µi−1 + o(1), i = 1, 2, , n , ε → +0 Hence, there exists a number ε such that the pencil {E + ε F, A} has n bounded eigenvalues in C− for ε ε2 By setting ε = min{ε1 , ε }, the above arguments imply that the parameterized pencil {E + ε F, A} should have exactly n = n + n eigenvalues in C− for each ε ∈ (0, ε] The proof of Proposition is completed Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 σ {E + ε F, A} = λ1 (ε), λ2 (ε), , λn +n (ε) 77 ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS r (E, A; B, C) = sup G s (t) −1 t∈iR , A = A11 − A12 A−1 22 A21 , C = C1 − C2 A−1 22 A21 , G S (t) = D + C(t E 11 − A)−1 B, B = B1 − A12 A−1 22 B2 , (4.2) D = −C2 A−1 22 B2 ; and r (F22 , A22 ; B2 , C2 ) = sup G F (s) s∈iR −1 , where G F (s) = C2 (s F22 − A22 )−1 B2 (4.3) We note that the stability radius corresponding to the quadruplet {F22 , A22 ; B2 , C2 } and that for {ε F22 , A22 ; B2 , C2 } are the same for all positive ε Similarly to the results in Dragan (1998) and Du & Linh (2005), we state the following theorem T HEOREM Let Assumptions A1–A3 hold Then the stability radius of the singular perturbation problem (1.1), (3.1) is asymptotically equal to the minimum of the stability radii of the reduced slow system and of the fast boundary-layer system for small ε, i.e lim r (E + εF, A; B, C) = min{r (E, A; B, C), r (F22 , A22 ; B2 , C2 )} ε→+0 Proof Let us fix a number ε provided by Proposition For ε ∈ (0, ε], a formula similar to (4.2) can be obtained for r (E + εF, A; B, C) as well However, it is too complicated and inappropriate for further examination It is more advantageous, first, to transform the system (1.1), (3.1) into an equivalent one with a block-diagonal leading term Applying the decomposition formula (2.6) to (E + εF), we have E + εF = In −1 F12 F22 In −1 F21 ] E 11 + ε[F11 − F12 F22 ε F22 In −1 F22 F21 In Let us denote the upper and lower triangular matrices in the above formula by P and Q, respectively, and introduce the new equivalent system with the modified coefficient matrices Eε = A = P −1 AQ −1 = −1 F21 ] E 11 + ε[F11 − F12 F22 In −1 −F12 F22 In , ε F22 (4.4) A11 A21 A12 A22 In −1 F21 −F22 In Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 R EMARK A similar idea, combined with the continuity of roots of polynomials, was used in the proof of Theorem in Du & Linh (2005) The argument used here is closely related to the positivity of the stability radius defined and analysed in Byers & Nichols (1993), see Remark of this paper Another proof of Proposition is also possible, if one transforms the matrix E + ε F into the diagonal form and then applies Proposition 3.1.1 in Dragan & Halanay (1999) to the transformed eigenvalue problem Let us denote the stability radius of the reduced slow system (corresponding to the quadruplet {E, A, B, C}) and that of the boundary-layer system (corresponding to the quadruplet {F22 , A22 ; B2 , C2 }) by r (E, A; B, C) and r (F22 , A22 ; B2 , C2 ), respectively Using Proposition and formula (2.6) for computing the inverses of block matrices, we obtain explicit expressions for r (E, A; B, C), r (F22 , A22 ; B2 , C2 ) as follows 78 N H DU AND V H LINH The perturbation structure is described by the new matrix pair B = P −1 B = −1 B1 − F12 F22 B2 B2 −1 C = C Q −1 = (C1 − C2 F22 F21 , C2 ) We will refer to the blocks of A, B, C marked with bar Accordingly, we have −1 F21 ) E ε,11 = E 11 + ε(F11 − F12 F22 −1 −1 A = A11 − A12 A22 A21 , B = B − A12 A22 B , −1 −1 C = C − C A22 A21 , D = −C A22 B , where A, B, C, D were defined in (4.2) Applying Proposition to the new system and taking into account that the stability radius corresponding to the quadruplet {(E + εF), A, B, C} and that corresponding to {E ε , A, B, C} are the same For ε ∈ [0, ε], the formula r (E + εF, A; B, C) = r (E ε , A, B, C) = sup G ε (t) −1 t∈iR , where G ε (t) = Dε (t) + Cε (t)(t E ε,11 − Aε (t))−1 Bε (t), Aε (t) = A11 + A12 (tεF22 − A22 )−1 A 21 , Cε (t) = C + C (tεF22 − A22 )−1 A21 , Bε (t) = B + A12 (tεF22 − A22 (4.5) )−1 B 2, Dε (t) = C (tεF22 − A22 )−1 B is valid As ε tends to zero, the function G ε (t) converges point-wise to G S (t) for t ∈ iR However, taking limit and supremum not commute, i.e it may happen that limε→+0 r (E + εF, A; B, C) = r (E, A; B, C) From now on, the proof advances quite analogously to that of Theorem in Du & Linh (2005) The key idea is that, on one hand, for sufficiently large |t|, the second expression in the formula of G ε (t) is arbitrarily small; therefore, the function G ε (t) is close to G F (s) with s = εt On the other hand, in a given bounded domain, the function G ε (t) can be approximated by G S (t) for small ε Indeed, since F22 is non-singular and A22 = A22 holds, we have sup (tεF22 − A22 )−1 = sup (s F22 − A22 )−1 < +∞ t∈iR s∈iR Hence, the functions Aε (t) , Bε (t) , Cε (t) and Dε (t) are bounded on iR and the bounds are independent of positive ε Due to Lemma 1, part (b), the function Cε (t)(t E ε,11 − Aε (t))−1 Bε (t) converges to zero uniformly w.r.t ε ∈ [0, ε] as |t| → ∞ Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 Now, we arrive at a robust-stability problem with data {E ε , A, B, C} which has the form very close to that analysed in Du & Linh (2005) By elementary calculations, it is easy to verify that 79 ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS Let us choose an arbitrary δ > We show that the inequalities max sup G S (t) , sup G F (s) t∈iR − 2δ s∈iR sup G ε (t) t∈iR max sup G S (t) , sup G F (s) t∈iR (4.6) +δ s∈iR hold for sufficiently small ε We recall that the variables t and s are considered on the line iR, only Cε (t)(t E ε,11 − Aε (t))−1 Bε (t) Therefore, for t with |t| δ, |t| T T , we have G ε (t) C (tεF22 − A22 )−1 B + δ Hence, we obtain sup C (tεF22 − A22 )−1 B + δ sup G ε (t) |t| T |t| T G F (s) + δ = sup |s| εT sup G F (s) + δ s∈iR (4.7) On the other hand, on the compact domain {(t, ε), |t| T, ε ε}, G ε (t) is continuous as a two-variable function, hence uniformly continuous, too Therefore, there exists a sufficiently small ε1 = ε1 (δ) such that for ε ε1 , we have sup G ε (t) |t| T Thus, for ε sup G S (t) + δ |t| T sup G S (t) + δ t∈iR ε1 , we obtain sup G ε (t) t∈iR max sup G S (t) , sup G F (s) t∈iR s∈iR + δ (b) Now, we prove the first inequality in (4.6) Analogously to (4.7), we have sup G ε (t) |t| T sup |s| εT G F (s) − δ Since G F (s) is continuous w.r.t s ∈ iR, there exists a sufficiently small ε2 = ε2 (δ) such that for ε ε2 , the inequality sup |s| εT G F (s) sup G F (s) − δ s∈iR holds Hence, we obtain sup G ε (t) |t| T sup G F (s) − 2δ s∈iR Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 (a) First, we prove the last inequality in (4.6) Due to the uniform convergence verified above, there exists a sufficiently large number T = T (δ) (T is independent of ε) such that 80 N H DU AND V H LINH On the other hand, since supt∈iR G S (t) is finite, there exists a number t0 = t0 (δ) ∈ iR such that G S (t0 ) sup G S (t) − δ t∈iR Moreover, because of the continuity of G ε (t0 ) as a function of ε, there exists a sufficiently small ε3 = ε3 (δ) such that for ε ε3 , we obtain sup G ε (t) t∈iR G S (t0 ) − δ sup G S (t) − 2δ t∈iR min{ε2 , ε3 }, the inequality sup G ε (t) t∈iR max sup G S (t) , sup G F (s) t∈iR s∈iR − 2δ holds Then, for ε min{ε1 , ε2 , ε3 }, the inequalities in (4.6) hold The proof of Theorem is complete R EMARK The results in this section let us conclude that under the conditions supposed here on F22 and for sufficiently small ε, the blocks F11 , F12 , F21 have almost no effect on the robust stability of the singular perturbation problem R EMARK Assumption A2 is sufficient to provide an index-change perturbation problem, only Index change may occur with a singular F22 as well A right question is that what we can in such a case? In the particular case F22 = 0, by transforming the leading term into the block-diagonal form as in (5.1) below, one can easily find sufficient conditions on the other blocks Fi j to ensure that the pencil {E + εF, A} be index-0 and remain stable for all sufficiently small ε However, if F22 is a non-zero and singular matrix, the situation becomes more complicated 4.2 The rank-change case A special problem of the rank-change case was discussed in Du & Linh (2005), where F11 , F12 , F21 are zero matrices The index of the parameterized system is invariant (and equal to 1) if index {F22 , A22 } = Then, under the stability condition A3, statements similar to Proposition and Theorem in the previous subsection were obtained in Du & Linh (2005) The restriction on the direction matrix F may be somewhat relaxed In this subsection, instead of A2, we suppose the following assumption A SSUMPTION A2# Matrix F is of block-triangular form, i.e F21 (or F12 ) is a zero matrix and index {F22 , A22 } = As well as in the index-change case, the stability condition A3 is supposed to be true First, the problem is transformed into an equivalent one with a block-diagonal leading term Under the assumption that F21 = (the case of F12 = is treated similarly), we have E + εF = with Qε = In E 11 + ε F11 0 Qε , ε F22 ε(E 11 + εF11 )−1 F12 In (4.8) Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 Therefore, for ε G ε (t0 ) ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS 81 Then, after modifying the data set as in the previous subsection, we obtain an equivalent problem with the leading term E 11 + εF11 Eε = ε F22 Let Aε , B ε , C ε denote the transformed matrices, respectively We have −1 Aε = AQ ε , B ε = B, −1 C ε = C Qε P ROPOSITION Assume that A1, A2# and A3 hold true For all sufficiently small ε, the pencil {E + εF, A} remains of index-1 and stable, i.e there exists ε such that index {E + εF, A} = and σ {E + εF, A} ⊂ C− hold for all ε ∈ [0, ε] Proof It is clear that the transformed pencil has the same index as the original one Since index {F22 , A22 } = 1, we have index {F22 , Aε,22 } = for all sufficiently small ε, see Remark Here, Aε,22 denotes the corresponding submatrix of Aε Then, one can easily verify that the transformed matrix pencil {E ε , Aε } is of index-1, too To verify the stability, one may repeat the arguments used in the proof of Proposition We note that, using (2.3), the number of generalized finite eigenvalues of pencil {E ε , Aε } is exactly rank E 11 + rank F22 for all sufficiently small ε Now we state an analogue of Theorem T HEOREM Let Assumptions A1, A2# and A3 hold As ε tends to zero, the stability radius of the singular perturbation problem (1.1), (3.1) converges to the minimum of the stability radii of the reduced slow system and of the fast boundary-layer system, i.e lim r (E + εF, A; B, C) = min{r (E, A; B, C), r (F22 , A22 ; B2 , C2 )} ε→+0 Proof The scheme of the proof of Theorem can be repeated without difficulties The only difference is that now the transformed matrices Aε , C ε contain O(ε) perturbations, too The uniform boundedness of the auxiliary functions Aε (t), Bε (t), Cε (t) and Dε (t) defined as in (4.5) remains valid Furthermore, we should also take into account, by invoking Proposition 3, the relation lim sup C ε,2 (s F22 − Aε,22 )−1 B2 = sup C2 (s F22 − A22 )−1 B2 ε→0 s∈iR s∈iR Similar to Remark 4, we underline again that (only) the block F22 plays a dominant role in the robust stability of the singular perturbation problem as the parameter tends to zero The regular perturbation problem 5.1 The case with a non-singular E Now, we are interested in the behaviour of the stability radius when the parameter perturbation does not change the algebraic structure of the original pencil First, we discuss the almost trivial subcase when E Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 Noting that Q ε = I + O(ε), ε → 0, hence Aε , C ε are simply obtained from the original ones by using O(ε) perturbations, only Furthermore, these two matrices are continuous functions in variable ε First, we establish the following statement 82 N H DU AND V H LINH is non-singular From Lemma 1, we know that the parameterized leading term remains non-singular for sufficiently small ε We obtain immediately the following proposition P ROPOSITION Assume that E is non-singular and {E, A} is stable Then the pencil {(E + εF), A} is of index-0 and stable for all sufficiently small ε, i.e there exist a positive number ε such that E + εF is non-singular and σ {E + εF, A} ⊂ C− for all ε ∈ [0, ε] T HEOREM Let the assumptions in Proposition hold Then the stability radius r (E + εF, A; B, C) tends to that of the reduced system as ε tend to zero, i.e lim r (E + εF, A; B, C) = r (E, A; B, C) ε→+0 5.2 The case with a singular E In this subsection, the case with a singular E is considered Since (E 11 + εF11 ) is non-singular for sufficiently small ε, using a decomposition of the form (2.6), we get E + εF = P ε E 11 + εF11 0 ε[F22 − ε F21 (E 11 + ε F11 )−1 F12 ] where Pε = In εF21 (E 11 + εF11 )−1 , In Qε = In Qε, (5.1) ε(E 11 + εF11 )−1 F12 In The rank-invariant assumption implies that for sufficiently small ε, the block ε[F22 − ε F21 (E 11 + εF11 )−1 F12 ] must be identically zero It holds true, for instance, when F21 , F22 are zero matrices (see also the characterization of admissible perturbations in Byers & Nichols, 1993) By expanding the expression [F22 − ε F21 (E 11 + εF11 )−1 F12 ] into a power series of ε, we obtain a necessary and sufficient condition for a regular perturbation P ROPOSITION Assume that the pencil {E, A} is of index-1 Then the pencil {E + εF, A} has the same index and the same number of finite eigenvalues as the reduced pencil {E, A} for all sufficiently small ε if and only if −1 −1 i F22 = 0, F21 E 11 (F11 E 11 ) F12 = 0, i = 0, 1, (5.2) Using the transformation technique as in Section 4, the original problem can be transformed into a new one with leading term E 11 + ε F11 Eε = 0 Furthermore, the new matrices Aε , B ε , C ε are obtained from the original ones by using O(ε) perturbations, only The following theorem characterizes the robust stability of this regular perturbation problem Downloaded from http://imamci.oxfordjournals.org/ at University of Birmingham on June 4, 2015 We note that this problem can be considered as a particular subcase of that discussed in the previous section, namely, where n = is taken The proof of Theorem can be repeated with n = 0, i.e the boundary-layer fast system and obviously the function G F (s) are no longer defined Alternatively, the parameterized implicit system can be transformed into an equivalent explicit one The new coefficient matrix Aε and the new perturbation-structure matrices B ε , C ε can be obtained from those of the reduced explicit system with O(ε) perturbations Then, one may refer to Proposition 2.2 in Hinrichsen & Pritchard (1990) as well and easily get the following theorem ON THE ROBUST STABILITY OF IMPLICIT LINEAR SYSTEMS 83 T HEOREM Let Assumption A1 hold again and the perturbation direction F satisfy (5.2) Then the parameterized pencil {E + εF, A} is stable for all sufficiently small ε Furthermore, we have lim r (E + εF, A; B, C) = r (E, A; B, C) ε→+0 Conclusion In this paper, the robust stability of implicit linear systems containing a small parameter in the leading term is analysed The problems are classified accordingly to structure changes of the pencil under the effect of parameter perturbations We have attempted to examine the asymptotic stability and the asymptotic behaviour of the stability radius as the small parameter tends to zero In the cases when some structure property changes, sufficient conditions on the parameter perturbation are given to ensure the asymptotic stability It is shown that the stability radius may be discontinuous as a function of the parameter Furthermore, as the parameter tends to zero, the stability radius of the parameterized system tends to the minimum of those of the reduced slow system and of the boundary-layer fast system In case of regular perturbation problems, i.e when the appearance of a sufficiently small parameter does not affect the pencil structure, it is proven that the asymptotic stability is preserved for all sufficiently small parameter In addition, the stability radius of the parameterized system is asymptotically equal to that of the reduced system, i.e the stability radius depends continuously on the parameter Comparing two kinds of perturbations we arrive at a conclusion that a small parameter perturbation affecting the ‘differential part’ of an index-1 system has almost no significance in the robustness analysis as the parameter tends to zero In contrary, a parameter perturbation affecting on the ‘algebraic part’ plays an important role in the robust stability Furthermore, the system affected by a small parameter perturbation has—in the best case—asymptotically the same stability radius as the reduced system The robustness analysis presented in this paper is devoted to linear time-invariant systems of index less than 1, only Extending the results to time-invariant systems of higher index or time-varying systems would be of interest A similar analysis for the (structured) real stability radius of implicit systems of the form (1.1) would be another interesting issue, too It is known that, in contrary to the complex one, the real structured stability radius has a more complicated formula (see Qiu et al., 1995) and may be a discontinuous function of data even in the case of explicit systems, see Hinrichsen & Pritchard (1990) At this moment, these problems are still open Acknowledgements This paper was written during the visit of V H Linh at the Computer and Automation Research Institute, the Hungarian Academy of Sciences, Budapest He is grateful to all persons arranging the visit The authors thank 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radius Automatica, 31, 879–890 Q IU , L & DAVISON , E J (1992) The stability robustness of generalized eigenvalues IEEE Trans Autom Control, 37, 886–891 T UAN , H D & H OSOE , S (1997) On a state-space approach in robust control for singularly perturbed systems Int J Control, 66, 435–462  ... i.e the stability radius depends continuously on the parameter Comparing two kinds of perturbations we arrive at a conclusion that a small parameter perturbation affecting the ‘differential part’... (E, A; B, C) ε→+0 Conclusion In this paper, the robust stability of implicit linear systems containing a small parameter in the leading term is analysed The problems are classified accordingly... based rather on some basic results in linear algebra and classical analysis We will show that the uniform convergence of associated artificial transfer functions on the imaginary axis as the parameter