- f? 6 - P f? Figure : P - loop literature, where stability of the system in Figure was studied for single-input single-output case, and is required t

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1 Stability Multipliers and Upper Bounds: Connections and Implications for Numerical Verication of Frequency Domain Conditions Y.S. Chou and A.L. Tits Department of Electrical Engineering and Institute for Systems Research University of Maryland at College Park College Park, MD 074 USA V. Balakrishnan School of Electrical Engineering Purdue University West Lafayette, IN USA June 9, 998 Abstract The main contribution of the paper is to show the equivalenc e between the following two approaches for obtaining sucient conditions for the robust stability of systems with structured uncertainties: (i) apply the classical absolute stability theory with multipliers; (ii) use modern theory, specically, the upper bound obtained by Fan, Tits and Doyle [IEEE TAC, Vol. 36, 5-38]. In particular, the relationship between the stability multipliers used in absolute stability theory and the scaling matrices used in the cited reference is explicitly characterized. The development hinges on the derivation of certain properties of a parameterized family of complex LMIs (linear matrix inequalities), a result of independent interest. The derivation also suggests a general computational framework for checking the feasibility of a broad class of frequency-dependent conditions, based on which bisection schemes can be devised to reliably compute several quantities of interest for robust control. Introduction A popular paradigm currently in use for robust control has a nominal nite-dimensional, linear, time-invariant system with the uncertainty in the feedback loop (see Figure ). Often additional information about the uncertainty is either known or assumed: diagonal or block-diagonal; sectorbounded memoryless, linear time-invariant or parametric, etc. In such cases, the uncertainty is called \structured". A fundamental question associated with this model is that of robust stability, i.e., \Is the model stable irrespective of the uncertainty, that is, with zero input, do all solutions of the system equations go to zero, irrespective of?" The origins of this question can be found in Russian This research was in part supported by NSF's Engineering Research Center No. NSFD-CDR , and in part by the Oce of Naval Research under contract No. N

2 - f? 6 - P f? Figure : P - loop literature, where stability of the system in Figure was studied for single-input single-output case, and is required to satisfy additional assumptions. This was known as the absolute stability problem []. This problem has received considerable attention over the years, and a number of sucient conditions for stability have been proposed; perhaps the most celebrated of these have been the circle and Popov criteria. These criteria have since been generalized to multi-input multioutput systems as the small-gain theorem (with loop transformations and multipliers) and the passivity theorem (with loop transformations and multipliers). An introduction to these methods can be found in the book by Desoer and Vidyasagar []. A second approach to the problem of robust stability of control systems with structured uncertainties is the modern (or structured singular value) approach, pioneered by Doyle [3, 4] and Safonov [5, 6]. This approach relies on deriving sucient conditions for the robust stability of the system in Figure through simple linear-algebraic techniques. Our main objective in this paper is to show explicitly and rigorously the connections between these two approaches in the case when is linear time-invariant and consists of both unmodeled dynamics and uncertain parameters. In this context, a sucient condition for robust stability of the loop of Figure, derived using the approach, is given in []; and a sucient condition based on the classical passivity theorem with multipliers can be found in [0,, 3, 4]. The former, which we will refer to as the standard mixed upper bound condition, can be recast as an LMI condition that must be satised at every boundary point of a \stability" region (the open unit disk or the open left half complex plane) in the complex plane. The latter passivity-multiplierbased robust stability condition in [0,, 3, 4] can be reinterpreted as the same LMI condition, with the additional restriction that the feasible solutions are themselves functions of a certain form. Therefore, the rst step in our treatment is to establish a general \interpolation" style result for a class of parameterized (by frequency) family of complex LMIs (many frequency-dependent conditions for stability and robustness [, 5, 6, 7, 8, 9] belong to this class). This result is of independent interest. Thus, we show that the standard mixed upper bound condition in [] is mathematically equivalent to the passivity-multiplier-based condition in [0,, 3, 4]. Concurrently, we explicitly characterize the relationship between the (D; G) scalings used in the standard mixed approach, and certain stability multipliers used in the passivity-multiplier-based condition in [0,, 3, 4]. Finally, our derivation also suggests a general computational framework for checking the feasibility Other authors have hinted at the connection [7], [8], [9], [0], but without proving equivalence.

3 of a broad class of frequency-dependent conditions. We rst carry out the analysis in the discrete-time context, then briey indicate how it extends to the continuous-time context. The organization of the paper is as follows. In Section, we present our study of a class of LMIs. Section 3 is concerned with the connections between the standard mixed upper bound condition and a passivity-multiplier-based robust stability condition. Implications of the results of Sections and 3 on computation are discussed in Section 4. In Section 5, we outline the continuous-time case results. Section 6 contains the conclusions. For clarity of presentation, all proofs have been relegated to the appendix. Preliminary versions of some of the results presented here appeared in [0] and []. The notation and terminology are standard. R e denotes the set of extended real numbers R [ fg, C? the open left-half complex plane fz C : Refzg < 0g, and D the open unit-disk fz C : jzj < the unit circle, and cld the closure of D. RH nn denotes the set of n n real rational proper transfer function matrices with no poles in the complement of D. Given a complex matrix/scalar M, we denote by M the complex conjugate of M, by M the conjugate transpose of M, by He(M) the Hermitian part (M +M ) of M, and by Sh(M) the skew Hermitian part (M? M ) of M. Framework: Parametrized Families of Complex LMIs A linear matrix inequality (LMI) is a matrix inequality of the form L (x ; ; x`; M 0 ; ; M`) := M 0 + `X i= x i M i + M 0 + `X i= x i M i! > 0; () where x C` is the variable, and M i C nn ; i = 0; : : : ; ` are given data. The inequality symbol indicates positive deniteness. LMIs () are widely encountered in system and control theory and their applications; see, for example []. In this section, we consider a parametrized family of LMIs, every element of which is of the form (). The variable z parametrizing this family takes on values on the unit circle. Using arguments from the theory of complex variables, we show that if every member of this parametrized family of LMIs is feasible, then a solution to the entire family exists in the form of a complex-valued function with certain strong properties. We will see later that such statements help establish the equivalence between two classes of numerical procedures for checking (approximately) the feasibility of the parametrized family of LMIs: The rst procedure relies on \sampling" resulting in a number of independent LMIs of the form (). The second procedure involves searches for a function (of the parametrizing variable z) of a certain form, whose values sampled serve as feasible points for the parametrized family of LMIs; this results in a single LMI of the form (). Let N be the set of nonnegative integers, and let n o FD := F C nn : F is continuous, with F (z) = F (z) ; 3

4 and H fir := ( NX i=0 a i z i + b i z?i ) : N N; a i ; b i R : The next proposition establishes properties of solutions of the complex parameterized LMI when F i FD, i = 0; : : : ; `. L (x ; ; x`; F 0 (z); ; F`(z))) > 0; () Proposition. Let F i FD, i = 0; : : : ; `. Let S be the set of functions s mapping cld to C, which are continuous on cld, analytic in D, and satisfy s(z) = s(z) for all z cld. The following conditions are equivalent: (a) for every there exist complex numbers x i, i = ; : : : ; `, such that () holds; (b) there exist s j i S, i = ; : : : ; `, j = ;, such that () holds for all with x i (z) = s i (z) + s i (z), for all i = ; : : : ; `; (c) there exist x i H fir, i = ; : : : ; `, such that () holds for all A similar result can be proved with D replaced by an essentially arbitrary complex domains, S dened accordingly, replaced by the extended e [ fg when is unbounded. The appropriate modication of statement (c) is obtained by replacing polynomials in z by linear combinations of functions from an arbitrary countable basis ' i for S and polynomials in z? by linear combinations of functions i such that i (z) = ' i (z) e. See [] for details. Remark: The equivalence of Statements (a) and (b) in Proposition. that LMI () is feasible for all if and only if there exist functions with certain smoothness properties whose values lie in the feasible set taken together with approximation theory, has important consequences. To wit, it implies that, when LMI () arises from the application of multiplier theory in robust stability analysis, the x i 's can be taken as a linear combination of largely arbitrary basis functions, provided the basis is rich enough. See [3] for various possible bases, Laguerre and Kautz bases in particular. As another example, Statement (c) uses an FIR basis. 3 Small Theorem and Classical Passivity Theorem We now apply the results of x to explore the connection between two popular sucient conditions for the robust stability of discrete-time systems with structured uncertainty. The rst is the standard mixed upper bound condition, given in []. This condition is derived using linear-algebraic methods, and is usually stated as an LMI condition that should be satised on the unit circle. The derivation of the second condition involves augmenting the system with multipliers (that are This is an generalization of Lemma.3 of [5]. A complete proof is provided for completeness and ease of reference. Parts of this proof, derived independently of the proof in [5], appeared in [0]. 4

5 introduced to take advantage of the structure and the nature of the uncertainty), and then applying the classical passivity theorem [0,, 3, 4]. Proposition. then helps establish the equivalence between these two conditions. A key tool in establishing the equivalence is a sucient condition for canonical factorization, rst stated and proved by A. Ran and his coauthors [4, 5], with the proof building on ideas from [6]. Related results can be found in [7] and [, xvi.9.4]. Proposition 3. [4, 5] Suppose that M is an m m transfer function matrix, not necessarily proper, with no poles on the unit circle, such that the frequency domain condition M(z) + (M(z) > 0 (3) holds. Then there exist M ; M RH mm, such that M? ; M? RH mm, and M = M M. Remark: In addition to the hypotheses of Proposition 3., suppose further that M(z) is Hermitian for all Then there exists M RH mm such that M? RH mm, and M = M M. This is the spectral factorization of M; it is well-known that condition (3) is both necessary and sucient for such a factorization (see, for example, [7]). Note however that condition (3) is only sucient for a canonical factorization of M. Before proceeding further, we need the following denitions and notation that are standard in analysis. Given a matrix M C nn and three nonnegative integers m r, m c, and m C with m := m r + m c + m C n, a block structure K of dimensions (m r ; m c ; m C ) is an m-tuple of positive integers, K := (k r ; : : : ; kr m r ; k c ; : : : ; kc m c ; k C ; : : : ; kc m C ), such that P m r i= kr i + P m c i= kc i + P m C i= kc i = n. Let n c := P m c i= kc i, and n C := P m C i= kc i (. Dene diag(d r ~D := ; : : : ; Dr m r ; D c; : : : ; Dc m c ; d CI k ; : : : ; d C C m C I k C mc ) : Di r = (Dr i ) C kr i kr i ; Di c = (Dc i ) C kc i kc i ; d C o i D := nd D ~ n : D > 0 G := diag(g r; : : : ; Gr m r ; 0 k c ; : : : ; 0 k m c c ; 0 k C ^(M) := inf 0;DD ;GG : MDM + GM? MG? D < 0. R o ; : : : ; 0 k C mc ) : G r i =?(Gr i ) C kr i kr i Next, let RF denote the set of real-rational functions and, given X = C or RF, dene o S r (X) := ndiag(s ; : : : ; S mr ) : S i X kr i kr i ; i = ; : : : ; m r o S c (X) := ndiag(s ; : : : ; S mc ) : S i X kc i kc i ; i = ; : : : ; m n c o S C (X) := diag(s I k C; : : : ; s mc I k C mc ) : s i X; i = ; : : : ; m C S c;c (X) := fdiag(s c ; S C ) : S c S c (X); S C S C (X)g S r;c;c (X) := fdiag(s r ; S c ; S C ) : S r S r (X); S c S c (X); S C S C (X)g S r;0;0 (X) := fdiag(s r ; 0 nc ; 0 nc ) : S r S r (X)g Finally, given a matrix/scalar function P continuous dene kp k^ := sup z@d ^(P (z)) (note that it is not a norm) P is dened by P (z) := (P (=z)) ~P := (I + P )(I? P )? ) 5

6 We now state Theorem 3., which essentially establishes that the standard mixed upper bound condition is mathematically equivalent to a passivity-multiplier-based stability condition: strict positive realness of W? ~P W? with \multipliers" W and W belonging to a certain subset of RH nn (see Fig. where ~ := (I + )(I? )? ). 3 In addition, the relationship between - h - W?? 6 - ~P - W? W ~ W h? Figure : Passivity framework (W ; W ) and the (D; G) scalings used in the mixed analysis is explicitly characterized. Theorem 3. 4 (a) Let P RH nn. The following conditions are equivalent: kp k^ < ; (b) (I? P )? is in RH nn and there exist a positive integer N, and n n, real matrices Q i S r;c;c (C), and U i S r;0;0 (C), i = ; : : : ; N, such that with the inequalities hold for all D(z) = G(z) = NX i=0 NX i=0 (Q i z i + Q T i z?i ); (4) (U i z i? U T i z?i ); (5) D(z) > 0; (6) P (z)d(z)(p (z)) + G(z)(P (z))? P (z)g(z)? D(z) < 0 (7) 3 Recall that a rational transfer matrix T is strictly positive real if it is in RH nn and He(T (z)) > 0 for all 4 Many researchers have been aware of this result for some time (see, e.g., [8]). (Our proof already appears in []; a less general version can be found in [0].) To our knowledge however, equivalence between (a) and any of the other conditions has never been rigorously established by other authors in the open literature, nor has been the continuous-time case counterpart. Implication (c))(b), which is of little importance to us and, in our proof, follows from other implications, is immediate from xed order multiplier results [9]. Implication (c))(d) is proved in [8] (which also includes a proof of equivalence between (b), (c), and (d) in the constant matrix case). The correspondence between (D; G) scalings and passivity multipliers (last statement in our theorem and an immediate consequence of the equivalence between (a), (b), (c), and (d)) has been alluded to by several authors, e.g., [9]. Finally, while the implication (d))(c) would follow, in the continuous-time case, from results in [], a key result there is incorrect, namely, the necessary part of Lemma 6: M(s) = diag(; s 4 + ) provides a simple counterexample. 6

7 (c) (I? P )? is in RH nn and there exist a positive integer N, and n n, real matrices R i; S i S r;c;c (C), i = ; : : : ; N, such that with W (z) = NX i=0 (R i z i + S i z?i ); (8) the lower right (n c + n C ) (n c + n C ) submatrix W cc (z) of W (z) is Hermitian for all and the inequalities hold for all He(W (z)) > 0; He( ~ P (z)w (z)) > 0; (d) (I? P )? is in RH nn and there exist transfer function matrices W = diag(w r ; W c ; W C ) and W = diag(w r ; W c ; W C ), where W r and W r, W c and W c, and W C and W C are in S r (RF), S c (RF), and S C (RF), respectively, satisfying (i) W, W, W?, and W? are in RH nn, W c W c = I, and W C W C = I, (ii) W W is strictly positive real, (iii) W? ~P W? is strictly positive real. Next, let D and G be as in (b) and let W = D + G; then W is as in (c). Also, if W is as in (c) then W = W? W for some W and W which are as in (d). Finally, let W and W be as in (d) and let D = He(W? W ) and G = Sh(W? W ); then D and G satisfy conditions (6) and (7) in (b). In addition to establishing the precise equivalence between modern mixed -theory based suf- cient condition for robust stability and the classical passivity-multiplier-based condition, Theorem 3. also states that the mutlipliers can be chosen to be of a particularly simple form, given in (4) and in (5). This fact has important ramications for the numerical verication of these robust stability conditions; we describe this briey next. 4 State-Space Verication of Frequency Domain Conditions Many frequency-domain conditions for stability and robustness [, 5, 6, 7, 8, 9] belong to the class of parameterized families of complex LMIs that were studied in Section. From the viewpoint of numerical optimization, these conditions amount to innite-dimensional convex feasibility problems, with the optimization variables being functions of frequency. There have been traditionally two approaches towards (approximately) reducing such an innite-dimensional feasibility problem to the feasibility of a nite number of LMIs. The rst is to verify the frequencydependent condition approximately by checking that it holds at a nite number of frequencies; the choice of the \frequency-grid" is typically based on engineering intuition. This frequency-gridding technique, though conceptually simple, in general only provides guaranteed method for verifying the infeasibility of the underlying innite-dimensional feasibility problem: If the feasibility test fails at 7

8 any of the points on the frequency-grid, then the original frequency-domain condition is infeasible; but no conclusions can be drawn in the case when feasibility holds over the frequency-grid (for an exception, see, e.g., [30]). The second approach towards numerically verifying the innite-dimensional convex feasibility problems that arise from frequency-domain conditions is a \state-space" approach, based on the Kalman-Yakubovich-Popov (KYP) Lemma (see for example [] and the references therein). The rst step here is to restrict the functions of frequency that are the optimization variables in the original frequency-domain condition to lie in a nite-dimensional subspace. Then, an application of the KYP Lemma yields a single LMI whose feasibility is sucient for that of the original frequencydomain condition. (This is the so-called \basis function method" [, 3, 4].) Here, it is not known if and how the choice of basis functions aects the outcome of the stability test. In contrast with the frequency-gridding technique, the basis function method provides guaranteed method for verifying the feasibility of the underlying innite-dimensional feasibility problem: If the frequency-domain condition is feasible with the optimization variables restricted to lie in a certain subspace, then the original frequency-domain condition is feasible; but in general no conclusions can be drawn in the case when the basis function method fails to establish feasibility. A fundamental question that then arises is whether there is a \gap" between the basis function method and the frequency-gridding method. The main implication of the results of Section is that there is no such \gap": When the grid is ne enough with the frequency-gridding method, and when the basis is rich enough with the basis function method, the two numerical methods yields the \same" answer. Specically, Proposition. and the comments that follow imply that the frequency-gridding method and the basis function method are equivalent in the limit, as the number of frequency points and the basis elements respectively in each method goes to innity. Consequently, algorithms can be devised that reliably compute quantities of interest for robust control; For example, consider the mixed--norm upper bound kp k^. Lower bounds on kp k^ can be computed using frequency-gridding (for example, as with the (real-) toolbox of MATLAB [4]); upper bounds can be computed using the basis-function method, using Theorem 3.. A simple bisection scheme can then be implemented, combining these bounds, to compute kp k^ to any arbitrary accuracy. 5 Synopsis of the Continuous-Time Case Continuous-time analogues of Proposition. and Theorem 3. are easily obtained. Among other possibilities, H fir can be replaced with H udr dened by ( NX ) a i H udr := (? z) + b i : N N; a i ( + z) i i ; b i R : i=0 (The subscript \udr" stands for unit-decay rate; the two-sided inverse Laplace transforms of the elements of H udr decay exponentially with a unit rate of decay.) Proposition. then holds with FD replaced with FC?, dened as n o FC? := F : jr e! C nn : F is continuous, with F (z) = F (z) ; 8

9 cld by the closure cl e C? = clc? [ fg of the extended left-half by the extended imaginary axis jr e, and H fir by H udr. And Theorem 3. holds with, in addition, the appropriate denition of kp k^, z i replaced by =(? z) i, z?i by =( + z) i and other obvious modications. These results are easily proved by making use of the bilinear transformation (z) = +z that maps?z cld to the one-point compactication cl e C? of the closed left-half complex plane. 6 Conclusions It has been shown that the standard mixed upper bound condition is mathematically equivalent to a passivity-multiplier-based stability condition in [0,, 3, 4]. Concurrently, the relationship between the scaling matrices widely used in mixed theory and certain stability multipliers used in absolute stability theory has been explicitly characterized. These connections provide a computational framework for checking the feasibility of a class of frequency-dependent conditions of interest in robust control (e.g., [, 5, 6, 7, 8, 9]). A Appendix A. Proof of Proposition. Before proving Proposition., we introduce three lemmas which will be used later. The rst lemma, Lemma A., extends a result of Packard and Doyle (Theorem 9.0 in [3]), which explores an intrinsic solution property for a class of parameterized family of complex LMIs. Lemma A. Let M i, i = 0; : : : ; `, be given complex matrices. Suppose that there exist complex numbers x i, i = ; : : : ; `, such that L(x ; ; x`; M 0 ; ; M`) > 0. Then the inequality also holds with M i and x i, i = ; : : : ; ` replaced with M i and x i, respectively. Furthermore, when the M i 's are real, the inequality holds with x i, i = ; : : : ; `, replaced with (x i + x i ), i = ; : : : ; `. Proof: Recall that, given a Hermitian matrix A, A > 0 if and only if y Ay > 0 for all y 6= 0. Taking complex conjugates on both sides of the resulting inequality proves the rst claim. The second claim is a direct consequence of the rst one. Lemma A. below is a key to prove Proposition.. With a direct application of the Dirichlet problem on cld we are able to establish the lemma, which implies the existence of a combination of functions of certain type that matches a given complex-valued function x satisfying certain properties. Lemma A. Let x C be a continuous function satisfying x(z) = x(z) for all Then there exist s i : cld! C, i=,, such that, for i=,, s i is continuous on cld, analytic in D, and satises s i (z) = s i (z) for all z cld, and x(z) = s (z) + s (z) for all Proof: 5 Let x C be given as assumed. Let Refxg and Imfxg denote its real part and imaginary part, respectively. Then both Refxg and Imfxg are real-valued, continuous functions 5 An alternative proof to that provided here can be found in [0] where explicit (Poisson) formulae for s and s are given. The advantages of the present proof are its conceptual simplicity and the fact that it readily extends to other domains of the complex plane. 9

10 dened Consider the Dirichlet problem on D with the real-valued, continuous boundary function Refxg. It is known that, given a simply connected set X and a real-valued continuous function g dened there exists a function u which is harmonic inside X and matches g (see, e.g., Corollary 4.8, pp.74, in [3]). Moreover, for any function u which is harmonic in a simply connected set X, there exists a harmonic conjugate v such that u + jv is analytic in X (see, e.g., Theorem.(j), pp.0, in [3]). Thus there exists a function analytic in D, whose real part matches Refxg Dene f 0 (z) = f(z) for all z cld. It is easy to verify that f 0 is analytic in D and its real part matches Refxg Let s = (f +f0 ). Then s is continuous on cld, analytic in D, with s (z) = s (z) for all z cld, and Refs g(z) = Refxg(z) for all Similarly, there exists a function s, continuous on cld, analytic in D, s (z) = s (z) for all z cld, and Imfs g(z) = Imfxg(z) for all Let s = (s + s ) and s = (s? s ). It follows that, for all x(z) = Refxg(z) + jimfxg(z) = (s (z) + s (z)) + (s (z)? s (z)) = (s (z) + s (z) + s (z)? s (z)) = s (z) + s (z): Lemma A.3 below, a slight extension of Mergelyan's Theorem (e.g., Theorem 0.5 in [33]), gives conditions for uniform approximation, on a bounded set, of a class of complex-valued functions by polynomials with real coecients. Lemma A.3 Suppose that s : cld! C is continuous on cld and analytic in D. Assume that for all z, s(z) = s(z). Then given > 0, there exists a polynomial p with real coecients, such that sup j s(z)? p(z) j< : zcld Proof: Given > 0, by Mergelyan's theorem, there exists a polynomial p 0 = p + jq where p and q are real-coecient polynomials, such that sup zcld j s(z)? p 0(z) j< =: (9) We show that p is as claimed. First, since for all z cld, s(z) = s(z), it follows from (9) that, for all z cld, j p 0 (z)? p 0 (z) j = j p 0 (z)? s(z) + s(z)? p 0 (z) j j p 0 (z)? s(z) j + j s(z)? p 0 (z) j < : On the other hand, for all z cld, j p 0 (z)? p 0 (z) j = j p(z)? jq(z)? (p(z) + jq(z)) j : Since, for any real-coecient polynomial m(z), m(z) = m(z) for all z, it follows that j p 0 (z)? p 0 (z) j = j q(z) j : 0

11 Thus, for all z cld, j q(z) j< =. Again using (9), it follows that, for all z cld, j s(z)? p(z) j = j s(z)? (p(z) + jq(z)) + jq(z) j j s(z)? p 0 (z) j + j q(z) j < : This proves the claim. We are now ready to prove Proposition.. Proof of Proposition.: We show that (a)) (b))(c)) (a). First the implication (a))(b). Assuming that (a) holds, we show that there exist continuous functions x i C, i = ; : : : ; `, such that, for all x i (z) = x i (z), i = ; : : : ; `, and () holds; assertion (b) will then follow from Lemma A.. The problem of constructing a continuous function based on data x i 's reduces to that of construction a continuous function on the closed interval [?; ] with same value assigned at the endpoint? and. The following argument is similar to that in [5]. For each (?; ], let x i, i = ; : : : ; `, satisfy and let x? i L(x ; ; x `; F 0 (e j ); ; F`(e j )) > 0 = x i, i = ; : : : ; `. By Lemma A., the x i 's can be chosen such that, for all [0; ], i = ; : : : ; `, x? i = x i and, in particular x0 i and x i are real. Since the F i 's are continuous, for every [0; ] there exists an open neighborhood I R of such that L(x ; ; x`; F 0(e j0 ); ; F`(e j0 )) > I : We now show that there exists a nite open cover fi ; ; I m g fi ; [0; ]g with I = I 0 and I m = I, where for i = ; : : : ; m, I i = ( i ; 0 i ), < 0 < < < m < 0 < < m? 0 m, and each I i is not a subset of the union of the other I j 's. Without loss of generality, assume that [0; ]n(i 0 [I ) is not empty. Let fi ; ; I k g fi ; [0; ]g be a nite open cover of the compact interval [0; ] n (I 0 [ I ), and replace each I i by its intersection with (0; ). For i = ; : : : ; `, let x j i, j = ; : : : ; k, be the associated x i 's. Let I 0 := I 0, I k+ := I and, for i = ; : : : ; `, x k+ i := x i. Then fi 0 ; ; I k+ g is a nite open cover of the compact interval [0; ]. Repeatedly take out any interval I i which is a subset of the union of the remaining I j 's (clearly I 0 and I will not be taken out in this process) and denote by m the number of remaining intervals. Finally, relabel the I i 's so that I i = ( i ; 0 i ) with i's, i = ; : : : ; m being in increasing order and, for i = ; : : : ; `, relabel the x j i 's accordingly. It is readily checked that the nite cover obtained possesses the desired properties. Now, for each i = ; : : : ; m, let V i := (x i ; ; xì ). Then V and V m are real. Dene, for [0; ], ( Vi I V () := i? [ j6=i I j ; i = ; : : : ; m: V i + (? )V i+ = i+ + (? ) 0 i ; (0; ); i = ; : : : ; m? : Then V : [0; ]! C` is continuous. Note that, since > 0 and 0 m? <, V (0) and V () are real. Extending V (:) by the formula V (?) = V () to the interval [?; 0] results in a continuous function dened on [?; ] with the same value V () assigned at the endpoints? and. This completes the proof of the implication (a))(b).

12 To show the implication (b))(c), it suces to show that given s S and any > 0, there always exists a polynomial p with real coecients which uniformly approximates the given function s over cld within. This result follows immediately from Lemma A.3. The implication (c))(a) is obvious. This completes the proof of Proposition.. A. Proof of Theorem 3. Proof of Theorem 3. (a))(b): Suppose that kp k^ <, i.e., suppose that, for every there exist D ~ D and G G such that D > 0; P (z)d(p (z)) + G(P (z))? P (z)g? D < 0: Since this condition is stronger than the mixed condition, it follows from the Small Theorem (e.g., [34]) that (I? P )? RH nn. By Proposition., there exist ^D(z) and ^G(z) whose entries are in H fir, such that the inequalities ^D(z) > 0; P (z) ^D(z)(P (z)) + ^G(z)(P (z))? P (z) ^G(z)? ^D(z) < 0; hold for all Let D = ^D + ^D and G = ^G? ^G. Since z = z? for all it is easy to verify that D(z) and G(z) have the forms as described and satisfy (6) and (7). (b))(c): Let D(z) and G(z) be as in (b). Then D(z) D, and G(z) G for all Dene W (z) = D(z) + G(z) = P N i=0 ((Q i + U i )z i + (Q T i? Ui T )z?i ) with real matrices Q i S r;c;c (C), and U i S r;0;0 (C). Clearly the submatrix W cc (z) of W (z) is in S c;c (C) and is Hermitian for all Moreover, since for all D(z) is Hermitian and positive denite, and G(z) is skew Hermitian, we have for all He(W (z)) = D(z) > 0: On the other hand, replacing D and G by (W + W ) and (W? W ), respectively, in the second inequality in (b) yields, for all P (z)(w (z) + (W (z)) )(P (z)) + (W (z)? (W (z)) )(P (z))? P (z)(w (z)? (W (z)) )?(W (z) + (W (z)) ) < 0; i.e., (I + P (z))w (z)(i? P (z)) + (I? P (z))(w (z)) (I + P (z)) > 0: (0) Note that (0) implies that I? P (z) must be nonsingular for all Thus it is equivalent to (I? P (z))? (I + P (z))w (z) + (W (z)) (I + P (z)) (I? P (z))? > 0; i.e., ~P (z)w (z) + ( ~ P (z)w (z)) > 0;

13 i.e., This proves the claim. He( ~ P (z)w (z)) > 0: (c))(d): Suppose that (c) holds. Write W as diag(w r ; W c ; W C ), where W r, W c, and W C have dimensions n r n r, n c n c, and n C n C, respectively. Then, W cc = diag W c ; W C is Hermitian for all Now note that, under the assumption that (d)(i) holds, strict positive realness of W W is equivalent to the condition He and strict positive realness of W? (W (z))? (W (z)) > 0 ~P W? is equivalent to the condition He ~P (z)(w (z))? (W (z)) > 0 (both of these conditions are obtained via congruence transformations). To complete the proof of the implication (c))(d) it is thus enough to factorize W into W = W? W where W and W satisfy (d)(i). The desired factorization of W is possible, in view of Proposition 3. and the well-known symmetric factorization result mentioned in the remark following that proposition. It remains to apply this result to the block entries of W r, W c, and W C (where W r, W c, and W C all satisfy (3), and both W c (z) and W C (z) are Hermitian for all and dene W and W in the obvious way. This completes the proof of the implication (c))(d). (d) ) (a): Suppose that (d) holds and let W, W satisfy (i)-(iii). In particular W? (z) ~ P (z)w? (z) + (W? (z)) ( ~ P (z)) (W? (z)) > 0 or, equivalently (via congruence transformation), ~P (z)w? (z)(w (z)) + W (z)(w? (z)) ( ~ P (z)) > 0 Multiplying on the left by (I?P ) and on the right by (I?P ) (another congruence transformation) we get P (z)d(z)(p (z)) + G(z)(P (z))? P (z)g(z)? D(z) < 0 with, for all D(z) = He( (W (z))? (W (z)) ), and G(z) = Sh( (W (z))? (W (e j )) ). It is easy to check that, for all D(z) = (D(z)) and G(z) =?(G(z)), and condition (ii) implies that D(z) > 0 for all Since the left-hand side and D(z) are both continuous over the compact it follows that, for (0; ) close enough to and for all P (z)d(z)(p (z)) + G(z)(P (z))? P (z)g(z)? D(z) < 0: Since (0; ), this implies that kp k^ <. This completes the proof of the implication (d))(a). The remaining assertion follows directly from the same congruence transformations and the proofs of the implications (b))(c) and (c))(d). The proof of Theorem 3. is complete. Acknowledgment The authors wish to thank Jie Chen for helpful discussions, as well as Associate Editor Anders Rantzer and the anonymous referees for their many valuable suggestions. 3

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