Stability radii of linear discrete{time systemsand symplectic pencilsD. HinrichsenInstitut f�ur Dynamische SystemeUniversit�at BremenD-2800 Bremen 33F.R.G.N. K. SonInstitute of MathematicsP.O.Box 631, Bo Ho, HanoiVietnam

AbstractIn this paper, we introduce and analyze robustness measures for the stabilityof discrete-time systems x(t+1) = Ax(t) under parameter perturbations of theform A ! A+ BDC where B;C are given matrices. In particular we charac-terize the stability radius of the uncertain system x(t+1) = (A+BDC)x(t), Dan unknown complex perturbation matrix, via an associated symplectic penciland present an algorithm for the computation of that radius.1 IntroductionConsider a dynamical system described by the following linear di�erence equationx(t + 1) = Ax(t); t 2 N (1)where A is an n� n matrix and x(t) denotes the n{dimensional state vector.Suppose that the system is C 1 -stable, i.e. the spectrum �(A) of A lies in the openunit disk C 1 = fs 2 C ; jsj < 1g. It follows from the continuity of the spectrum thatthis property is preserved under su�ciently small perturbations of the entries of A.An important problem of robustness analysis is to determine to what extent stabilityis preserved when the entries of the nominal system matrix A are subjected to largeparameter perturbations. In this paper we assume that the perturbed system matrixhas the form A + BDC where B 2 K n�m , C 2 K p�n are �xed matrices de�ning thestructure of the perturbations and D 2 Km�p is an unknown disturbance matrix,K = R or C . The perturbed system may be formally interpreted as a closed loopsystem with unknown static linear output feedback (Fig. 1).By choosing B;C appropriately, di�erent perturbation structures for the matrixA can be studied. In particular, if B = C = In (\unstructured case") then allthe elements of A are subjected to independent perturbations whereas by di�erentchoices of B, C the e�ect of perturbations of individual elements, rows or columns of1

x(t + 1) = Ax(t) +Bu(t); y(t) = Cx(t)Du yFigure 1: Feedback interpretation of the perturbed systemA can be investigated. Since B and C are chosen to re ect the structure of the modeluncertainty, it is not natural to assume controllability or observability properties forthe pairs (A;B) and (A;C), respectively.Robustness measures for perturbations of the above kind have been introducedand analyzed extensively in the continuous{time case, see [14], [8], [9], [20], [15],[1],further references can be found in the survey [7]. The robustness of discrete{timestate space systems has received much less attention, see [5], [19], [6].This paper is a revised and expanded version of the conference paper [10]. Theaim is to derive counterparts of a number of results in [8], [9], [4] for the discrete{time case. Some of these counterparts are obtained by straightforward translation,but others require special constructions and show special features in the discrete-timecase. We will also investigate some phenomena which have not been studied in thecontinuous{time case.As a robustness measure for the family of perturbed systemsx(t + 1) = (A+BDC)x(t); t 2 N ; (2)we de�ne the (discrete{time) structured stability radius of A with respect to theperturbation structure (B;C) by:r1K = r1K (A;B;C) = inf nkDk;D 2 Km�p ; �(A+BDC) 6� C 1o (3)Here kDk is the operator norm of D with respect to an arbitrary pair of norms k �kKpand k � kKm on K p resp. Km . The unstructured stability radiusd1K (A) = r1K (A; I; I)measures the distance of A from the set of unstable matrices in the normed space(K n�n ; k � k).For each real triple (A;B;C) we obtain via (3) two stability radii, the complexone (K = C ) and the real one (K = R). While, at �rst sight, it may seem arti�cial toconsider complex perturbations of a real system it has been shown that the complexstability radius | besides being a lower bound for the real one | yields valuableinformation about the robustness of a system with respect to wider classes of per-turbations (time-varying, nonlinear and/or dynamic), see [7] for the continuous timecase and [6] for the discrete time case. 2

We proceed as follows. In the next section we discuss some elementary propertiesof stability radii. In particular, we analyze the behaviour of the unstructured stabilityradii on similarity orbits and study the relationship between real and complex stabilityradii under various conditions. In Section 3 we derive characterizations of the complexstability radius in terms of a) optimal control, b) the existence of Hermitian solutionsof a discrete time parametrized algebraic Riccati equation and c) the spectra of anassociated family of symplectic pencils. In Section 4 we study the spectral behaviourof the family of symplectic pencils in dependency on its parameter. In Section 5,�nally, we describe an algorithm for the computation of the complex stability radiusand illustrate the results by some numerical examples.2 Elementary propertiesThroughout the paper we assume that the nominal system (1) is asymptoticallystable. For the sake of simpli�cation, we provide the spaces K p and Km with theusual 2-norms so that the corresponding operator norm in (3) is the spectral normkDk = smax(D) where smax(D) denotes the maximal singular value of D. A part ofthe results in this section can be extended to arbitrary operator norms.By de�nition, the in�mum of the empty set is in�nite so that r1K (A;B;C) =1 i��(A+BDC) � C 1 for all D 2 Km�p (cf. (3)). On the other hand, if r1K (A;B;C) <1an easy compactness argument shows that there exists a minimum norm destabilizingperturbation D 2 Km�p such that kDk = r1K and �(A+BDC)\ @C 1 6= ; (where @C 1is the unit circle). The following characterization of the complex stability radius hasbeen proved in [5] (for arbitrary operator norms).Proposition 2.1 If G(s) = C(sI � A)�1B is the transfer matrix associated with(A;B;C) then r1C (A;B;C) = " max�2[0;2�] kG(e{�)k#�1 (4)where, by de�nition, 0�1 =1.Remark 2.2 Suppose thatkG(e{�max)k = max�2[0;2�] kG(e{�)kand choose u 2 C m , kuk = 1 such that kG(e{�max)k = kG(e{�max)ukCp . Then it is easyto check [7, Remark 4.2] that D = (r1C )2uu�G(e{�max)� (5)is a minimum norm destabilizing perturbation for the triple (A;B;C). In general,even if the matrices A;B;C are real, this D will be complex and it will not be possibleto �nd a real destabilizing perturbation of the same norm. Hence, in general, thefollowing (obvious) inequalityr1C (A;B;C) � r1R(A;B;C) (6)3

will be strict. If, however, the above D is real (e. g. if �max = 0 or �max = �) thenthe real and the complex stability radii of (A;B;C) coincide.As a special case of (4) we note a formula for the unstructured complex stabilityradius: d1C (A) = min�2[0;2�] smin(e{�I � A) = min�2[0;2�] smin(I � e{�A) (7)A computable formula for the real stability radius exists, as yet, only for thecase m = 1 (or p = 1). To state the result, let d(y; zR) denote, for any y; z 2 Rp ,the (Euclidean) distance of the point y from the linear subspace zR = f�z;� 2 Rgspanned by z: d(y; zR) = min�2R ky � �zkRpThen d2(y; zR) = ( kyk22 � hy; zi2=kzk22 ; z 6= 0kyk22 ; z = 0 ; (8)see Fig. 2.������������*.....................PPPPPPi PPPPPPPPPPPPPPP0 � yz d(y; zR) = jjyjj sin �

Figure 2: The distance d(y; zR) in (Rp ; k � k2) if z 6= 0The following formula is a specialization of a result in [5].Proposition 2.3 If m = 1 and GR(s), GI(s) are the real and imaginary parts ofG(s) = C(sI � A)�1B, thenr1R(A;B;C) = " max�2[0;2�] d(GR(e{�); GI(e{�)R)#�1 : (9)In general the map s 7! d(GR(s); GI(s)R) is not continuous at the zeros of GI(�).As a consequence, search methods cannot be applied directly in order to obtain themaximum of � 7! d(GR(e{�); GI(e{�)R) on [0; 2�]. Instead one must investigate thejump points separately. Let� = f� 2 [0; 2�];GI(e{�) = 0g (10)4

Then GR(e{�) = G(e{�) for � 2 � andr1R(A;B;C) = min8><>:�max�2� kG(e{�)k2��1 ; 264 sup�2[0;2�]�=2� d(GR(e{�); GI(e{�)R)375�19>=>; (11)In the case of a scalar transfer function (m = p = 1) the second term on the RHS of(11) is always in�nite and sor1R(A;B;C) = �max�2� jGR(e{�)j��1 : (12)Fig. 3 illustrates the di�erence between the real and the complex stability radius viathe polar plot of G(e{�) = C(e{�I � A)�1B; � 2 [0; 2�]. In particular, we see thatstrict inequality may occur in (6) . Re

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.............................................................................................................................................................................................................................................................................................................................. (r1C )�1

(r1R)�1Figure 3: Real and complex stability radii in a polar plotA time{domain characterization of the complex stability radius is obtained byconsidering the discrete convolution operatorL : l2(N ; C m)! l2(N ; C p)de�ned by (Lu(�)) (t) = t�1Xk=0CAt�1�kBu(k) (13)where l2(N ; C m) and l2(N ; C p) denote the Hilbert spaces of square summable se-quences of vectors in C m and C p , respectively. L is the input output operator of thesystem (A;B;C) and it is well known that its operator norm is given bykLk = max�2[0;2�] kG(e{�)k (14)Hence we get the following corollary of Proposition 2.15

Corollary 2.4 r1C (A;B;C) = 1=kLk (15)As another consequence, we obtain a property of the complex stability radius r1C whichis of particular interest in the context of decentralized control and model reduction.Corollary 2.5 (Decomposition property) Let A1 2 C q�q and A2 2 C r�r be sta-ble matrices and letA = " A1 00 A2 # ; B = " B1 00 B2 # ; C = " C1 00 C2 # :where Bi and Ci are of compatible formats. Thenr1C (A;B;C) = min nr1C (A1; B1; C1); r1C (A2; B2; C2)o (16)An analogous result does not hold for the real stability radius. Here we have onlythe trivial inequality:r1R(A;B;C) � min nr1R(A1; B1; C1); r1R(A2; B2; C2)o (17)In fact, the following relationship between complex and real stability radii shows thatstrict inequality may occur in (17).Remark 2.6 If A;B;C are real, �(A) � C 1 then [7]r1R(" A 00 A # ; " B 00 B # ; " C 00 C #) = r1C (A;B;C) (18)In the rest of this section we discuss some properties of unstructured real and complexstability radii. Since smin(I �A) and smin(I +A) are the distances of A 2 Rn�n fromthe set of matrices with eigenvalues � = 1 resp. � = �1 in Rn�n , it follows for anyC 1 -stable real matrix A that0 < d1C (A) � d1R(A) � min fsmin(I � A); smin(I + A)g (19)As in the continuous{time case, the real and complex stability radii of A coincide ifA is normal.Proposition 2.7 If A 2 Rn�n is normal and stable with spectral radius�(A) = max�2�(A) j�jthen d1C (A) = d1R(A) = dist (�(A); @C 1) = 1� �(A) (20)6

Proof: Let D = (1� �(A))=�(A)A. Then A +D has the spectral radius 1, henceis not C 1 -stable. On the other hand, kDk = 1 � �(A) because D is normal. Thisproves the inequalities \�" in (20). Because of (19) it only remains to show d1C (A) �1 � �(A). Since d1C (A) is, by de�nition, invariant with respect to unitary similaritytransformations we may assume that A is diagonal. But then this inequality followsdirectly from (7).If A is not normal, the distance of �(A) from the unit circle can be a very misleadingindicator of robustness, even if the real and the complex stability radii of A coincide.This is illustrated in the following example.Example 2.8 Let A� = " 0 �0 0 #. An easy calculation shows that smin(e{�I � A�)is independent of �. Hence, making use of (19) and (7), we obtaind1C (A�) = smin(I � A�) = d1R(A�):Since the perturbation D� = " 0 0��1 0 # destabilizes A� we see thatr1C (A�) = r1R(A�)! 0 as �!1:On the other hand, the distance of �(A�) from the unit circle is equal to one for all�.The following proposition shows that the similarity orbit of any C 1 -stable n�n-matrixA =2 RI comes arbitrarily close to instability. Moreover, it shows that, among all C 1 -stable matrices having a given spectrum, the normal ones are of optimal robustness(as measured by the complex stability radius).Proposition 2.9 Let A be a real C 1 -stable n� n matrix. Thend1C (A) � 1� �(A) = dist(�(A); @C 1) (21)If A is not a multiple of the identity matrix then there exists, for every " > 0, a pairof nonsingular real matrices T1; T2 such thatd1R(T1AT�11 ) � " ; d1C (T2AT�12 ) � 1� �(A)� " (22)Proof: If �(A)e{� 2 �(A) then D = [1� �(A)]e{�I satis�ese{� 2 �(A+D) ; kDk = 1� �(A)hence (21). The �rst inequality in (22) follows from (7) since there exists T1 2Rn�n ; det T1 6= 0 such that smin(I � T1AT�11 ) = 1=kT1(I � A)�1T�11 k < ". Thesecond inequality follows from Prop. 2.7 and the continuous dependence of d1C (A) onA since the similarity orbit [A] = fTAT�1;T 2 Gln(R)gcontains a sequence converging to a normal matrix having the same eigenvalues asA. 7

In terms of realization theory Prop. 2.9 can be interpreted as follows. Let G(s) 2R(s)p�m be any given strictly proper transfer matrix with McMillan degree n andpoles �j = �je{�j ; 0 � �n � � � � � �1 < 1. Then there exist realizations (A;B;C) ofG(s) whose system matrices come arbitrarily close to instability. On the other handthere is no realization of G(s) of larger distance from instability than 1� �1 and thisdistance is achieved by any realization (A;B;C) with normal system matrix A (if itexists).3 Characterizations of the complex stability ra-diusIn this section we establish a relation between r1C (A;B;C) and the spectrum of anassociated parametrized matrix pencil. This relation will be used as basis of analgorithm for computing r1C (A;B;C). Moreover, we characterize r1C via an associatedparametrized optimal control problem and the corresponding Riccati equation.Throughout this section we suppose that (A;B;C) is a real or complex triple with�(A) � C 1 . We consider the associated family of matrix pencils [18]W�(�) = M� � �N ; � 2 C (� � 0) (23)where M� = " A 0�2C�C I # ; N = " I BB�0 A� # : (24)Given � � 0, a number � 2 C is called a characteristic value of the pencil W� ifdet(M� � �N) = 0. For each � � 0, the set of all characteristic values of the pencilW� is denoted by �(W�) and is called the spectrum of the pencil.The following symmetry property of �(W�) is well known: For every � 2 C ; � 6= 0:� 2 �(W�)() 1=�� 2 �(W�): (25)We say that 1=�� is obtained from � by re ection at the unit circle. SinceW�(�) = " A� �I ��BB��2C�C I � �A� # (26)we see that for all � � 0 0 2 �(W�)() 0 2 �(A): (27)W� is said to be singular at �0 � 0 if the polynomialp(�; �) = det(M� � �N) (28)is the zero polynomial (in �) for � = �0, otherwise it is said to be regular at �0. IfW� is regular at �0 then W�0 has at most 2n characteristic values. W� is singular at�0 i� �(W�0) = C .By (26), W� is regular for su�ciently small � � 0. For every � 2 C , p(�; �) is anontrivial polynomial of degree � n in �2 so thatW� is singular for at most n positivevalues of the parameter �. 8

Proposition 3.1 Let � > 0; � 2 C ; � 6= 0 and suppose that �; 1=� =2 �(A). Then� 2 �(W�)() 1=�2 2 � �G(1=��)�G(�)� (29)Proof: An easy calculation shows thatdet(W�(�)) = (�1)n det(�I � A) det(I � �A�)�det[I � �2B�(��1I � A�)�1C�C(�I � A)�1B]: (30)This proves (29).From (30) we see that detW�(1) = 0 only if 1=� is a singular value of G(1). Hence,if W� is singular at � = �0 > 0 then 1=�0 is necessarily a singular value of G(1). Inparticular, there exist at most minfm; pg values of � at which W� becomes singular.As immediate consequences of Proposition 3.1 we obtain the following corollaries.Corollary 3.2 Let � > 0. Then e{� is a characteristic value of W� if and only if1=� is a singular value of G(e{�).Corollary 3.3 The parametrized pencil W� is singular at �0 > 0 if and only ifs(�) � 1=�0 is a constant branch of singular values of the transfer matrix G(e{�),� 2 [0; 2�] on the unit circle.We now return to the problem of characterizing the complex stability radius r1C =r1C (A;B;C).Proposition 3.4 For any �0 2 [0; 2�] and any � 2 (0; kG(e{�0)k�1],� < r1C () �(W�) \ @C 1 = ;: (31)In particular, if r1C <1 thenr1C = minf� 2 R+ ; �(W�) \ @C 1 6= ;g: (32)Proof: It follows from Prop. 2.1 and Cor. 3.2 that, for � 2 (0; r1C ) the parametrizedpencil W� has no characteristic values on the unit circle:�(W�) \ @C 1 = ; ; � 2 (0; r1C )Indeed, if e{� 2 �(W�) then 1=� is a singular value of G(e{�), and therefore 1=� �smax(G(e{�)) = kG(e{�)k. This implies� � kG(e{�)k�1 � " max�2[0;2�] kG(e{�)k#�1 = r1C :Conversely, since the map � ! kG(e{�)k is continuous, it follows that for any � 2[r1C ; kG(e{�0)k�1], there exists � 2 [0; 2�] such that � = kG(e{�)k�1, or, equivalently,1=�2 = kG(e{�)k2 2 �(G(e{�)�G(e{�)). Therefore, by Cor. 3.2, e{� 2 �(W�) and thisconcludes the proof. 9

Thus r1C is that value of � for which the spectrum of the matrix pencil W� hits theunit circle for the �rst time as � increases from 0 to1. This characterization will beused in Section 5 to design an algorithm for the computation of r1C .Remark 3.5 Prop. 3.4 is not completely analogous to its continuous-time counter-part, see Prop. 2.3 in [4]. In the continuous-time case, the parametrized HamiltonianH� associated with a stable triple (A;B;C) has always at least one eigenvalue on theimaginary axis if � � r�C . In the discrete-time case it is, however, not true that thesymplectic pencil W� always has at least one characteristic value on the unit circle if� � r1C . In general, the set of all � for which �(W�) \ @C 1 6= ; is a disjoint union of�nitely many intervals. This fact will be illustrated in Section 4.Another characterization of r1C is obtained by considering the following parametrizedLinear Quadratic Destabilization Problem (LQDP�):min J�(x0; u(�)) = 1Xt=0 �ku(t)k2 � �2ky(t)k2�subject tou(�) 2 l2(N ; C m)x(t + 1) = Ax(t) +Bu(t); t 2 N ; x(0) = x0y(t) = Cx(t)Contrary to the usual linear quadratic regulator problem, the deviation of the outputfrom zero is here not penalized but awarded. As a consequence we expect the resultingfeedback law to deteriorate the stability of the system. It is not surprising that theabove parametrized optimization problem can be related to the stability radius whichmeasures the \e�ort" it takes to destabilize the system.Proposition 3.6 The following three statements are equivalent:(i) J�(0; u(�)) � 0 for all u(�) 2 l2(N ; C m)(ii) � � r1C(iii) I � �2G(e{�)�G(e{�) � 0 for all � 2 [0; 2�]Proof: Since x0 = 0 we havey(t) = C t�1Xk=0At�1�kBu(k) = (Lu(�))(t):Hence J�(0; u(�)) = kuk2l2(N;Cm ) � �2kLuk2l2(N;C p ) :Thus the �rst equivalence follows from Cor. 2.4, the second follows from Prop. 2.1.10

The discrete algebraic Riccati equation corresponding to the above linear quadraticoptimal control problem has the following form:P � A�PA+ �2C�C + A�PB(I +B�PB)�1B�PA = 0 (DARE�)A Hermitian matrix P is said to be a solution of (DARE�) if (I+B�PB) is invertibleand (DARE�) holds. P is said to be stabilizing if�(AP ) � C 1 where AP = A� B(I +B�PB)�1B�PA: (32)The existence of Hermitian solutions of discrete algebraic Riccati equations and theirrelation to discrete{time linear quadratic optimal control problems have been studiedin the literature under various assumptions, see e. g. [16], [18], [12]. Fortunately, theresults of [16] are applicable in our situation and yield the following characterizationof r1C in terms of the parametrized equation (DARE�).Proposition 3.7 The parametrized Riccati equation (DARE�) has a stabilizing Her-mitian solution P satisfying I +B�PB � 0 if and only if � < r1C .Proof: If there exists a Hermitian solution of (DARE�) such that I + B�PB � 0then (s) de�ned by [16, (4)] satis�es(e{�) = I � �2 hC(e{�I � A)�1Bi� C(e{�I � A)�1B � 0; � 2 [0; 2�] (33)by the complex version of Thm.2 in [16], and hence � < r1C by Prop. 2.1. Converselyif � < r1C then (33) holds and since A is stable this implies the existence of a solutionwith the required properties, see [16, Thm.2].Because of the positivity condition I + B�PB � 0 this proposition is not anexact counterpart of the corresponding characterization in the continuous{time case[9] where the existence of a stabilizing solution alone implies � < r�C . This fact wasoverlooked by us in [10]. As a consequence the statement of Prop. 3.7 in [10] is notcorrect. The following one{dimensional example shows that neither the positivitycondition I + B�PB � 0 nor the stabilization condition (32) can be dispensed within Prop. 3.7.Example 3.8 Consider the C 1{stable scalar triple (A;B;C) = (a; 1; 1); jaj < 1. Itfollows from Prop. 2.7 that r1C (a; 1; 1) = d1C (a) = 1� jaj:(DARE�) has the form P � jaj2P + jaj2 P 21 + P + �2 = 0 (34)To study this equation we may assume a � 0; � � 0. P 2 R is a solution of (34)(satisfying the invertibility condition 1 + P 6= 0) i�P 2 + (1� a2 + �2)P + �2 = 0 and 1 + P 6= 0: (35)11

Condition (32) takes the formj [1� P=(1 + P )] a j = a=j1 + P j < 1: (36)First, assume a = 0. Then condition (36) is satis�ed for all solutions P� =��2; � 2 R+ nf1g of (34) whereas 1+P� > 0 is satis�ed i� �2 < 1 = r1C (a; 1; 1). Thusthe positivity condition is the critical condition if a = 0. As another di�erence fromthe continuous time case [9], note that there does not exist a solution of the Riccatiequation (34) (with 1 + P 6= 0) at the limit value � = r1C (0; 1; 1) = 1.Now assume a > 0. The solutions of (34) have the formP�� = a2 � 1� �22 � 12q(a2 � 1� �2)2 � 4�2and these solutions are real i� j1� �j � a, i.e. � 2 R+ n [1� a; 1 + a]. If � increasesfrom 0 to r1C = 1� a the two solutions P�� coalesce:P+r1C = P�r1C = a� 1:Thus (contrary to the case a = 0) the positivity condition 1+P�� > 0 is automaticallysatis�ed at � = r1C . But now the stabilization condition becomes critical, i.e. thestabilizing solution P+� ; � < r1C of (34) loses the stabilization property at � = r1C (asit always does in the continuous time case). Note however that | contrary to thecontinuous time case | there exists a stabilizing solution P�� of (34) for � > 1+a > r1C .This solution, however, does not satisfy 1 + P�� > 0 .4 The characteristic loci of the symplectic pencilIn view of the above results it is of interest to investigate the behaviour of �(W�) as� increases from 0 to 1.Perturbation theory tells us that the eigenvalues of W� depend piecewise analyt-ically on the real paramter �. More precisely we have (see [11, II.1 and II.5])Lemma 4.1 The eigenvalues of the matrix family (W�)�2R+ can be written in theform �1(�); � � � ; ��(�) where � � 2n is constant and the functions �k : R+ ! C ,k = 1; � � � ; � are continuous. Moreover, the eigenvalues �k(�); k = 1; � � � ; � aredistinct and of constant algebraic multiplicity except at isolated "exceptional" valuesof �. They depend analytically on � in the intervals between the exceptional points.In the sequel, the � curves �k(t); t � 0; (k = 1; � � � ; �) will be called the characteristicloci of (W�)�2R+, see Fig. 4.For any Hermitian matrix H, let �min(H) and �max(H) denote the smallest andthe largest eigenvalue of H, respectively. Applying results from the perturbationtheory of Hermitian operator functions [11, II.6] to H(�) = G(e{�)�G(e{�) we have thefollowing lemma (cf. [17]). 12

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Figure 4: The characteristic loci of W�Lemma 4.2 (i) The eigenvalues of the family of Hermitian matricesH : R ! C m�m ; � ! G(e{�)�G(e{�)can be written in the form �1(�); � � � ; �h(�), h = const: � m such that each ofthe 2�-periodic functions �k : R ! R+ , k = 1; � � � ; h is analytic .(ii) The functions �min : [0; 2�]! R+ ; � ! �min(H(�))�max : [0; 2�]! R+ ; � ! �max(H(�))are continuous and piecewise analytic.(iii) If �max (resp. �min) has a local maximum (resp. minimum) at �0 then it isdi�erentiable at �0 and one of the branches �k(�) has a local maximum (resp.minimum) at �0.As a consequence of this lemma, the maximal singular value and the minimal singularvalue of G(e{�)smax(�) = smax(G(e{�)) = kG(e{�)k; smin(�) = smin(G(e{�)) (37)are continuous piecewise analytic functions on [0; 2�]. Each singular branchsk : [0; 2�]! R+ ; �! q�k(�) (38)has only �nitely many local extrema on [0; 2�] if it is not constant. The rangesk[0; 2�] of sk is a closed interval, and this interval is reduced to a point if and onlyif sk is a constant branch. The graph of sk is symmetric with respect to � = �, i.e.sk(� � �) = sk(� + �), � 2 [0; �].Our aim is to describe the movements of the characteristic values of W� in termsof the singular branches sk. As � increases from 0 to1 we observe that these charac-teristic loci approach the unit circle in symmetric pairs, one from the inside and one13

from the outside of C 1 , they collide and begin to run along the unit circle in oppositedirections, see Fig. 4. These collision points on @C 1 are called \arrival points" ofthe corresponding pair of loci. (It is obvious how to formalize this de�nition). Wealso observe characteristic loci moving along the unit circle, colliding with others andsplitting away from the circle, one locus moving into C 1 and the other to the outside,see Fig. 4. These points will be called \departure points" of the corresponding loci.(Again we omit a formal de�nition). As � tends to1, some of the characteristic lociapproach 0 while an equal number of loci go o� to in�nity. Other loci converge tocertain points in the punctured complex plane.Fig. 5 pictures the distance of the spectrum �(W�) from the unit circle as afunction of �. The zero set Z of this function consists of those parameter values �for which W� has at least one characteristic value on the unit circle. Z is the disjointunion of not more than h � m intervals (Cor.3.2, Lemma 4.2).

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0 1 2 3 4 5 6 7Figure 6: The singular values of G(e{�) on [0; 2�]14

The following proposition summarizes how these movements of the characteristicvalues of W� are determined by the singular values of the associated transfer matrixon the unit circle (compare Fig. 6 and Fig. 4).Proposition 4.3 Let [ak; bk] = sk([0; 2�]) where sk is de�ned by (38), 1 � k � h.Then(i) If 0 2 �(A) then 0 2 �(W�) for all � � 0.(ii) If �0 2 [0; 2�] is a local maximum of sk, sk is not constant and sk(�0) 6= sj(�0)for j 2 h; j 6= k then e{�0 is an arrival point of a pair of characteristic loci ofW�.(iii) If �0 2 [0; 2�] is a local minimum of sk, sk is not constant and 0 < sk(�0) 6=sj(�0) for j 2 h; j 6= k then e{�0 is a departure point of a pair of characteristicloci of W�.(iv) If ak < bk and 1=� decreases from bk to ak then the corresponding characteristicvalues of W� (i.e. the complex numbers e{� such that sk(�) = 1=�, see Cor. 3.2)�ll up the whole unit circle.(v) If 1=r1C = kG(e{�0)k then e{�0 is a \�rst arrival point" (corresponding to thesmallest � for which the spectrum of W� hits the unit circle).(vi) If a = min1�k�h ak = min�2[0;2�] smin(�) > 0 and a = smin(�0) then e{�0 is the\�nal departure point" so that no more characteristic values of W� are left onthe unit circle for � > 1=a.(vii) If sk(�0) = 0 and sk 6� 0 then e{�0 is a limit point of a pair of characteristicloci of W� moving on the unit circle from opposite sides of e{�0 towards e{�0 as�!1.(viii) Assume m = p and detG(s) 6� 0. If a characteristic locus of W� converges to�0 2 C , �0 6= 0 as �!1 then �0 or 1=��0 is a transmission zero of G(s).Proof: (i) follows directly from (26).(ii)Since sk(�) is analytic, non{constant and di�erent from all other singular branchesat �0 there exists � > 0 such that for all � 2 [�0 � �; �0 + �] we have(a) sk(�) < sk(�0) if � 6= �0;(b) sj(�) =2 [1=�0; 1=(�0 � �)] for all � 2 [�0 � �; �0 + �] and j 2 h,where sk(�0) = 1=�0. By Cor. 3.2 e{�0 2 �(W�), and for � > �0; � � �0 su�cientlysmall there are at least two distinct characteristic values e{�i 2 �(W�), i = 1; 2 suchthat �0� � < �1 < �0 < �2 < �0+ �. Hence � = �0 is one of the exceptional parametervalues where splitting occurs. Let U be a small neighborhood of e{�0 symmetric to theunit circle which does not contain any other characteristic value of W�0 and satis�esU \ @C 1 � fe{�; � 2 [�0 � �; �0 + �]g. Choose " > 0, " � � su�ciently small so that:15

(c) �k(�) 2 U for � 2 [�0 � "; �0 + "];(d) Every locus �j(�) intersecting U for some � 2 [�0 � "; �0 + "] belongs to the�{group of e{�0 [11], i.e. satis�es �j(�0) = e{�0 .By (b) and Cor. 3.2 there does not exist a characteristic value of W�, � 2 [�0� �; �0)in the sector fe{�; � 2 [�0 � �; �0 + �]g of the unit circle. On the other hand, by (c),�(W�)\U 6= ; for � 2 [�0��; �0]. Since the characteristic values ofW�, � 2 [�0�"; �0)in U do not ly on the unit circle they occur in symmetric pairs and by (d) they allcollide for � = �0 at e{�0 .Altogether we see that there exists at least one symmetric pair of characteristicvalues of W� colliding (with increasing �) at e{�0 for � = �0 and after this collisionthere occurs a splitting with at least one pair of eigenvalues running along the unitcircle in opposite directions. Hence e{�0 is an arrival point.(iii) is shown analogously as (ii).(iv) is just a restatement of the trivial equalityfe{�; � 2 [0; 2�]; ak � sk(�) � bkg = @C 1 .(v) follows from (ii) and Prop. 3.4.(vi) follows from (iii) and Cor 3.2.(vii) By a similar argument as in (ii) �0 is a strict local minimum of sk(�). Hence,for any sequence (�i), �i � 1 there exist two sequences �1i < �0 < �2i convergingtowards �0 such that sk(�1i ) = sk(�2i ) = 1=�i. The statement now follows from Lemma4.1 and Cor. 3.2.(viii) If �j(�) ! �0 6= 0 as � ! 1 then by Prop. 3.1 and the continuousdependence of the spectrum on the matrix entries we get 0 2 �(G(1=��0)�G(�0)), i.e.detG(1=��0) = 0 or detG(�0) = 0. This concludes the proof.In the special case of 1-parameter perturbations (m = p = 1) we obtain thefollowing more precise picture of the behaviour of �(W�) as �!1.Proposition 4.4 Suppose m = p = 1, G(s) = C(sI � A)�1B = p(s)=q(s) 6� 0,q(s) = det(sI � A), r = deg p(s) and let k0, l0 be the multiplicities of the root � = 0of q(s), p(s), respectively. Then we have the following behaviour of �(W�) (countingmultiplicities) as �!1:(i) ko of the characteristic values of W� remain at 0 for all � 2 R+ .(ii) d = n � r + l0 � k0 of the characteristic values of W� tend to in�nity alongasymptotic directions given by the d-th roots of the identity in C . Correspond-ingly, d of the characteristic values (obtained by re ection at the unit circle)tend to zero.(iii) The 2(r � l0) remaining characteristic values of W� tend to the nonzero rootsof p(s) and their re ections at the unit circle.(iv) Each zero of the transfer function is a limit point of a characteristic locus ofW�. 16

Proof: Applying (30) we getdet(W�(�)) = (�1)nq(�)�nq(1=�) h1� �2G(1=�)G(�)i= (�1)n hq(�)~q(�)� �2�n�rp(�)~p(�)i :where ~q(�) = �nq(1=�), ~p(�) = �n�rp(1=�) are real polynomials of degree n� k0 andr � l0, respectively. Hence, for � > 0� 2 �(W�)() ��2q(�)~q(�)� �n�rp(�)~p(�) = 0Note that the polynomial �(�) = �n�rp(�)~p(�) has the degree d� = n + r � l0 anda root at 0 of multiplicity n � r + l0. �(�) = q(�)~q(�) has degree d� = 2n � k0 anda root of multiplicity k0 at 0. Now apply a well known lemma from cheap controltheory concerning the roots of ��2�(�)��(�) as �!1, see [21, Lemma 13.2]. Sinced = d��d� and since �(s) has 2(r� l0) nonzero roots the statements (ii) { (iv) followdirectly from this lemma. (i) is an immediate consequence of (27).5 An algorithm and some numerical examplesWe assume G(s) = C(sI � A)�1B 6� 0 so that r1C (A;B;C) is �nite by Prop. 2.1.Starting from initial estimates ��0 and �+0 of r1C (A;B;C) such that, with k = 0,0 � ��k � r1C (A;B;C) � �+k <1 (39)one can compute successively better estimates by the following bisectional algorithmbased on Prop. 3.4. Suppose that in the k{th step estimates ��k and �+k are given asabove. Consider � = 1=2 ���k + �+k � :If W� has eigenvalues on the unit circle then set ��k+1 = ��k and �+k+1 = �. Otherwiseset ��k+1 = � and �+k+1 = �+k . Increase k.As a lower bound we take ��0 = 0. The initial upper bound has to be chosencarefully. To apply Prop. 3.4 we must secure that �+0 � kG(e{�)k�1 for some � 2[0; 2�]. Since the values of kG(s)k at s = �1 are of special importance (see Remark2.2) and since it is to be expected that the maximum of kG(e{�)k will be attainednear the poles of G(s) it is natural to choose as an upper bound�+0 = min nkG(1)k�1; kG(�1)k�1; kG(�0=j�0j)k�1o (40)where �0 is a nonreal eigenvalue of A which has the smallest distance from the unitcircle (if it exists | otherwise the term is omitted). In many examples this upperbound �+0 has proved to be an excellent estimate of r1C (A;B;C).The above procedure is a counterpart of the algorithm stabrad1 described in [4]for the continuous-time case. As a bisectional procedure, it is guaranteed to converge,but convergence is rather slow. 17

A more e�cient algorithm is obtained by exploiting the interplay between thetwo characterizations of r1C given in Prop. 3.4 and Prop. 2.1. This idea leads toan algorithm (dstabrad2, [6]) which yields a discrete time counterpart to a nu-merical method developed recently by various authors for the computation of theH1{norm of a proper rational stable transfer matrix (see e.g. [3], [2]) . dstabrad2produces a decreasing sequence of approximate upper bounds �j which converge to-wards r1C (A;B;C). It proceeds as follows. Let " > 0 be the required absolute accuracyand suppose that �j has been computed, j � 1. Determine�(W�j ) and �j = maxz2�(W�j )\@C 1 kG(z)k:Let 0 � �j1 < : : : < �jkj < 2� be the sequence of angles �jh 2 [0; 2�) ( �jh 2 [0; �) in thereal case) which satisfy kG(e{�jh)k = �j and e{�jh 2 �(W�j):Then de�ne �j+1 = min8<:�j; " max1�h�kj�1 kG(e{ (�jh+�jh+1)=2)k#�19=;� 2" (41)and set j := j + 1.The algorithm ends if �(W�j ) \ @C 1 = ; or kj = 1 yielding the approximate valuer1C (A;D;E) � �j + ".For the e�ciency of the algorithm the choice of the initial value �1 is of greatimportance. We use the same idea as in (40), but reduce this upper bound in thesame way as in (41): �1 = �+0 � 2"In many applications no further iterations are required to determine r1C with theprescribed accuracy.Note that the above algorithm is di�erent from a direct application of the methodin [3], [2] to the Cayley tranform of the system (A;B;C).The following numerical examples have been computed via the above algorithm(implemented on a SUN workstation in double precision arithmetic, using the PRO-MATLAB package).Example 5.1 Consider unstructured perturbations (B=C=I) of the matrixA = " 0 0:90:3 �0:15 #The characteristic loci of the corresponding pencilW�(�) = " A� �I ��I�2I I � �A> #18

are shown in Fig. 4, the singular values of G(e{�) = (e{�I � A)�1 are illustrated inFig. 6 and the graph of the function �! dist(�(W�); @C 1) is plotted in Fig. 5. Fig.4 shows that the �rst arrival point of the characteristic loci of W� is �1. Hence, as aconsequence of Remark 2.2, the complex and the real stability radii of A coincide. Infact one computes d1C (A) = 0:367763 = smin(I + A) and this implies d1C (A) = d1R(A)by (19).Example 5.2 The following 7�7 matrix is the closed loop system matrix of a heatedrod controlled by a dead{beat controller [13].2666666664 �0:1373 0:2139 �0:2831 0:2792 �0:2177 �0:1298 0:06660:0002 �0:0163 �0:0438 �0:0657 �0:0669 �0:0473 �0:02750:0469 0:0718 0:0896 0:0782 0:0493 0:0224 0:00740:0373 0:0712 0:1124 0:1292 0:1124 0:0712 0:03730:0074 0:0224 0:0498 0:0782 0:0896 0:0718 0:0469�0:0275 �0:0473 �0:0669 0:0657 �0:0438 �0:0163 0:0002�0:0666 �0:1298 �0:2177 �0:2792 �0:2831 �0:2139 �0:13733777777775The robustness of the discrete-time system described by this matrix has been ana-lyzed in [19]. Three lower estimates for the unstructured real stability radius werecomputed, the best being d1R(A) � 0:4451:The complex stability radius of the system isd1C (A) = 0:68066which yields a better lower bound of d1R(A).Dead-beat control results in a nilpotent system matrix. The spectra of nilpotentmatrices have the largest distance (= 1) from the unit circle among all C 1 -stablematrices. In Example 2.8 we have seen that this does not necessarily imply goodrobustness. Poor robustness may also result from the presence of large Jordan blocksin the Jordan normal form of the nilpotent matrix. This is illustrated in the followingexample.Example 5.3 Let Nk denote the nilpotent Jordan block of order k. Computing theunstructured complex stability radii of these matrices we obtain:k 2 3 4 5 10 15r1C (Nk) 0.6180 0.4450 0.3473 0.2846 0.1495 0.1013The characteristic loci of the pencils W� corresponding to the triplets (Nk; I; I)are all constant. Moreover, the pencils W� become singular at � = r1C .Example 5.4 In this example we compute the structured complex stability radiusof a stochastically generated triplet (A;B;C) given byA = 100�2 2666666664 16 �37 20 �21 �5 40�15 3 5 �10 46 �3929 �30 34 40 �47 42�44 �42 �3 �12 �26 3645 18 �50 47 44 �251 14 16 �46 37 �48

377777777519

B = 100�2 2666666664 �70 83�68 0�96 6637 44�84 3813 �323777777775 ;C = 100�2 264 66 88 �36 �92 �45 �2448 49 �44 �97 59 �479 95 1 19 �44 �30 375The complex stability radius is r1C (A;B;C) = 0:318787. The above triplet hasthe property that the distance of the spectrum of the associated symplectic pencilW�(�) from the unit circle does not decrease monotonically to zero as � increases.The spectrum �rst withdraws from the unit circle before approaching it. Only about10 percent of the 200 stochastically generated examples which we have examined,showed | to a lesser degree | such a non{monotonous behaviour.AcknowledgementThe algorithm dstabrad2 has been elaborated and implementedby L. Schwiedernoch. We would like to thank J. Schalth�ofer and L. Schwiedernoch forperforming the necessary numerical tests and computing the examples in this paper.References[1] R.M. Biernacki, H. Hwang, and S.P. Bhattacharyya. Robust stability with struc-tured real parameter perturbations. IEEE Transactions on Automatic Control,AC-32(6):495{506, 1987.[2] S. Boyd and V. Balakrishnan. A regularity result for the singular values of atransfer function matrix and a quadratically convergent algorithm for computingthe L1-norm. SIAM J. on Matrix Analysis and Applications. to appear.[3] N. A. Bruinsma and M. Steinbuch. A fast algorithm to compute the H1-normof a transfer matrix. Systems & Control Letters, 14:287{293, 1990.[4] D. Hinrichsen, B. Kelb, and A. Linnemann. An algorithm for the computation ofthe structured stability radius with applications. Automatica, 25:771{775, 1989.[5] D. Hinrichsen and A. J. Pritchard. New robustness results for linear systems un-der real perturbations. In Proc. 27th IEEE Conference on Decision and Control,pages 1375{1379, Austin, Texas, 1988.[6] D. Hinrichsen and A. J. Pritchard. On the robustness of stable discrete timelinear systems. In New Trends in Systems Theory, Proc. Conf., Genova, 1990.To appear.[7] D. Hinrichsen and A. J. Pritchard. Real and complex stability radii: a survey.In Proc. Workshop Control of Uncertain Systems, Bremen 1989, volume 6 ofProgress in System and Control Theory, pages 119{162. Birkh�auser, 1990.20

[8] D. Hinrichsen and A.J. Pritchard. Stability radii of linear systems. Systems &Control Letters, 7:1{10, 1986.[9] D. Hinrichsen and A.J. Pritchard. Stability radius for structured perturbationsand the algebraic Riccati equation. Systems & Control Letters, 8:105{113, 1986.[10] D. Hinrichsen and N.K. Son. The complex stability radius of discrete-time sys-tems and symplectic pencils. In Proc. 28th IEEE Conference on Decision andControl, pages 2265{2270, Tampa, 1989.[11] T. Kato. Perturbation Theory for Linear Operators. Springer Verlag, Berlin-Heidelberg-New York, 1976.[12] P. Lancaster, A. C. M. Ran, and L. Rodman. Hermitian solutions of the dis-crete algebraic Riccati equation. International Journal of Control, 44(3):777{802,1986.[13] B. Lenden. Multivariable dead{beat control. Automatica, 13:185{188, 1977.[14] C. Van Loan. How near is a stable matrix to an unstable matrix? ContemporaryMathematics, 47:465{478, 1985.[15] J.M. Martin and G.A. Hewer. Smallest destabilizing perturbations for linearsystems. International Journal of Control, 45:1495{1504, 1987.[16] B. P. Molinari. The stabilizing solution of the discrete algebraic Riccati equation.IEEE Transactions on Automatic Control, AC-20:396{399, 1975.[17] M. Motscha. An algorithm to compute the complex stability radius. Interna-tional Journal of Control, 48:2417{2428, 1988.[18] T. Pappas, A. J. Laub, and N. R. Sandell. On the numerical solution of thediscrete-time algebraic Riccati equation. IEEE Transactions on Automatic Con-trol, AC-25:631{641, 1980.[19] L. Qiu and E. J. Davison. A new method for the stability robustness determi-nation of state space models with real perturbations. In Proc. 27th IEEE Conf.Decision and Control, pages 538{543, 1988.[20] L. Qiu and E.J. Davison. New perturbation bounds for the robust stabilityof linear state space models. In Proc. 25th IEEE Conference on Decision andControl, pages 751{755, Athens, 1986.[21] W.M. Wonham. Linear Multivariable Control: A Geometric Approach. SpringerVerlag, Heidelberg, 2nd edition, 1979.21