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1 Stability radii and spectral value sets for real matrix perturbations Introduction D. Hinrichsen and B. Kelb Institut fur Dynamische Systeme Universitat Bremen D-859 Bremen, F. R. G. A desirable property of a mathematical model (e.g. stability of a dynamical system) can sometimes be expressed by the requirement that the spectrum (A) of a matrix A be contained in a given open subset C g 6= ; of the complex plane. Suppose that A is uncertain or subjected to perturbations of the form A ; A() where represents a vector or matrix of parameter deviations. The question arises whether (A()) C g for all perturbations of norm less than a given upper bound. More specically two problems emerge:. The stability radius problem: Determine the largest > such that kk < implies (A()) C g.. The spectral value set problem: For a given uncertainty level > determine the set of all C to which a spectral value of A() can be moved by some disturbance of norm kk <. The two following denitions formalize these two problems for ane perturbations of the static output feedback type A() = A + DE. Let A K nn ; D K n`; E K qn be given matrices (K = R or K = C ) and C = C g _[ C b ; C g open () a nontrivial partition of the complex plane. The size of a perturbation K `q is measured by the operator norm kk of : K q 7! K ` induced by a given pair of norms k k K q, k k K`. Denition. The stability radius of A with respect to the perturbation structure (D; E) and the partition () is dened by n o = r K (A; D; E; C b ) = inf kk; K `q ; (A + DE) \ C b 6= ; () r K Denition. The spectral value set of A with respect to the perturbation structure (D; E) and uncertainty level > is dened by [ K (A; D; E; ) = (A + DE) () K`q ; kk< r C and C are more easily analyzed than r R and R. An overview of results concerning the complex stability radius r C can be found in [4]. Spectral value sets for unstructured complex perturbations have been studied (under the names of \pseudospectra or \spectral portraits) in the context of numerical analysis and uid dynamics [8], [], [9]. In this paper we only deal with real perturbations and present a selective overview of recent results concerning r R and R. For references concerning the real stability radius see [],[6], [4], [7]; for references concerning spectral value sets under real perturbations see [5], [].

2 The real stability radius Suppose A R nn ; (A) C g ; D R n`; E R qn and set G(s) = E(sI n? A)? D = G R (s) + {G I (s) C q`; s C n (A) (4) where G R (s); G I (s) R q`. For s C n (A); R`q s (A + DE), det(i`? G(s )) = : (5) Equivalently, there exists u = u + { C `; u ; R`; u 6= such that G(s )u = u, i.e. in real terms GR (s )?G I (s ) u u u = ; 6= (6) G I (s ) G R (s ) Dene, for any y ; y R q ; u ; R` (y ; y ; u ; ) = inffkk ; R`q ; y = u and y = g: (7) Proposition. ([]) b = C g \ C b denote the boundary of C b in C. Then r R (A; D; E; C b ) = min s@c b min u ; R` (G R (s)u? G I (s) ; G R (s) + G I (s)u ; u ; ) (8) (u ; )6=(;) Although the function (see (7)) is easily computable [4] the above formula is dicult to evaluate. For ` = a computable characterization of r R is obtained as follows. If y; z R q we denote by dist(y; R z) = min R ky? zk R q the distance between y and the linear subspace R z in (R q ; k k R q). Proposition. ([]) If ` = then r R (A; D; E; C b ) =? max dist(g R (s); R G I (s)) : (9) s@c b The problem to nd a computable characterization for r R in the general case remained open for some time. In 99 Qiu and Davison [6] found a new lower bound for r R in the unstructured case (D = E = I n ) and conjectured this bound to be in fact equal to r R. Their lower bound is based on the observation that, for any R`q and every >, the existence of a nontrivial solution of the equation (6) is equivalent to the existence of a nontrivial solution of G R (s )?G I (s ) u u u? = ; 6= () G I (s ) G R (s ) Very recently, the conjecture of Qiu and Davison was proved to be true in the six authors' paper [7]. The basic result is: Theorem. ([7]) Suppose that M = X + {Y C q`; X; Y R q`, let k k denote the spectral norm, i () the i-th singular value (in decreasing order) of the matrix X?Y P () =? () Y X If there exists R`q such that det(i? M) = then h? minfkk ; R`q ; det(i? M) = gi = inf () () (;] The function () in () is a unimodal function of on (; ].

3 6 5 4 If u Figure : Singular values i () and the corresponding i () = =k i ()k, i ; v v are normalized right and left singular vectors of P () corresponding to i () > and () y denotes the pseudoinverse then i () = i ()? [u ][v v ] y R`q () satises det(i? i ()M) =. It can be shown that = i (^) minimizes the expression in brackets in () if ^ (; ) minimizes () on (; ] and (^) is a simple singular value of P (^). Figure shows the largest three singular values i () of a -matrix P () and k i ()k? for the associated perturbations i (); i as functions of. Note that k i ()k? touches i () at all the points (; ) where i () possesses an extremum. As a consequence of Theorem. the following computable general formula for r R is obtained. Corollary.4 Suppose that R q,r` are provided with their Euclidean norms. Then? G r R (A; D; E; C b ) = sup inf ( R (s)?g I (s) ) s@c b (;]? (4) If ` = formula (9) yields a geometrical interpretation of the inmum of the RHS in (4): GR (s)?g inf ( I (s) (;]? ) = lim GR (s)?g I (s) (!? ) = dist(g R (s); R G I (s)) (5) Spectral value sets under real perturbations Suppose A R nn ; D R n`; E R qn and let R q`; R`q be provided with the operator norm k k (spectral norm). Theorem. enables us to determine and visualize the spectral value sets R (A; D; E; ); > for an arbitrary number ` of disturbance inputs (see [] for the case ` = ). Note that in the special case ` = ; q =, if the triplet (A; D; E) is controllable and observable, the spectral value set R (A; D; E; ) is exactly the root locus

4 of the transfer function G(s) = E(sI? A)? D = p(s)=q(s) under output feedback with gain k (?; ) R (A; D; E; ) = f C ; 9k (?; ) : q() + kp() = g: (6) Hence R (A; D; E; ) can be viewed as an extension of the root locus to multivariable systems. Let P (; s) (reps. i (; s)) denote the matrix P () (resp. the i-th singular value i () of P ()) obtained by setting X = G R (s); Y = G I (s) in (). The function f : C n (A)! R ; s 7! inf ( (;] G R (s)? G I (s)?g I (s) G R (s) ) (7) is upper semi-continuous on C n(a). If max scn(a) rk G I (s) (respectively max sc n(a) rk G I (s) = ), f is continuous on the open set C n ((A) [ R G ) where R G = fs C n (A); rk G I (s) g; ( resp. R G = fs C n (A); G I (s) = g): The following theorem extends Proposition.4 of [] to arbitrary ` and provides the basis for a visualization of R (A; D; E; ) by programs of the MATLAB package. Because of space limitations we omit the details. Theorem. (i) For every >, R (A; D; E; ) is a bounded subset of C, R (A; D; E; )n ((A) [ R G ) is open and R (A; D; E; ) = (A) _[ fs C n (A); f(s) >? g: (8) (ii) The closure of R (A; D; E; ) in C is given by cl( R (A; D; E; )) = (A) [ fs C n (A); f(s)? g: (9) (iii) For every s C := fs C n (A); f(s) =? g there exists a perturbation R`q such that kk = and s (A+DE), and there is no smaller disturbance matrix with the latter property. (iv) R (A; D; E; ) is the union of (A), R G () := fs R G ; f(s) >? g and the connected components of C n cl(c [ R G ) which contain a point s satisfying f(s) >?. Those components of C n cl(c [ R G ) which contain a point s satisfying f(s) <? are disjoint from R (A; D; E; ). References [] S. K. Godunov, O. P. Kiriljuk, and W. I. Kostin. Spectral portraits of matrices. (Preprint, AN USSR, Sib. Branch, Inst. of mathematics, No. ), Novosibirsk, 99. (Russian). [] D. Hinrichsen and B. Kelb. Spectral value sets: a graphical tool for robustness analysis. Syst. Contr. Let., :7{6, 99. [] D. Hinrichsen and A. J. Pritchard. New robustness results for linear systems under real perturbations. In Proc. 7th IEEE Conference on Decision and Control, pages 75{79, Austin, Texas,

5 [4] D. Hinrichsen and A. J. Pritchard. Real and complex stability radii: a survey. In D. Hinrichsen and B. Martensson, editors, Control of Uncertain Systems, volume 6 of Progress in System and Control Theory, pages 9{6, Basel, 99. Birkhauser. [5] D. Hinrichsen and A. J. Pritchard. On spectral variations under bounded real matrix perturbations. Numerische Mathematik, 6:59{54, 99. [6] L. Qui and E. J. Davison. On the Real Structured Stability Radius. To appear in Proc. IFAC, 99. [7] L. Qiu and B. Bernhardsson and A. Rantzer and E.J. Davison and P.M. Young and J.C. Doyle. A formula for computation of the real stability radius. Report Nr. 6, Institute for Mathematics and its Applications, Minneapolis 99. [8] L. N. Trefethen. Pseudospectra of matrices. In D. F. Griths and G. A. Watson, editors, Numerical Analysis, volume 9, pages 4{66. Longmann, 99. [9] L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll. A new direction in hydrodynamic stability: beyond eigenvalues. Technical report, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, USA, 99. 5