where m r, m c and m C are the number of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. A. (D; G)-

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1 1 Some properties of an upper bound for Gjerrit Meinsma, Yash Shrivastava and Minyue Fu Abstract A convex upper bound of the mixed structured singular value is analyzed. The upper bound is based on a multiplier method. It is simple, it can exploit low-rank properties and it is shown to be less conservative than the well-known (D; G)-scaling. A direct relationship with (D; G)- scaling is given. The upper bound can be modied to one that is continuous with an explicit Lipschitz constant. Keywords Mixed structured singular values, linear matrix inequalities, multipliers. I. Introduction One of the nagging problems of robust control is to determine whether or not a system remains stable and retains satisfactory performance qualities under variations and uncertainties of some sort. In an attempt to get a handle on the problem, Doyle [8] introduced in 198 the structured singular value. The denition of the structured singular value, or for short, is such that robust stability=robust performance of the control problem is equivalent to being less than 1. Originally was dened for problems where the performance is measured by an H 1 -norm and the uncertainties come in the form of structured complex matrices, such as due to neglected dynamics. It was soon realized, however, that the denition could be extended to handle real parametric uncertainties as well. This extension of is known as the mixed structured singular value [9]. Unfortunately, in its full generality the test \ < 1?" is N-hard [5], [17], and, as such any attempt to develop polynomial-time algorithms for the general case seems fruitless. At present ecient algorithms work for special cases only, or, if they do work for the general case then they provide only sucient conditions (see Young, Newlin and Doyle [0] and the survey papers by ackard and Doyle [16] and Barmish and Kang []). The role of upper bounds of is that they provide such sucient conditions. Of the computable upper bounds of that have been reported the best at present that can handle the general problem are those by Fan, Tits and Doyle [9] and Fu and Barabanov [11]. In this note we examine an upper bound of based on what is called the multiplier method. The idea of the multiplier method is that nonsingularity of a family of matrices T is ensured if for some matrix C we have that CT + T C > 0 for all T T. Note that the condition CT + T C > 0 is convex in the variable C. The idea is not really new, although its application for the general problem was only recently realized by Fu and Department of Electrical and Computer Engineering The University of Newcastle, University Drive, Callaghan N.S.W. 308, Australia. hone , Fax , meinsma@pascal.newcastle.edu.au Barabanov [11]. The multiplier approach is extremely simple and, as Fu and Barabanov showed, it is readily translated using the S-procedure into linear matrix inequalities (LMIs) and generalized eigenvalue problems (GEVs). LMIs and GEVs are nowadays ecient to solve (see Nesterov and Nemirovsky [15] and Boyd et.al. [4]). The simplicity of the multiplier method may give the impression that the upper bounds of derived from it are conservative. Interestingly, however, the best advocated upper bound of by Fan, Tits and Doyle [9] can be seen as an example of the multiplier method and turns out to be more conservative than or, at best, equal to any of the upper bounds derived in [11] from the multiplier method. We collect in this note some general properties of the multiplier method and the upper bound of it induces. Section II introduces notation and some well established results. Section III discusses the multiplier method. Firstly we show that the upper bound of Fan, Tits and Doyle [9] is more conservative than the general multiplier method. This fact was realized already by Fu and Barabanov [11], but here we provide an explicit relationship. In Section IV we show that many variations of the multiplier method are equivalent. This result allows us to exploit low-rank conditions to reduce the computational cost without any conservatism. We briey comment on the lack of continuity of and its upper bounds, and how the multiplier method may be modied to recover continuity. Owing to the simplicity of the method we obtain Lipschitz continuity with an explicit Lipschitz constant. II. reliminaries In this section we summarize material that we need later. The mixed structured singular value is dened and we briey review (D; G)-scaling and the generalized eigenvalue problem. The norm kt k of a matrix T is in this note the spectral norm. T is the complex conjugate transpose of T and the Hermitian part He T of T is dened as He T = 1 (T + T ) Given a subset X of C nn we use BX to denote the unit ball in X BX = f X kk 1 g Given X the mixed structured singular value of M, denoted by X(M), is dened as the inmal value of > 0 for which all elements of the set I? 1 B XM are nonsingular. Obviously X(M) depends on the \structure" X. Invariably, the structures considered are of the form X = diag (RI k1 ; ; RI kmr ; (1) C I l1 ; ; C I lmc ; C f1 f1 ; ; C fm C fm C );

2 where m r, m c and m C are the number of repeated real scalar blocks, repeated complex scalar blocks and full complex blocks, respectively. A. (D; G)-scaling, LMIs and GEVs Let H q denote the set of q q Hermitian matrices and denote its subset of positive denite elements by q. Given the structure X of (1), the sets DX and GX are dened as DX = diag ( k1 ; ; kmr ; l1 ; ; lmc ; I f1 ; ; I fmc ); GX = diag (H k1 ; ; H kmr ; 0 l1l 1 ; ; 0 lmc l mc ; 0 f1f 1 ; ; 0 fmc f mc ) Note that for all X, D DX and G GX we have that D = D; D 1= = D 1= ; G = G = G Given M we dene (D; G) as (D; G) = M DM + j(gm? M G)? D This notation is a bit dierent from that of [9]. Fan, Tits and Doyle [9] showed that X(M) < if > 0 and (D; G) < 0 for some D DX and G GX. So, in particular the that solves the constrained minimization problem minimize subject to (D; G) < 0; > 0, D DX, G GX () is an upper bound of X(M). This minimum is denoted as X(M). The minimization problem () is an example of a generalized eigenvalue problem (GEV). A GEV is a minimization problem of the form minimize R;xR qp subject to B(x)? A(x) > 0; B(x) > 0; E(x) > 0 where A(x), B(x) and E(x) are square symmetric matrices depending anely on the variable x R qp. For GEVs ecient polynomial-time algorithms exist [15], [4], [1]. A linear matrix inequality (LMI) in the variable x R pq is an inequality E(x) > 0 where E(x) is a square symmetric matrix ane in x. III. The multiplier method In this section we examine an upper bound of X that relies on the multiplier method as developed by Fu and Barabanov [11]. In particular we provide a connection with (D; G)-scaling. At the basis of the multiplier method lies the trivial observation that a square matrix T is nonsingular if there is a square matrix C such that He CT > 0. This is easy to see If T is singular with T v = 0, v 6= 0 we get for any C that v (CT + T C )v = 0, contradicting that CT + T C = He CT > 0. The matrix C is called multiplier because of its links with the opov multiplier, although it is not quite the same as that of opov [18]. opov style multipliers, in the modern setting, appear in the work of Safonov and others see [7], [14] and they are in way equivalent to (D; G)-scaling (see [1]). The approach we use here can be traced back to an idea by Brockett and Willems [6] from The following result is immediate. Theorem III.1 X(M) < for an n n matrix M if there is a single multiplier C C nn such that He C(I n? 1 B XM) > 0 (3) By (3) is meant that all elements of the set C(I n? 1 B XM) have positive denite Hermitian part. If C satises (3) we say that C is a feasible multiplier for the set I n? 1 B XM. The inmal for which such a feasible multiplier C can be found is thus an upper bound of X(M). For later reference we call this upper bound bx(m) bx(m) = inff > 0 9C s.t. He C(I n? 1 B XM) > 0 g (4) Even though the inequality (3) is convex in C, computation of bx is generally not possible due to the innite family 1 B X. There is an exception. If the structure X has real components only, that is, X = diag (RI k1 ; ; RI kmr ); (5) then by convexity (3) holds for multiplier C i the nitely many LMIs He C(I n? M) > 0 8 = 1 diag (I k 1 ; ; I kmr ); (6) are satised. Minimizing > 0 over all C subject to (6) is a GEV. Thus it seems that computation of bx is well suited for the real case. This is in contrast with (D; G)- scaling which is primarily useful for the non-real case. In any case, real or not, the following holds. Theorem III. For every structure X and matrix M we have X(M) bx(m) X(M) In particular, if (D; G) < 0 with D DX, G GX then C = D + jm G satises (3). roof Let C = D + jm G. It is readily veried that for every X we have He C(I? M) = C(I? M) + (I? M) C =? (D; G) + M D 1= (I? )D 1= M (7) + (D 1=? D 1= M) (D 1=? D 1= M)

3 3 Therefore He C(I? M) > 0 for all 1 B X if (D; G) < 0. In (7) we used the fact that for every X we have D 1= = D 1= and G = G. So (D; G)-scaling can be interpreted as being a restrictive special case of the multiplier method. On the other hand, while X is ecient to compute, computation of bx is generally possible only for real-valued structures (5) and even then care has to be taken since the computational cost grows exponentially with the number of real blocks m r. In [11] it is explained how to combine the advantages of the multiplier method with the advantages of (D; G)- scaling. Loosely speaking the idea is to apply the multiplier method on the rst, say, 5 real blocks, and to apply (D; G)- scaling on the remaining real blocks (and complex blocks). As shown in [11] these combined computable upper bounds of X are less conservative than X. It remains to be seen how these upper bounds compare to (D; G)-scaling in combination with a gridding of the real parameter space, which is an alternative method to reduce conservatism of the upper bound X. We end this section with an example that shows that bx is generally strictly less than X. Example III.3 We derive in this example the values of X(M), X(M) and bx(m) for the case that X = R 0 0 R and M = 0 1 j 0 For all = diag (1; ) X the determinant det(i?m) equals det 1? 1 = 1? j1?j 1 This is nonzero irrespective of the values of 1 R and R, hence X(M) = 0. For this example DX is the set of diagonal positive denite matrices, and GX is the set of real diagonal matrices. With D DX and G GX written as D = diag (d1; d) and G = diag (g1; g) we get that (D; G) = M DM + j(gm? M G)? D d =? d1 jg1? g?jg1? g d1? (8) d Based on the diagonal elements of (8) it can be seen that (D; G) < 0 for some D DX only if > 1. On the other hand, (8) is negative denite for every > 1 if we choose d1 = d = 1 and g1 = g = 0. Hence X(M) = 1. For C = I and = diag (1; ) X we have that He C(I? M) = 1 (j? 1)= (?j? 1)= 1 This is positive denite for all real i satisfying j i j < p. Hence bx(m) 1= p. A further technical analysis shows that bx(m) is in fact equal to 1= p. So, for this example none of the bounds are the same 0 = X(M) < bx(m) = 1= p < X(M) = 1 IV. Equivalent multipliers The multiplier method has many variations. For example, instead of trying to nd a multiplier C such that (3) holds, it might perhaps be advantageous to try to nd a scalar c for which He c det(i? M) > 0 for all 1 B X (9) If such a c exist then again X(M) <. In this section we discuss a few of these variations. We show that certain variations are equivalent in that they induce the same upper bound of X. The class of equivalent multiplier methods includes one that exploits low-rank properties of M, but it does not include the one in (9). Numerous robust stability problems require computation of X(M) in which the matrix M has low rank. In fact, M is often constructed from tall matrices and Q as M = Q ; with and Q readily given by the problem at hand Example IV.1 Let fa0; ; A m g be a collection of q q matrices and let A be any q q matrix of the form A = A0 + 1A1 + + m A m ;?1 < i < 1 It is well known that the uncertain system _x = Ax is asymptotically stable for all such matrices A i A0 has all its eigenvalues in the open left-half plane and for all frequencies! R we have that mx I q? i A i (j!i q?a0)?1 is nonsingular 8j i j < 1. (10) i=1 The above condition (10) is a low-rank problem. To see this we dene = diag (1I q ; ; m I q ) and we dene the tall matrices and Q! as = 6 4 I q. I q ; Q! = 4 A 1.. A m 3 5 (j!iq? A0)?1 m This way the matrix I q? i=1 ia i (j!i q? A0)?1 of (10) can be written as I q? Q!. By the determinant rule det(i? AB) = det(i? BA) it now follows that (10) holds for all frequencies i with respect to structure X = diag (RI q ; ; RI q ) we have X(Q! ) < 1 for all frequencies! R. The matrix M = Q! is qm qm but only has rank q. In general, robust nonsingularity of the n n set I n? 1 B XQ

4 4 is the same as robust nonsingularity of the \smaller" q q set I q? 1 B XQ The obvious question is whether or not the multiplier method applied to the smaller I q? 1 B XQ provides a more conservative upper bound of X(Q ) than bx(q ). They are equivalent Theorem IV. Let M = Q. The following four upper bounds of X(M) are the same. bx(m) = inff > 0 9C s.t. He C(I n? 1 B XM) > 0 g; bx(; Q) = inff > 0 9C s.t. He C (I q? 1 B XQ) > 0 g; bx3(; Q) = inff > 0 9C 3 s.t. He (I q? 1 B XQ)C 3 > 0 g; bx4(m) = inff > 0 9C 4 s.t. He (I n? 1 B XM)C 4 > 0 g roof We prove in four steps that bx bx = bx3 bx4 = bx, which shows they are all equal. (bx bx) Suppose > bx and that C satises He C(I q? 1 B XQ) > 0. Consider C = C + "EE with E and " yet to be determined. Then C(I n? M) = ( C + "EE )(I n? Q ) C = [ E ] (I q? Q) 0?"E Q "I Therefore its Hermitian part equals E He C(I n? M) = He C [ E ] (I q? Q)? " Q E? " E Q "I E (11) For any E for which [ E ] has full row rank there is a small enough " > 0 such that (11) is > 0 for all 1 B X. Hence bx <. Since this can be done for any > bx we have that bx bx. (bx = bx3) Take C3 = C?. (bx3 bx4) Suppose > bx4 and that C4 satises He (I n? 1 B XM)C4 > 0. Consider C3 = C4 + "E E with E and " yet to be determined. Then (I q? Q)C3 = (I q? Q)( C4 + "E E) = [ E (In? ] M)C4?"QE 0 "I E Therefore its Hermitian part equals He (I q? Q)C3 = [ E He (In? ] M)C4? " QE? " EQ "I E (1) For any E for which [ E ] has full row rank there is a small enough " > 0 such that (1) is > 0 for all 1 B X. Hence bx3 <. Since this can be done for any > bx4 we have that bx3 bx4. (bx4 = bx) Take C = C? 4. The equivalence of bx and bx4 shows that ipping the multiplier to the other side of the nonsingularity set has no eect on the upper bound it induces. A further variation of the multiplier method which is easily veried is that bx(qr?1 ) equals inff > 0 9C5 such that He C5(R? 1 B XQ) > 0 g Example IV.3 Consider Example IV.1. For a xed frequency!, the upper bound bx(q! ) which involves a qm qm multiplier is the same as the inmal > 0 for which a C5 C qq exists such that mx He C5[(j!I q? A0)? i A i ] > 0 for all i [? 1 ; 1 ]. i=1 The equivalence of the various multiplier upper bounds generally breaks down if I n?m or I q? Q is replaced with one that no longer depends anely on. For example, we know that I n?m is nonsingular i det(i n?m) is nonzero. However the existence of a multiplier C for I n? M satisfying (3) does not necessarily imply that a scalar c exists such that He c det(i? 1 B XM) > 0 (13) As an example, suppose we have R 0 X = and M = 0 R j 0 0 j Then X(M) = bx(m) = 0 because for any > 0 we have He (I? 1 B XM) = fig > 0. However, for, say, = 1 we can not nd a c that satises (13) because for the four s 1 0 = B X the determinant det(i? M) = (1 j)(1 j) takes the values?3 4j and 5 (twice) and these do not lie in one half-space f s + = > arg s >? = g. It is an important asset of the multiplier method that it can exploit the low rank properties of M. The reduction in size when going from the n n matrix I n? M to the equivalent q q matrix I q? Q is most extreme if M = Q has rank 1. Lemma IV.4 bx(m) = X(M) if M has rank 1. roof Let and Q be column vectors such that M = Q. By Theorem IV. we have that bx(m) = bx(; Q). Suppose > X(M). Then by denition of X all members of 1? ( 1 B X)Q are nonzero. Since 1? ( 1 B X)Q is convex it must, therefore, lie in some half-space f s? = < arg s < + =g. Then He e?j (1? 1 B XQ) = Re e?j (1? 1 B XQ) > 0; so that bx(; Q) <. Since this holds for any > X(M) the result follows.

5 5 Young [19] showed that also (D; G)-scaling is exact for rank 1 matrices M. The multiplier method is also exact if X = C nn in which case we can simply take the multiplier to be the identity. V. Continuity In typical robust stability applications the nonsingularity property that needs to be tested is I? 1 B XH(j!) is nonsingular for all! R [ 1? (14) In other words, checking at each frequency! whether or not X(H(j!)) <. This is usually impossible and one has to rely on a nite frequency grid. The problem however is that X and also bx and X are discontinuous which greatly impairs the use of gridding. Example V.1 Suppose X = R. If x is a real then X(x) = jxj. X is discontinuous because for every (arbitrarily small) y 6= 0 we have that X(x + jy) = 0. Since X = bx = X for this case (See Lemma IV.4) also bx and X are discontinuous. Generally X(H(j!)) and its upper bounds are discontinuous at some isolated frequencies only. An even more severe discontinuity problem that can occur is that the in- mal for which (14) holds true may be discontinuous as a function of H (measured in, say, the H 1 norm). This problem was rst demonstrated in [10] and subsequently rigorously studied in [3]. The multiplier method upper bound can be adjusted at the expense of worsening the bound to one that is continuous Lemma V. Let be some (small) positive number < 1. If the multiplier C in (4) is restricted to the convex set kck < 1 and He C > I; (15) then C has condition number less than 1= and bx is continuous jbx(m + E)? bx(m)j 1 kek (16) The proof is trivial. Note that the condition kck < 1 alone has no eect on the upper bound bx(m) since the multiplier may be scaled to be like that. Secondly note that any feasible multiplier C is nonsingular and satises He C > 0. However as the bound approaches its optimal value bx(m) the associated feasible multiplier C may become ill-conditioned in the sense that He C approaches a singular matrix. An advantage of the continuity condition (16) over that of Lee and Tits [13] is that (16) is a Lipschitz continuity condition with known Lipschitz constant 1=. Example V.3 Suppose the multiplier C is restricted to (15) for some 0 < < 1. For X = R it may be veried that p 1? bx(x + jy) = maxf0; jxj? jyj g, (cf. Example V.1.) The LMI (15) is easily incorporated in the convex upper bounds of Fan, Tits and Doyle [9] and Fu and Barabanov [11] as all of them generate a feasible multiplier C. (The condition (15) is also an LMI in (D; G) if we use C = D + jm G.) References [1] V. Balakrishnan, Y. Huang, A. ackard, and J. Doyle. Linear matrix inequalities in analysis with multipliers. In roc. American Control Conference, pages 18{13, [] B. R. Barmish and H. I. Kang. A survey of extreme point results for robustness of control systems. Automatica, 913{35, [3] B. R. Barmish,.. Khargonekar, Z. C. Shi, and R. Tempo. Robustness margin need not be a continuous function of the problem data. Systems & Control Letters, 1591{98, [4] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, hiladelphia, [5] R. D. Braatz,. M. Young, J. C. Doyle, and M. Morari. Computational complexity of calculation. IEEE Trans. Aut. Control, 39(5)1000{100, [6] R.W. Brockett and J.C. Willems. Frequency domain stability criteria{part I. IEEE Trans. Aut. Control, 1055{71, [7] R. Y. Chiang and M. G. Safonov. User's Guide, Robust Control Toolbox. The MathWorks, Natick, Mass., USA, 199. [8] J. C. Doyle. Analysis of feedback systems with structured uncertainties. IEE roc., art D, 194{50, 198. [9] M. Fan, A. Tits, and J. Doyle. Robustness in the presence of joint parametric uncertainty and unmodeled dynamics. IEEE Trans. Aut. Control, 36(1)5{38, [10] M. Fu. olytopes of polynomials with zeros in a prescribed region. In M. Milanese, R. Tempo, and A. Vicino, editors, Robustness in Identication and Control. lenum ress, [11] M. Fu and N. E. Barabanov. Improved upper bounds of the mixed structured singular value. Technical Report EE9461, University of Newcastle, Australia, Also appeared in roc. IEEE Conf. Decison and Control, New Orleans, LA, December, 1995, pp. 3115{310. [1]. Gahinet and A. Nemirovsky. LMI Lab a package for manipulating and solving LMIs, reprint. [13] L. Lee and A. Tits. On continuity / discontinuity in robustness indicators. IEEE Trans. Aut. Control, 38(10)1551{1553, [14] J.H. Ly, M.G. Safonov, and R.Y. Chiang. Real/complex multivariable stability margin computation via generalized opov multiplier{lmi approach. In roc. American Control Conference, pages 45{49, [15] Y. Nesterov and A. Nemirovsky. Interior oint olynomial Methods in Convex rogramming. SIAM, hiladelphia, [16] A. ackard and J. Doyle. The complex structured singular value. Automatica, 9(1)71{109, [17] S. oljak and J. Rohn. Checking robust nonsingularity is nphard. Math. Control, Signals and Systems, 61{9, [18] V.M. opov. Absolute stability of nonlinear systems of automatic control. Automat. Remote Control, 857{875, 196. [19]. M. Young. The rank one mixed problem and 'Kharitonovtype' analysis. Automatica, 30(1)1899{1911, [0]. M. Young, M.. Newlin, and J. C. Doyle. ractical computation of the mixed problem. In roc. American Control Conference, pages 190{194, 199.

Rank-one LMIs and Lyapunov's Inequality. Gjerrit Meinsma 4. Abstract. We describe a new proof of the well-known Lyapunov's matrix inequality about

Rank-one LMIs and Lyapunov's Inequality Didier Henrion 1;; Gjerrit Meinsma Abstract We describe a new proof of the well-known Lyapunov's matrix inequality about the location of the eigenvalues of a matrix