Faculty of Mathematical Sciences University of Twente University for Technical and Social Sciences P.O. Box 17 75 AE Enschede The Netherlands Phone: +31-53-48934 Fax: +31-53-4893114 Email: memo@math.utwente.nl Memorandum No. 1471 On stability robustness with respect to LTV uncertainties G. Meinsma, T. Iwasaki 1 and M. Fu November 1998 ISSN 169-69 1 Dept. of Control Systems Engineering, Tokyo Institute of Technology, -1-1 Oookayama, Meguro, Tokyo 15, Japan Dept. of Electrical and Computer Engineering, University of Newcastle, Callaghan, NSW 38, Australia
On stability robustness with respect to LTV uncertainties Gjerrit Meinsma Department of Systems, Signals and Control Faculty of Mathematical Sciences University of Twente P.O. Box 17, 75 AE Enschede The Netherlands g.meinsma@math.utwente.nl Tetsuya Iwasaki Department of Control Systems Engineering Tokyo Institute of Technology -1-1 Oookayama, Meguro, Tokyo 15 Japan iwasaki@ctrl.titech.ac.jp Minyue Fu Department of Electrical and Computer Engineering University of Newcastle Callaghan, NSW 38 Australia. eemf@ee.newcastle.edu.au Abstract It is shown that the well-known.d; G/-scaling upper bound of the structured singular value is a nonconservative test for robust stability with respect to certain linear time-varying uncertainties. Keywords Mixed structured singular values, Duality, Linear matrix inequalities, Time-varying systems, Robustness, IQC. 1991 Mathematics Subject Classification: 93B36, 93C5, 93D9, 93D5. 1
1 Introduction v + 1 + y u M + + v 1 Figure 1: The closed loop. Is the above closed loop stable for all 1 s in a given set of stable operators B? That, roughly, is the fundamental robust stability problem. There is an intriguing result by Megretski and Treil [4] and Shamma [8] which says, loosely speaking, that if M is a stable LTI operator and the set of 1 s is the set of contractive linear time-varying operators of some fixed block diagonal structure 1 = diag.1 1 ;1 ;:::;1 mf /; (1) that then the closed loop is robustly stable that is, stable for all such 1 s if and only if the H -norm of DMD 1 is less than one for some constant diagonal matrix D that commutes with the 1 s. The problem can be decided in polynomial time, and it is a problem that has since long been associated with an upper bound of the structured singular value. The intriguing part is that the result holds for any number of LTV blocks 1 i, which is in stark contrast with the case that the 1 i s are assumed time-invariant. Paganini [6] extended this result by allowing for the more general block diagonal structure 1 = diag.ž 1 I n1 ;:::;Ž mc I nmc ;1 1 ;:::;1 mf /: () A precise definition is given in Section. Paganini s result is an exact generalization and leads, again, to a convex optimization problem over the constant matrices D that commute with 1. In view of the connection of these results with the upper bounds of the structured singular it is natural to ask if the well known.d; G/-scaling upper bound of the mixed structured singular value also has a similar interpretation. In this note we show that that is indeed the case. The.D; G/-scaling upper bound of the structured singular value was originally defined as a means to provide an easy-to-verify condition that guarantees robust stability with respect to the contractive linear time-invariant operators 1 of the form 1 = diag. Ž 1 Iñ1 ;:::; Ž mr Iñmr ;Ž 1 I n1 ;:::;Ž mc I nmc ;1 1 ;:::;1 mf /; (3) with Ž i denoting real-valued constants [1]. It is known that for general LTI plants M this sufficient condition is necessary as well if and only if,.m r + m c / + m F 3:
(See [5].) In this note we show that the.d; G/-scaling condition is in fact both necessary and sufficient for robust stability with respect to the contractive LTV operators 1 of the form (3) with now Ž i denoting linear time-varying self-adjoint operators on `. A precise definition follows. Paganini [7] has gone through considerable trouble to show that for his structure () one may assume causality of 1 without changing the condition. In the extended structure (3) with self-adjoint Ž i this is no longer possible. Notation and preliminaries ` := {x : Z R : k Z x.k/ < }. The norm v of v ` is the usual norm on ` and for vector-valued signals v `n the norm v is defined as. v 1 + + v n /1=. The induced norm is denoted by. So, for F : `n `n it is defined as F := sup u `n Fu = u. For matrices F C n m the induced norm will be the spectral norm, and for vectors this reduces to the Euclidean norm. F H is the complex conjugate transpose of F, andhef is the Hermitian part F defined as He F = 1.F + FH /. An operator 1 : `n `n is said to be contractive if 1v v for every v `n. Lower case Ž s always denote operators from `1 to `1. Then for u; y `n the expression y = ŽI n u is defined to mean that the entries y k of y satisfy y k = Žu k. An operator Ž : ` ` is self-adjoint if u;žv = Žu;v for all u;v `. The M and 1 throughout denote bounded operators from `n to `n and M is assumed linear time invariant (LTI). Bounded operators on `n are also called stable. Hats will denote Z-transforms, so if y ` then ŷ.z/ is defined as ŷ.z/ = k Z y.k/z k. To avoid clutter we shall use for functions ˆf of frequency the notation ˆf! := ˆf.e i! /:.1 Stability The closed loop depicted in Fig. 1 is considered internally stable if the map from [ 1 ] [ v to uy ] is bounded as a map from l n to l n. Because of stability of M and 1 the closed loop is internally stable iff.i 1M/ 1 is bounded. The closed loop will be called uniformly robustly stable with respect to some set B of stable LTV operators if there is an >such that [ [ [ ] u v1 v1 y] 1 B ; v ] `n : (4) v We only consider 1 s with norm at most one and stable M. In that case (4) holds if and only if there is an ž> such that.i 1M/u ž u 1 B ; u `n :. The 1 s and the.d; G/-scaling matrices Throughout we assume that 1 : `n `n and that 1 is of the form 1 = diag. Ž 1 Iñ1 ;:::; Ž mr Iñmr ;Ž 1 I n1 ;:::;Ž mc I nmc ;1 1 ;:::;1 mf / (5) 3
with Ž i : ` ` LTV, self-adjoint and Ž i 1; Ž i : ` ` LTV and Ž i 1; 1 i : `qi `qi LTV and 1 i 1. (6) The dimensions and numbers ñ i ; n i ; q i ; m r ; m c ; m F of the various identity matrices and 1 i blocks are fixed, but otherwise 1 may vary over all possible n n LTV operators of the form (5), (6). Given that, the sets D and G of D and G-scales are defined accordingly as D = {D = diag. D 1 ;:::; D mr ; D 1 ;:::;D mc ; d 1 I q1 ;:::;d mf I qmf / :< D = D T R n n }; G = {G = diag. G 1 ;:::; G mr ; ;:::;; ;:::;/ : G = G H jr n n }: Note that the D-scales are assumed real-valued and that the G-scales are taken to be purely imaginary. As it turns out there is no need to consider a wider class of D and G-scales. 3 The discrete-time result Theorem 3.1. The discrete-time closed-loop in Fig. 1 with stable LTI plant with transfer matrix M is uniformly robustly stable with respect to 1 s of the form (5, 6) if and only if there is a constant matrix D D and a constant matrix G G such that M H! DM! + j.gm! M H! G/ D <! [; ³]: (7) The existence of such D and G can be tested in polynomial time. The remainder of this paper is devoted to a proof of this result. Megretski [3] showed this for the full blocks case (1), Paganini [6] derived this result for the case that the 1 s are of the form (). The proof of the general case (5) follows the same lines as that of [6] and [5]. A key idea is to replace the condition of the contractive 1-blocks with an integral quadratic condition independent of 1: Lemma 3.. Let u; y `q and consider the quadratic integral 6.u; y/ := ³ The following holds..ŷ! û! /.ŷ! +û! / H d! R q q : (8) 1. There is a contractive self-adjoint LTV Ž : ` ` such that u = ŽI q y if and only if 6.u; y/ is Hermitian and nonnegative definite.. There is a contractive LTV Ž : ` ` such that u = ŽI q y if and only if the Hermitian part of 6.u; y/ is nonnegative definite. 3. There is a contractive LTV 1 : `q 6.u; y/ is nonnegative. `q such that u = 1y if and only if the trace of 4
Proof. See appendix. A consequence of this result is the following. Lemma 3.3. Let u be a nonzero element of `n. Then.I 1M/u = for some 1 of the form (5, 6) if-and-only-if 6.u; Mu/ := ³.M! I/û! û H!.M! + I/ H d! (9) is of the form Z 1????????? Z???????.??..????????? Z 1????????? Z???? R n n ; (1).?????..????????? Z 1????????? Z?.????????.. with Z i = Z i T, He Z i, Tr Z i, and with? denoting an irrelevant entry. Here the partitioning of (1) is compatible with that of 1. Proof (sketch). The equation.i 1M/u = isthesameas u = 1Mu: With appropriate partitionings, the expression u = 1Mu can be written row-block by rowblock as u 1 = Ž 1 M 1 u u = Ž M u... u K = 1 mf M K u: By Lemma 3. there exist contractive Ž i, Ž i and 1 i of the form (6) for which the above equalities hold iff certain quadratic integrals 6 i have certain properties. It is not to difficult to figure out that these quadratic integrals 6 i are exactly the blocks on the diagonal of 6.u; Mu/, and that the conditions on these blocks are that they satisfy 6 i = 6 T i, He 6 i, or Tr 6 i, corresponding to the three types of uncertainties. Proof of Theorem 3.1. Suppose such D D and G G exist. Then a standard argument will show that there is an ž>suchthat.i 1M/u ž u for all u and contractive 1 of the form (5). This is the definition of uniformly robustly stable. 5
Conversely suppose the closed loop is uniformly robustly stable. For some ž>, then,.i 1M/u ž for every u of unit norm. Define W := {6.u; Mu/ : u = 1} R n n : (11) By application of Lemma 3.3, the set W does not intersect the convex cone Z defined as Z :={Z : Z is of the form (1) with Z i = Z T i, He Z i, Tr Z i }: In the appendix we show that in fact W is bounded away from Z. Remarkably the closure W of W is convex. This observation is from Megretski & Treil [4], and for completeness a proof is listed in the appendix, Lemma 5.1. Because W is bounded away from Z, also the closure W is bounded away from Z, so there is a > such that W also does not intersect Z := Z +{Z R n n : Z }: Both W and Z are convex and have empty intersection, and therefore a hyper-plane exists that separates the two sets [, p.133]. In other words there is a nonzero matrix E R n n (say of unit norm) such that 1 E;W E; Z : (1) As inner product take X; Y =Tr X T Y. In particular (1) says that E; Z is bounded from below. By Lemma 5.3 that is the case if and only if E is of the form E = diag.ẽ 1 ;:::;Ẽ mr ; E 1 ;:::;E mc ; e 1 I;:::;e mf I/ with Ẽ i + Ẽ T i, E i = E T i and e i R, that is, if and only if E D + jg. Inthat case inf E; Z =, and so a := inf E; Z < : From (1) we thus see that E;W a <. If u = 1, then ³ û H!( He.M! + I/ H E.M! I/ ) û! d! = Re Tr ³ E.M! I/û! û H!.M! + I/ H d! = E;6.u; Mu/ sup E;W a < : (13) This being at most a < foreveryu `n ; u = 1 implies that He.M! + I/ H.E + ži/.m! I/ <! [; ³]; (14) for some small enough ž>. Express E + ži as E + ži = D + jg for some D D and G G. Then Equation (14) becomes (7). 1 In (1) the expression E;W denotes the set {x : x = E; Y ; Y W } and the inequality in (1) is defined to mean that every element of the set on the left-hand side, E;W, is less than or equal to every element of the set on the right-hand side, E;Z. 6
4 The continuous-time result Analagous to the discrete-time case we say that a continuous-time system is uniformly robustly stable if there is a > such that (4) holds for all v 1 ;v L. Completely analagous to the discrete-time case it can be shown that: Theorem 4.1. The continuous-time closed-loop in Fig. 1 with stable LTI plant with transfer matrix M is uniformly robustly stable with respect to 1 s of the form (5) with Ž i : L L LTV, self-adjoint and Ž i 1; Ž i : L L LTV and Ž i 1; 1 i : L q i Lq i LTV and 1 i 1. if and only if there is a constant matrix D D and a constant matrix G G such that M. j!/ H DM. j!/+ j.gm. j!/ M. j!/ H G/ D < for all! R. 5 Appendix Proof of Lemma 3.. Items and 3 are proved in [6] (note that the Hermitian part of (8) is ³ ŷ! ŷ H! û!û H! d!, and its trace equals ³. y u /). If u := ŽI q y with Ž self-adjoint and contractive then (8) is easily seen to be Hermitian and. Conversely suppose (8) is Hermitian and nonnegative. Now let { f i } i=;1;; be an orthonormal basis of `, and expand y `q in this basis: y =.j/ f j ;.j/ R q : j=;1;::: We may associate with this expansion the matrix Y R q of coefficients 1././ q./ 1.1/.1/ q.1/ Y = 1././ q./ :.... The matrix U is likewise defined from u. In this matrix notation the expression u = ŽI q y becomes U = 1Y, and the quadratic integral (8) becomes 6.u; y/ =.Y T U T /.Y + U/: By assumption the above is Hermitian and nonnegative definite, that is, Y T U = U T Y and U T U Y T Y: (15) We may assume without loss of generality that the orthonormal basis { f j } was chosen such that the first, say p, elements { f 1 ;:::; f p } span the space spanned by the entries {y 1 ;:::;y q } of y. ThenY is of the form [ ] Ip Y = C for some full row rank C R p q : p 7
Then the second inequality of (15) is that U T U C T C.ThisimpliesthatU is of the form U = VC for some V. Partition V as [ V 1 ] V with V1 R p p. The two formulas of (15) then become C T V 1 C = C T V T 1 C and CT.V T 1 V 1 + V T V /C C T I p C: (16) As C has full row rank, (16) is equivalent to that V 1 = V T 1 and V T 1 V 1 + V T V I p : It is now immediate that U equals U = 1Y for 1 defined as 1 := [ V1 V T V V V 1.I V 1 / 1 V T ] : (17) It is easy to verify that 1 is contractive. Furthermore 1 is symmetric and so the corresponding operator Ž is self-adjoint. (It may happen that I V1 is singular. In that case the inverse in (17) may be replaced with the Moore-Penrose inverse.) Lemma 5.1. The closure of (11) is convex. Proof. The proof hinges on the fact that lim N u; N v =for every pair u;v `n and with N denoting the N-step delay. Let u;v `n both have unit norm, i.e., 6.u; Mu/; 6.v; Mv/ W.GivenN Nand ½ [; 1] define x as x := ½u + 1 ½ N v: Since 6 is linear in its two arguments, we have that 6.x; Mx/ =½6.u; Mu/ + 1 ½ ½6.u; M N v/ + 1 ½ ½6. N v; Mu/ +.1 ½/6.v; Mv/: As N the contributions of 6.u; M N v/ and 6. N v; Mu/ tend to zero, so lim 6.x; Mx/ = ½6.u; Mu/ +.1 ½/6.v; Mv/: N That this is an element of the closure of (11) follows from the fact that. lim N x = ½ u +.1 ½/ v = 1. Lemma 5.. Uniform robust stability implies that W is bounded away from Z. Proof. Suppose to the contrary that inf 6.u; Mu/ Z =: u `n ; u =1;Z Z This means that there is a sequence {u k ; Q k } k N `n Rn n such that 6.u k ; Mu k / + Q k Z; u k = 1; lim k Q k =: 8
For each k define y k := Mu k `n and take zk to be any element of `n whose entries are mutually orthogonal and have unit norm, z k i ; zk j =Ž ij, and whose entries are also orthogonal to all entries of u k and y k. With it define ū k : = u k + 1. Q k I n ȳ k : = y k + 1. Q k I n + The reason for this definition is that now 6.ū k ; ȳ k / = ³.ŷ k! û k! + 1 Qk Q k/z k ; 1 Qk Q k/z k : 1 Qk Q kẑ k!/.ŷ k! +û k! + Q k ẑ k!/ H d! = 6.u k ; y k / + Q k Z: So we see that 6.ū k ; ȳ k / is an element of Z and, hence, ū k = 1 k ȳ k for some contractive 1 k of the form (5,6). Finally consider.i 1 k M/ū k =ū k 1 k M.u k +.ū k u k // =ū k 1 k.y k + M.ū k u k // =ū k 1 k.ȳ k +.y k ȳ k // 1 k M.ū k u k / = 1 k.y k ȳ k / 1 k M.ū k u k /: (18) Using the fact that ū k u k = O. Q k /, ȳ k y k = O. Q k / and that lim k Q k =, we obtain from (18) that lim.i k 1k M/ū k = ; lim k ūk = 1: This contradicts uniform robust stability. Lemma 5.3. inf Z Z Tr E T Z is bounded from below for some E R n n if and only if E is of the form E = diag.ẽ 1 ;:::;Ẽ mr ; E 1 ;:::;E mc ; e 1 I;::: ;e mf I/ with Ẽ i + Ẽ T i, E i = E T i and e i. Proof. Suppose that inf Z Z Tr E T Z is bounded from below. The off-diagonal blocks of E are then zero for the following reason: Let F be equal to E but with its blocks on the diagonal equal to zero. The off-diagonal blocks of Z Z are not restricted in any way so Z := ½F is an element of Z for every ½ R. IfF is nonzero then Tr E T Z = Tr E T.½F/ = ½Tr F T F and this is unbounded from below as a function of ½. Therefore F must be zero, i.e., E is block-diagonal. The general form of a block-diagonal E is E = diag.ẽ 1 ;:::;Ẽ mr ; E 1 ;:::;E mc ; Ē 1 ;:::;Ē mf / Express Z as in (1). Then Tr E T Z = Tr Ẽ T i Z i + Tr E T i Z i + Tr Ē T i Z i : 9
Each block of Z Z can vary independently of all other blocks of Z, so the only way that the above is bounded from below is that all inf Tr Ẽ Z i = Z T i T i Z i ; inf Tr Ei T He Z i Z i and inf Tr Z i Tr ĒT i Z i are bounded from below. It is fairly easy to show that inf Z i = Z T i Tr ẼT i Z i > He Ẽ i inf He Z i Tr ET i Z i > E i = Ei T inf Tr Zi Tr Ēi T Z i > Ē i = e i I; < e i R: (This is considered in more detail in [5].) Acknowledgement: The authors would like to thank Fernando Paganini for his emails. References [1] M. Fan, A. Tits, and J. Doyle. Robustness in the presence of joint parametric uncertainty and unmodeled dynamics. IEEE Trans. on Aut. Control, 36(1):5 38, 1991. [] D. G. Luenberger. Optimization by Vector Space Methods. John Wiley, New York, 1969. [3] A. Megretski. Necessary and sufficient conditions of stability: A multiloop generalization of the circle criterion. IEEE Trans. on Aut. Control, 38(5), 1993. [4] A. Megretski and S. Treil. Power distribution inequalities in optimization and robustness of uncertain systems. J. Math. Syst. Estimation and Control, 3(3):31 319, 1993. [5] G. Meinsma, Y. Shrivastava, and M. Fu. A dual formulation of mixed ¼ and on the losslessness of.d; G/-scaling. IEEE Trans. Aut. Control, 4(7):13 136, 1997. [6] F. Paganini. Analysis of implicitly defined systems. In Proceedings of the 33rd CDC, pages 3673 3678, 1994. [7] F. Paganini. Sets and Constraints in the Analysis of Uncertain Systems. PhD thesis, Caltech, USA, 1996. [8] J. Shamma. Robust stability with time-varying structured uncertainty. IEEE Transactions on Automatic Control, 39(4):714 74, 1994. 1