130 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991
Local-Global Double Algebras for Slow H" Adaptation: Part I-Inversion and Stability George Zames, Fellow, IEEE, and Le Y. Wang, Member, IEEE
Abstract-In this two-part paper, a common algebraic framework is persistent disturbances, say in I", have to' be considered. H" introduced for the frozen-time analysis of stability and H"-optimization interpolation theory is usually set in a Hilbert space and in that in slowly time-varying systems, based on the notion of a normed algebra setting is restricted to transient or periodic disturbances. One on which local and global products are defined. Relations between local stability, local (near) optimality, lotal coprime factorization, and global possibility is to abandon H" altogether and turn to I' kernel versions of these properties are sought. optimization, but then most spectral information is lost, together The framework is valid for time-domain disturbances in 1". H"-be- with the extensive qualitative insight which spectral data pro- havior is related to 1" input-output behavior via the device of an vides. The alternative chosen here is to relate H" frequency-do- approximate isometry between frequency and time-domain norms. Part I elaborates the double-algebra concept for Volterra operators main properties with I' kernel behavior via the device of an which approximately commute with the shift. The main algebraic prop- approximate isometry between certain frequency and exponen- erties and norm inequalities are summarized. Local conditions for global tially weighted time-domain norms. (The results obtained here invertibility are obtained. Clalssical frozen-time stability conditions are for I" input-output disturbances can be modified for the simpler incorporated in relations between local and global spectra. case of I* disturbances which have been treated elsewhere [26] .) Part I sets up the basic notions of a double algebra in the time I. INTRODUCTION and frequency domains, and applies them to deduce relationships between local and global stability for feedback systems with HIS two-part paper considers the problem of adaptive opti- stable components. Classical frozen-time stability results, includ- Tmization using H" and frozen-time methods, continuing the ing a Nyquist-type frequency-domain condition [13] and an work started in [22]. example of Desoer's [5], are reinterpreted in terms of a relation- The ability to adapt or learn from experience is limited to ship between local and global spectra. those aspects of data which persist or, at most, vary slowly with Part II is concerned with adaptive sensitivity reduction foi time. It seems worthwhile, therefore, to single out features of stable plants, based on local H" interpolation. An explicit optimization which are peculiar to slowly time-varying systems. ' (double) algebraic expression for adaptive design is obtained and There are results from the 1960's on the stability of such is applied to an example involving rejection of narrowband systems [13], [5], which were obtained using contraction and disturbances of uncertain frequency in which the local optimum monotonicity ideas and doing frozen-time analysis on a case by depends Lipschitz continuously on the data. Notions of adaptive case basis. In the course of trying to extend those ideas to H" and robust (nonadaptive) sensitivity minimization are intro- optimization, it became apparent that a common framework for duced, and adaptive minimization is shown to give better sensi- these problems, extracting their recurrent algebraic properties, is tivity than H" optimal robust minimization. possible. However, it is shown that, in peneral, H" optimal inter- The framework introduced here involves a space of input-out- polants do not depend Lipschitz continuously on data. On the put maps which approximately commute with the shift, and other hand, 6-suboptimal interpolants of the [ 11 central (maximal which are represented by Volterra operators in the time-domain entropy) type are shown to have a Lipschitz dependence on data, and "local" transfer functions in the frequency domain. Two and can serve as a basis for frozen-time design. prcducts are defined on this space, one global and the other Relations between local and global coprime-factorizations are local, creating the notion of a double algebra of operators. Local summarized in [25] and will be reported in a separate article. and global versions of properties such as spectrum, stability, Previous Results: Stability conditions for systems with slowly optimal sensitivity, and coprime factorization can be defined in time-varying parameters go back to the 1960'~-1970's. The terms of these two products, and relations established between notion of frozen-time stability in an input-output setting, using them, enabling global properties to be deduced from the more moving window averaging, was introduced in Freedman and easily computed local ones, in systems that vary slowly. Zames [13]. Closely related conditions in a state-space setting In a nontrivial theory of adaptive optimization, the effects of were introduced by Desoer and students, (see, e.g., [5]) and are described in [8]. Algebraic approaches to input-output feedback Manuscript received December 4, 1989; revised June 26, 1990. Paper go back to the same period, culminating in generalized recommended by Associate Editor, J. Hammer. coprime-factorization [7] and operator-norm sensitivity mini- G. Zames is with the Systems and Control Group, Department of Elec- trical Engineering, McGill University, 3480 University Street, Montreal, mization [29]. Feintuch and Francis obtained an early result on P.Q.,Canada H3A 2Al. optimization of time-varying systems based on Arveson's theory L. Y. Wang is with the Department of Electrical and Computer Engineer- [lo]. The sensitivity and well-posedness of H" interpolants ing, Wayne State University, Detroit, MI 48202. with respect to data is discussed in a recent paper by Smith [ 181, fEEE Log Number 9041413. from a different point of view. Despite the existence of more general theories of optimization for time-varying systems, e.g., [FFI, which have yet to be exploited to produce For contemporaneous approaches to time-varying systems, see controllers causally dependent on data. Khargonekar and Poolla, [16] who have introduced a variety of
0018-9286/91/0200-0130$01,00 0 1991 IEEE ZAMES AND WANG: LOCAL-GLOBAL DOUBLE ALGEBRAS: PART I 131 algebraic and optimization results, e.g., [16],1171; Verma 1191 Be.These spaces will be specialized to convolution operators on who has studied relations between robustness and coprime fac- I"( - 00, 00) as follows. torization; and Ioannou and Tsakalis [15]. Ball, Foias, Helton, Let A denote the Banach space of @"-valued functions and Tannenbaum extended optimal interpolation to nonlinear I"( - a,=).(Later, in Section HI, A will be equipped with systems, e.g., in [3]. Recently, Dahleh and Dahleh [6] have certain equivalent auxiliary norms.) Stable systems will belong started using the notion of frozen-time system introduced in to the Banach space E3 of bounded causal linear operators K: 1987 1221, [23]. A -+ A which have convolution sum representations Parts of this paper were included in [221-[251. t (Ku)(t) = k(t,7)~(7), tEZ (2.1) NOTATION r= -01 @, 2 denote the reals, complex numbers, and integers. The W, where the kernel k: Z2 + Gflxnis assumed, for each t E 2, to complex conjugate of any x E @ is 2. satisfy k(t, -)ell(-m,~);supteZI)k(t, * )1111 =: IIKII, < E" and Rnxndenote n-tuples and x matrices over a n n 00; and k(t, 7) = 0 whenever t c 7. The norm on B will be ring R. @" is viewed as a Euclidean space; for x E @" the I1 - IIW conjugate transpose is X* and norm I x I = (x*x)'/'. For a Unbounded systems will belong to a linear extension of B, matrix KEGnX", I K 1 is its largest singular value. denoted by El,, defined as follows: Let A' be the subspace of I,"[ a, b], 1 Ip I00, U 1 1, denotes the space of sequences functions of finite support in A u(t), t = a, a + l;.., b, t €2,either of vectors in @" or matrices in Gnx" for which A' := {UEA:u(t) = o for t > tu, t < t:}
where t;, tueZ depend on U. (A'), is a linear space of sequences whose truncations II,u lie in A' for each tE2. Then Be is the linear space of operators in (A'), which have sup Iu(t)Iu'coo for p = 00. convolution sum representations of the form (2.1). tau, bl B can be viewed as a subspace of Be modulo the following equivalence: to each K E E3 assign the unique bounded operator The dimension n will usually be suppressed in notation. Occa- K,eB, obtained by first restricting K from A down to A', sionally, it will be emphasized by the symbols (I,P[a,b])" or and then extending to (A'),; the map K + K, is an equivalence (I:[ a, b])"'". between bounded operators in B and Be. H,P, 1 Ip s 00, U 2 1, denotes the HP space either of @"-vector or GnX"-matrix functions K(z) on the disc 1 z I < U A. Local Systems for which K(.)ll := K(u(*))Hp H,P is viewed as a 1) Hg 11 11 . If an operator K Bechanges greatly over time, it can not be subspace of L,P, the space of LP functions of the circle of E approximated globally by a single time-invariant system. In- radius U. stead, frozen-time analysis employs a sequence of time-invariant F( U) denotes the discrete Fourier transform of any U E systems, K,, t E 2, each of which approximates a sIowly vary- 13- 0O,=) ing K in a neighborhood of a particular t. Coupling between the 01 K, complicates the global perturbation analysis of K. Our F(u)(z):= c u(t)z', Iz1 = 0. formalism seeks to bring out the main features of this analysis. t= -01 The approach here is to chose each time-invariant K, to 9(U) will also be represented by Fi. produce outputs which coincide with those of K at t E Z2 If K F-'(K)€1: denotes the inverse transform of any KEL', is any linear operator defined by a convolution sum defined for tcZ by 01 (KU)(~)= k(t,e)u(e), tez (2.2) e= --oo then the local system of K at ~EZis the (time-invariant) operator K, with the same domain as K satisfying Functions in will be denoted by lower case letters, in H," I,P m by capitals, and operators in either space by boldface capitals. (K,u)(t):= k(7,7 - (t - e))@), tEz II,, tc2 denotes the truncation operator which maps any e= --OD linear-space sequence f( 7), 7 E 2, into f,(7), where f,( 7) = i.e., the kernel of K, is a Toeplitz matrix determined by a f(7) for 7 5 t, f,(7) = 0 elsewhere. single row of the matrix k(7,e). The terms local and frozen-time II. LOCAL-GLOBALALGEBRAS IN THE TIME-DOMAIN will be used interchangeably. Freezing will be applied to operations as diverse as inversion, Although in general the notation K, will denote the local factorization, optimization, etc. If performed on a case-by-case system of K E Be at time 7 E Z, an exception will be made in basis, freezing can become extremely unwieldy. The alternative the case of II,, which denotes a truncation operator at 7. taken here will be to extract the common features of the various B. The Normed Double Algebra B freezing operations, albeit at the cost of some extra abstraction and notation at the outset. We define two products on the space Ele: 1) The usual The standard setup for input-output stability [28],[8] will be composition product, which will be called the global product, used. Stable systems will be represented by elements of a 'Other choices of K,,involving combinations of averaging and freezing normed linear space of causal bounded linear operators denoted are possible (see Remarks 2.1, 2.2). but this one seems to give the simplest by B. Unstable systems will lie in a (larger) "extended" space theory where the sole information about K is that it vanes slowly. 132 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991
and denoted explicitly by e, although that symbol, as usual, will Inversion Lemma 2.1 mostly be suppressed in notation, i.e., F K = FK; and 2) a a) If K E A has a local inverse K 0E A, then it has a global local product, denoted by 8 and defined as follows: for any inverse K- ' E A whenever either 1) )I K OVK (1 < 1, in which F,KEBe, F@ K is the unique operator in Be whose local case operators satisfy
(FmK),= F,K, VtEZ. (2.3) A double algebra is any subspace of Be which is equipped with and both products and is an algebra with respect to each one. In particular, the space Beequipped with both products is clearly a double algebra. Henceforth, the symbol Be will denote this double algebra. or 2) 11 K VKO11 < 1, in which case A normed double algebra (NDA) is a double algebra on which local and global norms, 11 - (1 and 11 (1 g, are defined satisfying the inequalities
IIFKIlg 5 IIFllgllKllg (2.4a) and
II 8 K II I 5 II FII Ill KII I. (2.4b) In particular, the space B equipped with both products, and with local and global norms taken to be equal to 11 llB is an NDA. b) Part a) remains valid under an interchange of global norms, In the following text assume B to be a NDA so equipped. products and inverses with their local counterparts. (In subsequent developments, the local norm may be distinct (For an extension of this result to certain open-norm algebras, from and superior to the global norm, i.e., 11 )I 5 11 * 11 I, or see Lemma 2.2.) 0 vice versa.) Proof: If A is any (normed) double algebra, its restriction to one of a) If KOis in A, each matrix k(t, t), t E H, has an inverse its products (and norms) will be denoted by the prefixes L for in Gnxn,where k is the kernel of K. Therefore, K-' exists in local and @ for global. LA and @A will be called the local Be. Furthermore, from the identities (normed) algebra and global (normed) algebra, respectively. If LA and @A are both complete normed algebras then A is a K 0K = I + KOK - KO8 K = z + K 0VK (2.8) Banach double algebra. B is an example. K E A has a local inverse in A, denoted by K 0,if K 0 is we get after multiplication by K- ' on the right, K- ' = K an inverse of K in LA, i.e., - (K VK)K-'. Subject to the norm characterization prop- erty P1 and causality of K, the usual "small gain" argument K~KO=KO ~K=I applied in the global algebra @A gives and a global inverse denoted by K- ' if K- ' is an inverse of K in @A. Similarly, any object defined in LA (in @A) will be I~,K-'III(1 - 11 KOVK~~}-'~I,K~~~ termed the local (global) object in A. Relations between local and global properties will depend 5 (1 - IIK~VKIIJ-~~~KOII,ta(2.9) upon the following. provided 11 K 0VK 11 < 1 in which case, since the bound (2.9) C. The Local-Global Coupling holds for all t E Z, K- ' is in A. Expressions (2.6a), (2.6b) now This is expressed by the product difference binary operator follow from (2.8), (2.9). The proof of (2.7a), (2.7b) is obtained
V: Be x Be+ Bedefined by similarly by multiplying KKo by K-' on the left. b) The proof remains valid under the specified interchange. 0 FVK = FK - F8K. (2.5) The condition that K VG is small will be related to the smallness of the commutant of G with the shift, i.e., to slow We seek a relation between local and global invertibility in a variation of the kernel in the time domain, or, in Section III, of normed double algebra A, as this determines stability. Observe the transfer function in the frequency domain. First, however, first, that KEB~has a global inverse in Be if and only if we summarize some elementary algebraic identities involving k( t, t) is invertible in Gnxn for each t E H (for then K decom- shift invariant and memoryless operators. poses into the sum of a memoryless invertible operator and a strictly causal one). Conditions for local invertibility in Be are D. Algebraic Identities identical to global ones. However, in a general normed double The ensuing identities are obtained more or less immediately algebra A this will not be true, and we get the following from the definition of 8 and V. inversion results. Let T E B denote the shift, (Tu)(t)= u(t - l), t E Z. An be Let A any normed double subalgebra of Be (not necessar- operator G E Beis shift-invariant if its commutant TG - GT ily complete) with global norm 11 * 11 subject to the norm (i.e., with the shift) vanishes. From the definition of 8 characterization property PI: KEA@ (~,K)EA VtEZ and supIln,K\\ < 00, K@T-T@K=O (2.10) t where {n,},,, is the family of truncation operators, e.g., B is for all K E Be, implying that all operators are locally shift-in- certainly so characterized by its norm )I 11 B. variant. The term shift invariant will be reserved for the global ZAMES AND WANG: LOCAL-GLOBAL WUBLE ALGEBRAS: PART I 133 property. If G is shift invariant then Equation (2.15) suggests that F VK is small whenever K has a small commutant with T and the memory of F decays quickly GK = K @ G, K VG = 0. (2.11) enough, motivating the following development. From these identities, the commutant of K is precisely the E. Slowly Time- Varying Systems and Exponential difference between the local and global products of K and the Memories Shift TK-KT=TVK. (2.12) The (time) variation-rateof K E Bein any given norm 11 11, denoted by dll.,,(K)is The simplicity of this relation of the V operator to the commu- tant is noteworthy. dll . 11 (K):= 1) KT - TK 11. (2.16) For any F, K, and shift-invariant G in Be K will be said to commute approximately with the shift if (KVF)G= KV(FG). (2.13) dll. K)< 11 K 11. Actually, dll.11( K) will also be required to be small in relation to certain other constants as yet to be specified, Let (All,), 7~2,denote the family of projection operators, and the term "approximately" is included mainly as a re- (All,) := ll, - II,-'. An operator KEBe has no memory if minder of intent. yd,( K) is obtained' (Anr)K = (AnT)K(An,) 9 (An estimate of the form 11 F VK (1 s whenever the kernel of F satisfies the summability assumption i.e., (All,)(/:) is an invariant subspace of K. The identities y := sup! E,"==,1 rf(t, t - r)I c OD. However, this kind of
(2.1l), (2.13) which were previously shown to hold for G shift summability is not preserved under both (a, @ ) products. In- invariant, also hold for G arbitrary if K has no memory. In the stead, we focus on the stronger assumption that operators have last case, if F is arbitrary exponentially decaying I' memories.) For any U > 1, let 1) I)((T) be the function from operators (KF)VG = K(FVG). K E Beto R defined by t Any FEB can be expressed a Volterra sum or equivalently as ~~K~~~,,)= sup 1 )k(t,7)~('-~)I (2.17) as a linear combination of powers of T t& r= -oa W F - F(')T' (2.14) where k is the kernel of K. Equation (2.17) equals r=O sup,,,))k(t, t - (*))ll,;. Let E,, be the subspace of E3 in which F(') E B, r = 1,2, * a, are operators without mem- 0ry3 whose kernels satisfy f(r)(t,7) = f(t,t - r) when 7 = t, f(r)(t,7) = 0 elsewhere. Series convergence here is weak 1' which is a Banach space under 11 - \I(,,) as norm. The variation convergence, defined as follows: rate of K in this norm is d,(K). Definition: A sequence of operators K, E B weakly (I1) Proposition 2.2: The Banach space E, under the products E E OD,~) converges to K B iff given any U lm(- and any ( *, 63 ) is a Banach double algebra. For any FEE ,,,, K E E ,, , functional f (with kernel) in Ii( - OD, m), f: I"( - OD,OD) -* @, U, > U the sequence f( K,u) converges to f( Ku). FVK can now be expressed as a linear combination of II F VK I1 (,,) 5 .-'(e In (U, /U)) - II FII (0,) du(K). (2.18) comrrutants T' VK ( = T'R - KT') with coefficients in B. Pro08 It is shown in Appendix I, proposition Al. 1 that E ,, Proposition 2.1: T VK I oincides with the commutant (2.12) is a normed algebra under the local and global products, and and m therefore a normed double algebra (with local and global norms F VK = F")( T' VK) (2.15) equal to 1) (I(,,)). From the Volterra sum representation of r= 1 operators in E,, by kernels in the complete space I:, it can be where F(') has no memory, and the series weakly (1') con- deduced that E,, is complete, and is therefore a Banach double verges in B. algebra. Proof: Equation (2.12) follows the identities T VK = TK From (2.13, (2.14), we get for the kernel g of P VK -T@K (by definition of V) and T@K=K@T=KT OD (from the definition of @ and the shift invariance of T). The g(t,r)u('-l) = 1 f(t,t - r)[k(t- 19 r) expression (2.14) for F gives r= -w FVK = CF("Tr1 K - LXF(')Tr 1 @ K As F(r)has no memory, (F(r)Tr)@ K = F(')(T'@ K); also, weak (1') convergence in B of a sequence G, implies that of G,, 60 K; therefore F VK = F("(TrK - T'@ K) r which implies (2.15). 0
3The expression (2.14) means that @ is a module spanned by powers of T. 4This coincides with the usual weak operator convergence provided the domain of operators in B is taken to be A' (i.e., modulo the equivalence mentioned in Section I), the dual-space of A' being /'(-OD, OD). 134 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36. NO. 2, FEBRUARY 1991
and, as suprzo r8-' I(elno)-' for 8 > 1, the conclusion where g(t, t) equals G, (i.e., the matrix representing the no (2.18) follows. 0 memory system G,). The choice p = i(l + v), U, = i(l + p-') and observation that In U, > 1 - ut1 gives a sufficient F. Frozen-Time Stability in the Time Domain e(1 - 6)' Most classical frozen-time stability conditions can be: encom- condition for stability, d G . If v=: 1- passed in a statement relating the existence of local and global (l)( ) 2dl, + .,B) inverses. Time-domain conditions are contained in the follow- 2~,the rate bound is better than ee2j4y. ing. Remark 2.1: It would appear that frozen-time analysis is distinguished from usual perturbation analysis in that the object Corollary 2.1: If G and K are in E,,, U, > 1, and either G has no memory or K is shift-invariant, then existence of the perturbed is not any particular operator but rather a method of local inverse (I+ G €3 K)0 in E ,,,implies that of the global computing products. Double algebras capture this fact. inverse (I+ GK)-' in B, provided that Remark 2.2: Our particular choice of local system K, seems natural if K is assumed to be slowly varying in the sense of d(,,(G@K) (elnU,)lI(I+ G@K)O having a commutant norm (2.16) near zero. Other choices of K, might be better under other assumptions. For example, consider a!)Z- @[(I - a!G@KK]))cA) (2.19) the map { *},: B -+ B, K + { K},, where the kernel k,(-) of
{ K}, is an average of the past of the kernel k( e, * ) of K, for some OLE W. namely Proof: The assumption that G is memoryless or K is shift invariant implies that G €3 K = GK and, by Inversion Lemma 2.1, Corollary 2.1 is true provided that k,(6) = X c(r-r)k(7, 7 - 6) r=-w II(I+G~K)OV(I+GBK)II,C 1. (2.20) where 0 cc 1, AX:, -,-,,{('-') = 1. The choice of local As I has no memory and is shift invariant, it is true, in general, system { K}, would seem more appropriate if the slow variation that A VB = (A - a!Z)V(B - I);therefore (2.20) is equiva- assumption were valid for the averaged operator as opposed to lent to the original K.Whatever the choice of local system, the local product (2.3) and double algebra framework would be pre- II[(I+GNK)O €3 ((1 - a!)~ served. -a!G@K)] V(G@K))I,< 1. G. Nests of NDA 's By (2.18), (2.19) is sufficient for (2.20). 0 Unfortunately, Corollary 2.1 involves the estimation of the In preparation for the frequency domain, let us axiomatically 1'-kernel norm of an inverse, which is seldom an analytically introduce a recurrent concept which is exemplified by the tractable object, and we will therefore emphasize alternative parametrized family {E,,} of double algebras. methods in the frequency domain. First, however, an example of Definition: A continuous nest of NDA's is a set {A,,} of Desoer [5], which is nicely tractable, is included to illustrate the nontrivial normed double subalgebras of E3 depending on a symbolism. parameter U, U 2 1, with the following properties: Example 2.1: Stability of the difference equation 1) { A,} is monotone by containment, i.e., A,, C A, C B if 1 5 U IU,, containment being strict whenever 1 # U # U,. ~(t)= G,x(t - 1) + F,u(t), tE2 (2.21) 2) For KEA,,, the local and global norms, 11 KllL and x(t), u(t)EW"; G,, ER"^", is tobededucedfromitslocal 11 K llf, depend continuously on U and are monotone in U, i.e., if 1 < U c U, then properties. If U E P(- 00, Q))and G,, F, are bounded functions of t E Z, (2.21) can be expressed in operator form 11 K 11 5 const. 11 K 11 ," Iconst. 11 K )I Eo x = GTx + FU for every combination of local/global norm superscripts (a, b, c) which has the solution XE(Z- CT)-'FU, where (I- GT)-' chosen from (g, I), the constants being independent of K. E Be. The a priori assumption that the frozen-time system is 3) Each NDA A, is characterized by the global norm 11 * 11 f exponentially stable means that (I- G €3 T)O is in E,, according to property P1 of Section 11-C. whenever 1 < U, < v-l, where v is the supremum of the spec- A NDA is continuously nested if it is in the nest or is a tral radii of the matrices G,, ten. The actual system is union of nest elements of the form A := U ,>ol{A,}. 1"-stable if the global inverse (I- GT)-' is in B which, by The NDA's in this paper all satisfy an additional inequality, Corollary 2.1 (a! = l), is ensured whenever the variation rate of linking local-norm rates of change to global behavior, which, G satisfies however, is not part of the nest definition. For U U, d(,,(G) (elnuo)II(I- G€3 T)O€3GJ\;~,, (2.22) < IIKVFllf 5 const. IIKllfr, dll.,,L,(F). for some U,E(I,v-'). The norm in (2.22) can be estimated as An extension of the Inversion Lemma to certain open normed in [5], where it is shown that for any 0,v c p C 1, (as G has double algebras will be required. no memory and G, is finite dimensional), supr6zII(G/P)rII(1) Extended Inversion Lemma 2.2: If A = U ,,,,JA,,} is a =: y is a finite constant depending on 0.Therefore continuously nested NDA, then Inversion Lemma 2.1 holds W provided (1 - I( is identified with 11 - )I ![(I-G€3 T)O €3 GI1(,,) Isup 1 g(t, t)'+'I U: Proof: If K and KOare in A, they are certainly in A?,. tdi=O W By Lemma 2.1, K-' is in A,,, and satisfies the inequalities -.c ,i+1u; IyP(1 - Uop)-', uop < 1 (2.6), (2.7). All that remains to be shown is that K- is actually r=O in A. Under our hypothesis, K and KO are in some A,, ZAMES AND WANG: LOCAL-GLDBAL DOUBLE ALGEBRAS: PART I 135
U > U,, where 11 K VK@ 11 X, < 1 implies (1 k VK 011 < 1 I( * 11 (,) is E,. Let E, denote the space by continuity of (1 (1 :. Lemma 2.1 now implies that K- E A,, - E, := (KEB3: p,(R) < m}. (3.la) U > ul. and therefore K- ' E A. 0 These three spaces are related by the inclusions III. ~AL-GLOBALALGEBRAS IN THE FREQUENCY DOMAIN
For arly E El and 7 E &, the convolution kernel k( 7,7 - (e)) K Also, Buo C E, for U < uo. Of the three spaces (3.2), only E, of K, has a well-defined transform F(K,) is a normed double algebra with both time and frequency dom& Q3 characterizations. $(K,)(Z):= 1 k(7,~- e)& I z I < 1 (3.1) 8-0 A. An Auxiliary Time-Domain Norm ig H" called the transfer function of K, and denoted by 2,. There is now a double algebra E,, which (viewed as a space) K, will be called the local transfer functio? (local transform) has equivalent descriptions in the time domain via the kernel of K (of k(7,_7 - (.)) at 7. For KEE,,K, is in H,". norm (1 I[(,), and the frequency domain via the transfer-func- Notation: K will denote the sequence { Y(K,)},7 E U,of tion norm p,(.). However,these norms are incommensurate and local transfer functions of In the case of shift-invariant K. K, inconvenient for the estimation of I"( - 03, 00) time-domain all elements of that sequence ar_e identical. In that special case, behavior from local frequency-domain properties, unlike, e.g., to keep the notation simple, will also denote a particular K the time-invariant situation in 12( - 00, 03) where Parseval's element of the sequence in Ha. The context will be relied upon theorem provides an isometry between kernel and transform to resolve the ambiguity. representations. Instead, we introduce an auxiliary time-domain The local product is equivalent to a sequence of transfer norm on E3, denoted by 11 * which is equivalent to the n A+ function products, (G€3 K), = (G,)(K,). I"( - m, m)-induced operator norm on B,and produces a kind Although operators in B (in E,) have local transfer functions of approximate isometry for slowly varying operators, i.e., in H" (in H,"), the reverse is not true. B and E, have no subject to certain assumptions on uniformity of growth, 11 * 11 Nu) precise characterizations in terms of transfer functions. To deal will have the properties with operators initially specified in the frequency domain, we turn instead to subalgebras of B which have such a specification. Define the function p of operators KE@ where 0 -+ 0 as the variation rate p of I? approaches 0,' and
CY -+ 0 as p -+ 0 and U -+ 1. By making U depend on p and P\P)(k) := SUP I1 R, II Lg 7& approach 1 as p * 0, it is possible to ensure that (1 K I(a(u) -+ 2 5 p s m, (L,P = the LP space of the circle- I z 1 = U); In p,(X) as p -+ 0. This device of an asymptotic isometry formal- the caie p = 00, omit6 the superscript, i.e., p,(~)= pLm)(K). izes an idea originally used to obtain a BIB0 version of the Introduce two subspaces of operators in D, one specified (in circle criterion in [27]. the time domain) by Equip the space I"( - 00, m) =: A with the family of auxil- iary norms depending on the parameter U, 1 < U < 33 -E,= {KED: IIK(((oo)
p, = {KEB:pu0(k) < 00 for some U, > U}
and equip each with local andBlobal products. m u-2r 112 Proposition 3.1: E, and E, are identical double algebras. where K, := (1 - u-~)'/~,(= ) ), i.e., JIuIIN(,) 0 is the lanorm of a moving window average of U, with an exponential window normalized to have unit l2 norm. Proof: In Appendix I. The norms in this family are equivalent to each other (Henceforth, we will often not distinguish between E, and 1 U1 E,). Equip E,, with local and global norms lI~ll4(Ul) const. lltola(02) 5 const. ll4Ia(Ul)~ < < U,
I1 K II d = I1 K II (U) and to the I" norm
IIKII; = CL,(@. II~II~,)5 II~IIP4 Koll~IINu~
E,, is now a normed double algebra, and conforms to our the constants being independent of U. definition of being continuously nested, as E, = U ,{ 5, } , Each auxilliary norm on A induces an auxiliary operator Although the space E, is not closed in either norm, the extenled norm on the space of operators; for K E B version of the Inversion Lemma, Lemma 2.2 will apply never- theless. llKlla((,) := suP{ll~~Ila(,):ueA9 Ilull(I(g)5 1). p, will be the algebra of choice for problems initially speci- fied in the frequency domain. The two norms will serve to Assume the space E3 and global algebra QE3 to be equipped with compute local-norm approximants to global-norm behavior. the family of auxiliary operator norms 1) * ll(I(o), 1 < U 5 W. Remark 3.1: The closure of the space E, in the kernel norm The norms in this family inherit from A equivalence to each other and to the I"-induced operator norm on any K E B. That 6Note that for time invariant K, the notations ~y)(k)and 1) k 11 Lg coincide, and will be used interchangeably. 'The term "approaches" must be suitably interpreted (see Section III-D). 136 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991 last norm equals the I’ norm of the kernel k of K, i.e., The time and frequency-domain norms are related by inequali- ties. The main ones are listed in the next two propositions whose IIKII,= IIKIINm) = supIIk,II/1, KE@* (3-3) proofs, together with subsidiary inequalities are in Appendix II. 1 In Propositions < U 5 00, t E Z, and 5 p IW. Since @@ is a Banach algebra under the )I )I ,norm, we obtain 3.4-3.5, 1 2 the following proposifion: The constant K:”) is defined by Proposition 3.2:. global algebra @E3 under the auxiliary norm )I 11 Nu) is a Banach algebra. (However, as the constant in ifp=w the inequality 11 K €3 GJI IConst. 11 K 11 Nu)II GI1 Nu) differs KLp’ = from unity, the local algebra LE3 is not a normed algebra under II - II Nu).) In view of the eq~ivalenceof 11 11 ,and 11 - 11 Nu), we can employ 11 11 Nu) as the global norm for E3 in place of 11 11 (,,), Proposition 3.4 - Transfer-Function Bounds: and will do so in the rest of Section III. 1) Bounds on K E Eu:
II 8 II % if K is shift-invariant ~~K~~u(u;~)sPu(8t) + K:”’d:”)(8) if 8 is slowly varying Kufip\p)(8) IKu&pu(8) always. 2) Local Bounds on (GK + F) in E,,:I For F, K with slowly varying transfer functions, S := GK + F, S’ := G €3 K +F
(0 - 1)-1{Pu(8) a,($) + au
+ ~u(~)llKllNu) if e is slowly varying (3.10) always. (3.11)
The auxiliary norms are bounds on an operator which are uniform in time. Occasionally, we shall relate these to certain finer bounds emphasizing particular times. The (exponentially weighted) recent past seminorms 11 K 11 Nu; I) of K E E3 are defined by the equation (3.13)
IIKlla(,,;q := ~,‘u-~sup{Iln,KuJI,~:u~A,IJuJIN,,) 5 1). C. Radial Growth Inequalities The auxiliary norm is the supremum of these seminorms, If K f 0 is in H,“,,U, > 1, Hardy’s convexity theorem [9] II K II Nu) = SUPf II K II Nu; I)’ implies the radial growth condition B. Slowly Time- Varying Transfer Functions “”), 1) K )Ie/ 1) K 11 Hw Iv(”’ Definition: An operator K E E,, has a slowly varying local v := IIK~I~,/I~KIif~~15 U< 00- transferfunction with rate a:p)(8), 2 5 p 5 w, if KEEuo. has uniform (i.e., in t) radial growth with constant a:”)(P):=SUpIIk,,’ - 8,11Lg < pyJ)(k) (3.4) re2 V,,(K) Iff
:= 11 11 HG / 11 11 Ha“ < O0 and in case p = 00, write a,(#) := (@’)(8)will t€Z later be assumed small in relation to certain additional constants.) Proposition 3.3: For U, > U > 1, the transfer function and in which case operator (commutant) variation rates are related - (Ino/lnoo) pum5 CLI(~)VUO(K) duW IOK(u,/u) Proposition 3.5: If the operators S, S‘ defined in (3.8) are in and K commutes approximately with the shift whenever a,,,( 8)
< 0- ‘KTJ, U)II K II (U)’ *The bounds (3.13) remain valid for certain noncausal operators Go Prooj! Follows (A1.4) of Appendix I. (see the definition 2.1 in Part 11). ZAMES AND WANG: LOCAL-GLOBAL DOUBLE ALGEBRAS: PART I 137
Eu0, U, > 1, and S' has uniform radial growth, then E. Frozen-Time Stability in the Frequency Domain -I (In u/ln "0) Although the initial reason for introducing double algebras I It $11 (I(,,; t) - Pu(mI 5 Pl(m{ %O(S ) - 11 was to deal with adaptive optimization, their chief application in + ,(PI a(P) Part I is to obtain frozen-time conditions for stability in the ('I) -t 'I vK'I Nu)' (3.14) frequency domain. The main idea here is to introduce notions of Proof: In Appendix II. local spectrum and local resolvent to parallel the global (i.e., usual) ones. Most classical frequency domain conditions of the D. Remarks on Variable Rates frozen-time type can then be incorporated in a statement linking In adaptation problems, the variation rate of a system is often local and global resolvents. The spectrum Spec,(K) of an operator K in a normed adjustable, e.g., by reduction of the rate of change of its algebra is the set { X E has no inverse in The (memoryless) gain parameters, in a way which distorts time- A G:(XI - K) A}. set Res,(K) is the complement of the spectrum, and scales but in a certain sense preserves local values and, espe- resolvent cially, preserves operator norms. We wish to describe the the y-sublevel set of that resolvent is behavior of such a system as rates approach zero. The details of Res,;,(K) = {XEG: Il(XZ- K)-'lIAS y}. how rate variation is achieved can be complex in discrete time, and can be disregarded for our purposes except as follows. The local (global) spectrum of an operator in a normed double Definition: The rate d,(.) of an operator K E E,, is variable algebra A is its spectrum in the local algebra LA (global algebra towards 0 if K is an element of a set of operators K(p) @A); the local (global) resolvent set and its y-sublevel sets depending on a parameter p such that' are similarly defined. 1) p has values in R+ with zero as limit point; and for some Let G,K be operators in E,,L U > 1, where G has no PI, K(Pd = K memory and K is shift invcriant. [E,, is defined in (3.1a)l. The 2) d,(K(P)) 5 P local and global norms 0," E,, are p,,(-) and 1) * 1) and G is 3) the local norm 11 K(p)11 (,,) is invariant with p. locally normalized, p,,(G) = 1. The y-sublevel sets of G @ K AAsidar definition is made for the transfer function rate in the local algebra LE,, can be determined from a plot of the a,( K) under local norm CL,,(.)(replacing d,( K) and (1 1) (,,)). spectrum of K on a Nichols or Hall chart. The main result of Remark 3.2: In (3.14), if the operators G, K, F have rates Part I is the following. a,(.) variable towards 0 and approaching 0, and U 1, then Corollary 3.1: The local sublevel sets of 6 8 k are related (3.14) implies that the auxiliary time-dopain norm (( Slid,,) to the global resolvent of GK by the inclusion approaches the transfer function norm &SI); in this sense, the two norms are asymptotically isometric. ResLE,;7(6 @ k)C Res,,(GK) Remark 3.3: This asymptotic isometry can be deduced with- provided that out assuming uniform radial growth or using (3.14). An alterna- tive inequality based on the continuity of pu( e) versus U can be a,,@) < (U - 1)/YPu(R)(1 + YP,(k)). (3.18) Ebtained as follows. For S, S' defined as in (3.8) and in some Euo, U, > U, the inequalities Pro08 In Appendix 11. Note that for G with memory, Corollary 3.1 still holds with (3.18) replaced by
hold. However, if s: denotes the kernel of S:, we have stability test. For scalar systems and G memoryless, the follow- ing prescription is obtained. Plot the o-shifted dkcrete Nyquist diagram of the time-invariant operator K, i.e., K(ueie) versus 8, 0 E [ - T,a). (See [13] for details in the case-of continuous- time systems). The local spectrum of G @ K in E ,, is the union of the interiors of such diagrams of cK, evaluated for each constant c in the range of the kernel of the memoryless operator Equations (3.16) and (3.15) give G, i.e., cE{g(t,t): tEZ}. Let y, 0
138 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 7, FEBRUARY 1991
Although a particular approach to freezing was used, the key i.e., ISu is a normed algebra under @ . If K, M are actually in features of the double algebra structure, including the local lSU,then so is K @ M, by the reasoning of part a), so that E, is product and V operator, hold more generally. a normed algebra. 0 Proposition AI.2: For K E E,, with kernel k, and U C uo APPENDIXI llKll((I)5 K(uo/(I)fiP(Io(~) (Al.3) THEALGEBRAS E (I AND (I Proposition AI.1: The spaces EU and E, under the norm d,(K) 5 flK((Io/(I)J;; auo(4* (Al.4) (1 ((((I) and either a) the global product, or b) the local product, is a normed algebra, and Eu is a Banach algebra. Proof: Prod: First, we will show that for u1 1 1 a) Under the global product, for any K, M in E, it will be shown that KMEEu and lIktII,;l 5 fillk,llL%l. (Al.5)
I1 KM II ((I) 5 K I1 ((I) I1 M I1 (U) (A1 .l) II Indeed, by the matrix inequalities 1 G I 5 trace (G*G) s which implies that Eu is a normed algebra for each U 2 1. It nIGl2 then follows, from the representation of any K E EU by a Volterra sum with kernels in the complete space 1; of bounded 1;-norm, that the normed algebra EUis complete and therefore a I1 k*II & =
Banach algebra. If K, M are actually in the space Eu,they are 01 in some common ISoo, U, > U, which has been shown to be an I trace(k*(t, t - 7)k(t, t - 7))~:' algebra, and therefore KM E E (Io, which implies KM E r=O i.e., is a normed algebra under the global product. To prove (Al.l), let F := KM, and denote the kernels of trace (kF(u,eie)kt( uleie)) dB F, K, M by f,k, m. As k, m are in l:, the following bounds 2* --* and changes of order of summation are valid: by Parseval's Theorem m W n II i,II i& (KMW)= C k(-~?) C m(q,e)u(e) q=-m e=-W which proves (A1.5). For tEZ
W llktll,: = C I k(t9 f - 44 r=O
W 1 12 5 (FIk(tJ-r)u,'l2C r=O Iu/uo12~) r=O by Schwartz's inequality - - K(UO/U) J;; II kt II L2,0 by (A1.5)
5 K(oo/(I)J;;Puo(~)
which implies (A1.3). Equation (A1.4) follows from the inequal- ities
After taking suptsg of (A1.2), we conclude that SUP,^^ f(t,t -
(e)) 11 ,! 5 (1 K 11 1) M (1 ((I) which implies that KM E E (I and (Al. 1) holds. 0 b) Under the local product, for K, M in E., and ten, K, and M, are in E, and' Proof of Proposition 3. I: If K E Bu, the? for some U, > U, 1) K < ca. Therefore, for each t EZ, K,E~~satisfy (K@M), = K,M,. II kt II eo5 II kt II /bo 5 I1 K II (00) By part a), K,M, is in EU and satisfies )I K,Mtll((I)5 )I K,11 ((I) 11 M, )I((I). As F, := K,M, is time invariant, its kernel where k, is the kerne! of K,, which implies that puo(k>5 f, satisfies II Kll((Io)and that KEEV Conversely, if K E IZ, then for some uo > U, kuo(k)< ca. ft 1: = ((I) ((I) ((I) I1 II II Ft II II K II I1 MI1 For any ul, U < uI < U,, (1 K )I((Il) 5 Const. pUO(K)by (A1.3), and since this holds for all t, and K E 0 (11, which implies that K,E E (I. This proves that the spaces E,, are identic4. As E, is a normed double algebra by Proposition Al. 1, so is Bu. 0 ZAMES AND WANG LOCAL-GLOBAL DOUBLE ALGEBRAS: PART I 139
APPENDIX II 00 will be treated by separate approaches. Consider p = 00 first. FREQUENCY-DOMAININEQUALITIES n,(K - K,)can be resolved into a sum and then summed by Pas Before proving (3.5)- (3.13) certain subsidiary inequalities will be established, for KEE,, UEI~,tE2 t n,(~- K,) = - K,) r=-w1 II K II &U; 1) 5 II 2II G if K is shift invariant (A24
II Kn,ull/; 5 KUUfII a I1 GI1 UII Nu) where lim denotes a limit in the sense of weak (I1) operator
if K is shift invariant (A2.2) convergence. Such a limit implies that lim ,~ - o3 11 n,(K, - K,)ull = 0 for U E A, i.e., the limit is null (a fact which will also be used in the remaining inequality proofs). Therefore, for Also, it will be shown that the expressions G VK and GK - (G UEA 63 K),truncated by II, can be represented by operator series
by A2.2, which yields (A2.3) for p = 00. Next, suppose 2 Ip < 00. Denote (K- K,)u by y, and the kernel of K, by k,; ask, E I:, and U E I", we may write
= n,[ GK - (G €3 K),]
= (AU,)(GVK)
(by Schwartz's inequality, as k,, k, are in 1;) where (An,) := II, - n,-,, and the series of operators are weakly(P) convergent. Proof of inequalities A2.1 and A2.2: For tEZ, UEA, and k the kernel of K Now multiply both sides of the inequality by a' and sum, (n,~u)(~>a~= (II,K(II,U))(~)U~ (as K is causal) (E:=-m(*)2)1/2to get (A2.3) for 2 I p < 00. 0 I Proof of series expansions A2.4-A2.6: We have the identities
GK - (G€3K),] = n,[(G - G,)K G,(K - as k is shift invjtriant. This is a convolution of functions whose n,[ + K,)] transforms K,Y are in L:(= L" of the circle I z I = a), where k := f(k(-)), and P := F((nlu)(a)). By Parseval's Theorem
where U, has been resolved into B(An,). Now, for any FEB, (An,)F = (AIIJF,, so G and K can be replaced by G,, K, in the sums, which can be summed by parts to give (A2.4), after the observation that n,( G, - G,) and n,( K, - K,)both weakly (I1) operator converge to o as T + - 00. By definition of V n,( G VK) = n,t[ (GK and IlntuJI,;I K,,(J'JIUII~~), (A2.2) holds. 0 Proof of inequality A2.3: The cases p = 00 and 2 I p < -(G&K),) - (G63K- (GBK),)]. (A2.8) 140 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991
Resolution followed by partial summation gives The bound (3.6) on 11 K 1) (I(u) gives
+(a - l)-l a,(k)]}. (A2.11) Now, by the triangle inequality Equation (A2.9) applied to (A2.8) proves (A2.5). IInt(GK + F)ull1; Equation (A2.6) is obtained by multiplying (A2.4), (A2.5) by (An,), observing that (An,)X:=-,n,-, = 0, and equating 5 Ilnt(G@'++)tUII/; + Ilnt(GK the parts that do not vanish. 0 Proof of inequalities (3.5)-(3.7): Inequality (3.5) is the -(G@K)t)4;+ IlntP- 04l/;* same (A2.1). To prove (3.6), begin with as On the. right-hand side, the first norm is bounded by A ((n,Kul)/$5Iln,(K- Kt)ullr;+ Il~tKPll/$. K ,a'p,(G 63 K + 8)II u1( (I(u) using (A2.2); the second norm is bounded using (A2.11); and the third using (A2.3). After noting By (A2.3), the first norm on the right is that 11 )I(I(u;t) = K;'u-' ~~~,(*)~~Q,(3.8) is obtained. 0 Proof of inequality 3.9: -< KaKyU t a, (P) (~)ll~Il(I(,) IISII(l(u;t) - Pu(m IIS - S~II(l(u;f)+ 11s' - S:llo(o;f) and by the shift invariance of K, and (A2.2), the second norm is + IIs:II@u;r)- CL~(S:)
5 K ,Uf I1 kt II GI1 U II (I(,). 5 )I G VK 11 (I(u;,) -t K bP) a:P)(SI)
Division by K,U' gives (3.6). using S - S' = GVF, (A2.3), (A2.1), and (3.11), and giving To prove (3.7), note that (3.9). Proof of inequality (3.10): By definition of V and the triangle inequality
T= -w (In,(GVK)ull/; = Iln,(GK- G(=)ull,;
by (A1.5). Therefore
II KUII (I(,; I) 5 II Kull p" 5 SUP KUWkt I1 H,2 II UII (I(,) which implies (3.7), as p:) 5 &')(e) Ipy)(.). 0 Proof of inequality (3.8): From the series expansion (A2.4), and inequality (A2.2), we get
IPt[GK- (G@K),]uII/; t
KU~~6,~~~~a(7-1)ll(KT - KT-l)ullU(U) T= -w ZAMES AND WANG: LOCAL-GLOBAL DOUBLE ALGEBRAS: PART I 141
(A2.15) is a bound on 1) yI1 p 2 1) y 11 a(O), and (3.11) is true for as a,(FA@ @ FA)= 0. The terms appearing in (A2.18) can be 25pcm. 0 bounded as follows: Proof of inequalities (3.12), (3.13): These two inequali- ties are implied by the following ones, which hold for t E Z
We will show that S also has the lower bound REFERENCES IISIIN~;~)2 PI($:) - K:” aLp’(s‘)- IIGVKIIN.) (A2.17) V. M. Adamjan, D. Z. Arov, and M. G. Krein, “Infinite Hankel blocks matrices and related extension problems,” AMA Trans- which together with (A2.16) implies (3.14). lations, vol. 111, pp. 133-156, 1978. To prove (A2.17j observe that from the definition of pl(*) W. Arveson, “Interpolation problems in nest algebras,” J. Functional Anal., vol. 20, pp. 208-233, 1975. there are sets of exponepial inputs U,E I”( - CO, CO), U( t) := J. A. Ball, C. Foias, J. W. Helton, and A. Tannenbaum, “On a aexp(jOt), forwhich IIS:u(Ip = pl(S:)IIullp. For imaginary local nonlinear commutant lifting theorem,” preprint. exponentials the 11 * )INa) and I” norms coincide, and there H. M. J. Cantalloube, C. E. Nahum, and P. E. Caines, “Robust exists an exponential U for which adaptive control: A direct factorization approach,” to be pub- lished. C. A. Desoer, “Slowly varying discrete system xi+l = Aixi,” ll~tSi41Nu;t)2 ~l(~;)ll~ll@u).(A2-174 Electron. Lett., vol. 6, pp. 339-340, 1970. M. Dahleh and M. A. Dahleh, “On slowly time-varying From the definition of )I 11 No; t), we now get systems,” M.I.T. Tech. Rep. LIDS-P-1852, Feb. 1989. C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, “Feedback (1 SUI1 NO; 1) 2 K, utII ntSuIl‘$ system design: The fractional representation approach,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 399-412, 1980. 2 Kiluf{ ll~p$~ll,$- llnt(s‘- s:)Ulll~ C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. - 11 II,(s - s‘)U 11 ,;} (triangle inequality) W. L. Duren, Theory of HP Spaces. New York: Academic, 1970. 2 {PI($:) - K‘,“’a?)(J’) - ~~GvK~~@u)]~~u~~@~)A. Feintuch and B. A. Francis, “Uniform optimal control of linear systems,” Automatica, vol. 21, no. 5, pp. 563-574, 1985. by (A2.17a), (A2.3), and as (S - S’) = GVK. This proves B. A. Francis, “A course in Hm control theory,” in Lecture (A2.17). 0 Notes in Control and Information Sciences, vol. 88, New Proof of Corollary 3.1: Write FA - (XI - G @ K). If York: Springer-Verlag, 1985. XE ResLiu.r(G@ K) then pu(FA@) 5 y by definition of B. A. Francis and M. Vidyasagar, “Algebraic and topological aspects of the servo problem for lumped systems,” Dep. Eng. Res(. ;r). is shift invariant, = the Inver- As K GK G 8 K. By Appl. Sci., Yale Univ., New Haven, CT, S&IS Rep. 8003, 1980. sion Lemma 2.2, the resolvent conclusions will be true if M. Freedman and G. Zames, “Logarithmic variation criteria for 11 F~~VFII(1(,,) 1, for then (XZ - GK)-’ E B. Let us evaluate the stability of systems with time-varying gains,” SIAM J. this. Contr., vol. 6, no. 3, pp. 487-507, 1968. T. T. Georgiou, A. M. Pascoal, and P. P. Khargonekar, “On the robust stabilizability of uncertain linear time-invariant plants us- 11 FAo vFAII Nu) = 11 FA@ VG 8 11 NO) ing nonlinear time-varying controllers,” Automatica, vol. 23, no. 5, pp. 617-624, Sept. 1987. (as K and XIare shift invariant) P. A. Ioannou and K. S. Tsakalis, “Time and frequency domain uncertainty bounds in robust adaptive control,” preprint. P. P. Khargonekar and K. Poolla, “On polynomial matrix frac- tion representations for linear time-varying systems,” Linear Algebra Appl., vol. 80, pp. 1-37, 1986. -, “Uniformly optimal control of linear time-invariant plants: 142 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 2, FEBRUARY 1991
Nonlinear time-varying controllers,” Syst. Contr. Lett., vol. 6, George Zames (S’57-M’61-SM’78-F’79) re- no. 5, pp. 303-308, 1986. ceived the B. Eng. degree from McGill Univer- M. C. Smith, “Well-posedness of H” optimal control sity, Montreal, P.Q., in 1954, and the Sc.D. problems,” to be published. degree from the Massachusetts Institute of M. S. Verma, “Robust stability of linear feedback systems under Technology, Cambridge, in 1960. time-varying and nonlinear perturbations in the plant,’ ’ preprint. Currently, he is the Macdonald Professor of M. Vidyasagar, H. RSchneider, and B. A. Francis, “Algebraic Electrical Engineering at McGill University. and topological aspects of feedback stabilization,” IEEE Trans. Prof. Zarnes is a Fellow of the Royal Society Automat. Contr., vol. AC-27, pp. 880-894, Aug. 1982. of Canada and the Canadian Institute for Ad- L. Y. Wang, “Adaptive H“ optimization,” Ph.D. dissertation, vance Research. He has written many papers on McGill University, P.Q., Canada, 1989. systems and control theory, won many awards L. Y. Wang and G. Zames, “H” optimization and slowly for his research (for details see The IEEE Directory), and served on time-varying systems,” in Proc. 26th Conj. Dec. Contr., Dec. many journal editorial boards. 1987, pp. 81-83. -, “Slowly time-varying systems and N“ optimization,” in Proc. 8th IFAC Symp. Ident. Param. Est., vol. 1, Beijing, R.O.C. 1988, pp. 492-495. Le Yi Wang (S’85-M189) was born in Shang- 1241 -, “Local-global double algebras for slow H” adaptation,” hai, China, on July 5, 1955. He received the MTNS Conj., Amsterdam, The Netherlands, June 22-28, 1989. M.E. degree in computer control from the t251 - “Local-global double algebras for slow H” adaptation,” Shanghai Institute of Mechanical Engineering, in Proc. IEEE 28th CDC Conf., Tampa, FL, Dec. 13-15, in 1982, and the Ph.D. degree in electrical 1989. engineering from McGill University, Montreal, 1261 -, “Local-global double algebras for slow k? adaptation: P.Q., Canada, in 1990. The case of I* disturbances,” to be published. From 1982 to 1984, he taught computer t271 G. Zames, “Nonlinear time-varying feedback systems: Condi- courses in the Department of Automation, tions for L” boundedness derived using conic operators on Shanghai Institute of Mechanical Engineering. exponentially weighted spaces,” in Proc. 3rd Allerton Conj. While at McGill University, he held the Dalbir Circuit Syst. Theory, Univ. Illinois, Urbana, 1965, pp. 460-471. Bindra Fellowship and Alma Mater Fellowship, and worked as a 1281 -, “Input-output stability of nonlinear time-varying feedback Teaching Assistant and Research Assistant. Since April 1990, he has systems,” IEEE Trans. Automat. Contr., vol. AC-11, pp. been with Wayne State University, Detroit, MI, where he is currently 228-238 and pp. 465-477, 1966. Assistant Professor in the Department of Electrical and Computer -, “Feedback and optimal sensitivity: Model reference trans- Engineering. His research interests are in the areas of N“ optimization, formations, multiplicative seminorms, and approximate inverses,” robust control, slowly time-varying systems, system identification, and IEEE Trans. Automat. Contr.,vol. AC-26, pp. 301-320, 1981. adaptive systems.