E L S E V I E R Systems & Control Letters 25 (1995) 299-306

$YSN tS COUll L

LrrTmw

Suboptimal solutions to the time-varying model matching problem

A v r a h a m F e i n t u c h

Department of Mathematics and Computer Science, Ben Gurion University of the Neoev, P.O. Box 2053, Beer-Sheva, Israel

Received 3 April 1994; revised 6 August 1994

Abstract

Two approaches to the characterization of suboptimal solutions are given. The first, via Parrot's theorem, gives all solutions. The second uses J-spectral factorizations and gives a particularly simple parametrization of all solutions. Unlike the time- invariant case, however, J-spectral factorizations do not always exist.

Keywords: Time-varying systems; Model matching, Lower triangular operators, Distance formula

1. Introduction

There are many approaches to the problem of finding controllers for the simple "Nehari type" problems for linear finite-dimensional systems, which blend together methods from complex function theory, operator theory, state-space methods, etc. In fact, the problem was originally solved in the infinite-dimensional case in the classic paper of Adamjan et al. [1], and there is a large literature available on this problem (see [7, 9]).

Less can be said about the corresponding time-varying problem. Here the problem has its most natural formu- lation in the language of operator theory. Let C denote the algebra of "lower triangular" linear operators, on a Hilbert space H and let R be an arbitrary bounded linear operator, whose distance from C is less than one. Find all T c C which satisfy IIR - TII < 1. For the case that R is upper triangular and there is an added growth condition on the operators considered, this problem is solved in [8]. [ 13, 3, 4] present solutions in discrete time, as do [12, 10] in continuous time, using state-space methods, again under appropriate assumptions.

Here we present two approaches to this problem. The first and most complete approach is based on the well- known "Parrot theorem" [ 14]. This theorem has been used to solve the Nehari problem by means of the "one-step extension" idea. We use a similar idea here to generate the rows of those elements T E C for which IIR - TH < 1. The procedure described is an iterative one. Parrot's theorem has been used originally by Power [11] to obtain Arveson's distance formula and then by Feintuch-Francis [5, 6] for the solutions to the 2- and 4-block problems. Here the theorem is applied in a slightly different way.

The second approach presented here is that using J-spectral factorizations, following the ideas of [9]. We show that the existence of a J-spectral factorization for an operator determined by R allows us to give a parametrization of all sub-optimal solutions T. However, as opposed to the finite-dimensional time-invariant case [7], such J - spectral factorizations do not always exist. We give an example to illustrate this fact. We do not know the answer to this question for R E L°% We mention that the idea of applying J-spectral factorizations to such problems is due to [2]. We thank an anonimous referee for this reference as well as some others that escaped our attention.

0167-6911/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6911(94)00076-X

300 A. Feintuch/Systems & Control Letters 25 (1995)299-306

2. Preliminaries

H is the Hilbert space of square-summable complex (matrix) sequences:

(x0,xl . . . . ) " ~ Ixil 2 < ~ . i=0

For our purposes it makes no difference if xi are scalars or finite-dimensional square matrices. P. will denote the standard truncation projection.

P~(xo,xl . . . . . x . , x .+l . . . . ) = (Xo . . . . . xn, O, 0 . . . . ). L ( H ) will denote the algebra of all bounded linear operators on H and C will denote those elements T of L(H) which satisfy P . T P . = PnT for all non-negative integers n. C is identifiable with the algebra of stable linear time-varying systems on the input--output space H.

For R E L (H) , the distance from R to C is given by the well-known formula of Arveson [11] which is the time-varying analogue of the Nehari formula:

d(R, C) = sup tIP, T(1 - P,)ll . n>~0

M2(C) will denote the 2 × 2 operator matrices acting in the usual fashion on the Hilbert space H ® H, whose entries are in C. It is easily checked that a 2 x 2 operator matrix S with entries in L ( H ) is in M2(C) if and only Q, SQ~ = QnS for all n ~> 0 where

0 Pn "

3. The "Parrot theorem" approach

We state Parrot's theorem [14].

Theorem 3.1. Le t Hi, Ki (i = 1,2) be Hilbert spaces and let A, B, C be 9iven such that

[AB] "H2 -~ K, ~3 K2, [C,A] " H, ®1-12 --* K,

are contractions. Then there exists D : Hi ~ K2 such that

is a contraction. A l l such D are 9iven by D = - K A * L + (I - KK* )I /Zz(I - L ' L ) 1/2 where

K = B ( ! - A ' A ) -1/2 L = (I - A A * ) - I / 2 C

and Z is an arbitrary contraction.

We have slightly simplified matters by assuming that (I - A*A)I/2, (I - AA* )1/2 are invertible. Now suppose R c L ( H ) such that ]lenR(l - P.)ll < 1, n>~0. Denote PnR(I - Pn) by An. For n = 0, Ao can

be identified (since all other entries are zero) with the r o w [ r O l , r ' o 2 , r 0 3 . . . . ] , which we will continue to call Ao. Let too denote the entry to be placed to the left of rol so that the enlarged row [too, A0] will be a contraction. All such entries are characterized by too = (I - AoA~ )l/2zoo where ][zoo ]] ~ 1.

Now consider the matrix

A. Feintuehl Systems & Control Letters 25 (1995) 299-306 301

where, again, Al will denote the (2 × c~) matrix identified with P1R(I - Pi ). Parrot 's theorem determines all 1 × 2 matrices [tl0,tll] such that

it00 1 tlo tl~

is a contraction. In this case [q o, tll] is determined by [too, rol ] = [(I - AoA~ )l/2zoo, Aoeo] (where eo is the standard first unit vector written as a column) and A1, and depend on a parameter II[zl0,zll][[ ~< 1.

In the next stage we use Parrot 's theorem to determine all 1 x 3 matrices [t2o, t21, t22] such that

too ro I to2 7 rio tll r12 A2

t22 t2o t2

is a contraction. This iterative procedure is continued by induction and has a number of consequences:

(1) The infinite matrices S determined by this procedure define contractions on H [ 14, Lemma 15,13]. We have constructed in this way all contractions S such that

P~S(I - P~) = P~R(I - P~) for all n>~0.

(2) For each such S, V = R - S E C and IIR - VII = lIsII ~ 1, o n the other hand, if V C C and liB - VII ~< l, the P~R(I - P~) = P~(R - V)(I - P~), thus S = R - V is in the class constructed above. We thus have all V ~ C such that IIR - VII ~< 1.

(3) Since for V E C, P n R ( I - P , ) = P , ( R - V)(I - P ~ ) , thus I I R - vii > / I I P . ( R - v ) ( I - P. ) l l = IlP.R(Z - P . ) l l for all n>~0. It follows that d(R,C)>~ ][P,R(I - P. ) l l for all n~>0 and d(R,C)>~ supn IIP~R(I - P. ) l l . I f w e assume (without loss of generality) that sup~ ]IP~R(I - P~)ll = 1, then we have shown that there exists V E C such that [[R - VII ~< 1. This proves Arveson's distance formula for the given nest.

(4) It is o f interest to write down the particular solution obtained when the free parameters are all chosen to be zero. Let Dn be the nth row of this lower triangular infinite matrix. Then Do = 0 and

1 Do ]

..... ?o-'] - oi_, " This is often called the central solution.

4. J-spectral factorizations Suppose R E L (H) and let

G = 0 I "

The relationship between the distance problem and the existance of J-spectral factorizations is given in the next theorem. The proof is similar to [9, Theorem 2.1 ].

Theorem 4.1. R E L ( H ) and let

, o_,1 J = [ 0 - '

I f there exists W E M2(C) which is invertible on H G H with Wml invertible on H, such that G*JG = W*JW, then d (R ,C) < 1.

302

Proof. Consider

[Wjl W = W21

A. Feintuch/Systems & Control Letters 25 (1995) 299-306

w,2] W22

invertible in M2(C) with W~l invertible in C. (Recall that both in C and M2(C) invertibility as operators is equivalent to invertibility in the appropriate algebras.) Let V = W -1 and write it as an operator matrix

[Vlj Vl2 ] V = V21 V22

Since W11 is invertible,

W,1 W,2] = 1 01] [W,, O Since W is invertible and the two outer matrices in the above product are invertible, so is the middle matrix. Thus W22 - Wzl W1S 1Wl2 is invertible and a simple computation gives that V22 E C is invertible and V2~ 1 = W22 - W21WIlIWI2 E C. Thus Q = VlzV22 1 E Cand

[ V12 It follows that

0 1

Therefore,

( R + Q ) * ( R + Q ) - I = R Q j ,~ O

0 I * 0

• 0 1

= --(V22V2*2) -1 ~ 0.

Thus IIR+Qll < 1. []

Remark. G*JG = W*JW implies that Wl* 1 Wli - W~*l m21 ~0. It is easily seen that if Y is invertible in M2(C) and Y*JY = W*JW, then Y11 is invertible in C if and only if W11 is. Thus either all J-spectral factorizations of G*JG have this property or none do (see [9]).

We now show that the existance of such a spectral factorization gives a pararnetrization of solutions to the suboptimal time-varying problem. Again, the ideas in the proof are similar to those of [9].

Theorem 4.2. Suppose R E L(H) and that there exists W invertible in M2(C) with Wll invertible in C so that G*JG = W*JW. Then the set o f operators Q E C with ]]R + QH <~ 1 is 9iven by

Q = QIQ~J' Q2

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Proof. Again, let V = W -1 and recall that V22 is invertible in C. Take U E C with IlUII ~ 1 and let

We show that Q2 is invertible. Since G*JG = W*JW, we have that VJV* = G-IJG *-1 . Equating the (2,2) entries of these operator matrices gives

g2, v2*~ - g22 v2"2 = - I .

Therefore, [[ V221V21 [[ < 1 and so is II g2-2 1 g21 uII ~< I[ g ~ ' g2~ II. It follows that I + V22 1V21U is invertible and that its inverse is an uniformly convergent power series in V~ 1 V21U. In particular, (I + V~ 1 Vzl U) - l is in C. Thus Q2 = V22(V~ 1VzlU + I ) is invertible in C.

Now, for Q = Q1 Q21,

[;]*[ ] ( R + Q ) * ( R + Q ) - I = R Q i R + O I

= ( Q 2 1 ) . [ u * /] = (Q21) , [ U *

= (Q21)*[U*U -I]Q21 <~o.

Thus Ilg + QII ~< 1. Conversely, suppose Q E C such that [IR + QII ~< 1. Define

[U121:w[QI]:WG-I[R-~QIEc. Then,

V [ U1 =

gives that V21Ui + V22U2 = I, so that UI, U2 are fight co-prime and thus U~U1 + U~U2 > 0. Without loss of generality we can assume that UI* U1 +/-/2* U2 = I (otherwise by spectral factorization, there exists C E C such that U1* U1 +/-:2*/-/2 = C* C and then ( U1 C - t ). (/-/1 C - 1 ) _~_ ( U2 C -- 1 )* ( U2 C - - 1 ) = [ and C - 1 can be carried along in all the computations to follow). Then,

U ~ U ' - U ~ U 2 = [ R + Q I * J I R + Q ] I

gives U~ U1 <. U~ U2. Therefore, 2U~ U2 >>. U~ U2 + U~ U1 = I and U2 is left invertible in L(H). To complete the proof we show that U2 is in fact invertible. Let Y E L(H) such that ¥U2 = L In particular, Y can be chosen so that Y = 0 on R(U2) ±. We claim that IIg~ Yll ~< 1. For if there exists x E H with Ilxll = 1 such that Ilwl Yxll > 1, the x = U : forz e H with I I s : [ I = 1 and I[UI Y u : I I = IIulzll > 1. But g~u2 >~ U~Ul implies that HU:H/> IIS:l l for all z E H. It follows that IIUIYII ~ 1.

Now, U2 = g221 ( / - V21Sl) implies that V22 1 -=-(g221g21UlY+I)S2 . By the proof of Theorem 1, I[ V~ 1V211[ < 1 and therefore I[ V~ l g21UI Y[I < 1. It follows that g22 1 g21 e l Y + I is invertible and therefore so is U2. []

304 A. Feintuchl Systems & Control Letters 25 (1995) 299 306

5. An example

We show that the converse of Theorem 4.2 is false for time-varying systems by giving R E L ( H ) with d(R , C) < 1 but such that there exists no W E M2(C) with Wll invertible so that G * J G = W * J W .

Let

R = 0 ~ ... 0

I The verification that there is no appropriate W E M 2 ( C ) is done in two stages. In the first stage Then d(R, C) = 5. we show that the question reduces to one about M 2 ( C ) where C is the appropriate algebra of lower triangular matrices on C 2. The rest is arithmetic.

Now

[, R] G * J G = R* R * R - I '

where

, 0 iilI R*R = 0 0 , R R * = 0 0 .

Since G * J G and R*R - I are invertible, we write this as

I'0 lE ( ,

and observe that the entries in the matrix on the left are upper triangular and those on the right are lower triangular. Suppose there exists the appropriate W such that W * J W = G*JG. Rewrite this as

/ [w~2 -w$2 R* -(1OR*R) V2~ v22J The entries on the left side of this equation are upper triangular and those on the right are all lower triangular.

Thus they must all be diagonal. Looking at the ( 1,1 ) position gives

W~I + R ( I - R ' R ) - j W~2 = (I - R ' R ) - I Vlt .

Then (I - R ' R ) - l Vii diagonal implies Vjl is. The (2,1) entries give

W~2 = R* Vll - ( I - R*R)V21.

Thus W~2 is diagonal. This and the form of R ( I - R ' R ) i gives that Wl* 1 is the direct sum of a 2 x 2 upper triangular matrix and an infinite diagonal one. A similar argument using that VI~ is diagonal gives that V21 is of the same form as Wll.

Examining that (1,2) and (2, 2) entries in the same way give that VI2 and W2* 2 are diagonal and that W2* l and V2* 2 are as Wl* 1 .

We can thus reduce our problem to two parts, one for 2 x 2 scalar matrices and the second for infinite diagonal matrices. We look at the problem for 2 x 2 matrices.

WII - - [ a21 a22 ' b22 '

R = 0 W21 = cll 0 W22 = 1 0 ' C 2 1 C22 ' d22 "

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We have

W~, + R ( I - R * R ) - I W72 ---- ( I - R * R ) - I Vl l . (1)

W~2 = R* Vii - ( I - R*R)V21. (2)

By computation we have (verified by mathematica)

1 [ (az2d22 - b 2 2 d l l ) d l l 0 ]

L J VII = Z (b22c2l - a21d22)dll ( a l l d l l - bllCll)dz2

where A = det W = ( a l l d l l -- bllCll )(a22d22 - b22c22). Since Vll is diagonal, either d ll = 0 or b22c21 = a21d22.

Eq. ( 1 ) gives that d~l = 0 implies all = 0. This possibility is ruled out by the assumption that W11 is invertible 2 2 so that we have b22c21 = a21d22. Also by Eq. (1), a21 = - - g b 2 2 and therefore c2t = - ~ d 2 2 .

Also,

all = a22d22 - b 2 2 c 2 2 ) d l l ,

Using Eq. (2) in the same way gives

b;1 = l(a22d22-b22c22)Cll , b ~ 2 = 3 ( a l l d l l - b l l C 1 1 ) c 2 2 .

We also have the Equations

- -W~l -- R ( I - R ' R ) - l W22 = ( I - R ' R ) - I VI2 ,

- W ~ 2 = R* 1/12 - ( I - R * R ) V 2 2 .

These lead in the same way to

* = l ( a 2 2 d 2 2 - b 2 2 c 2 2 ) b l l , * ~A(a l ld l l - -b l lCl l )b22 . Cll C22 =

and

d ll = (a22d22 - b z 2 c 2 2 ) a l l ,

From these we obtain

. 4 , a22 ---- a l ld l l -- bl lCl l )d22 .

d22 = a l l d l l -- b l lCl l )a22.

(3)

(4)

Remark. We eleminated the possibility that d l l ----- 0 by showing this contradicted the assumption that WII is invertible. In fact the example shows that there is no invertible W even without this property. This takes a little more fiddling and is left to the reader.

We conclude this paper with an example which illustrates the difficulties involved with J-spectral factorizations in the time-varying case. Suppose T is an invertible operator, J = J* = j - 1 such that JPn = P n J for all n ~>0.

Jail[ = ] d l l [ , ]bill = [Cll).

and a~" 1 clt = b~'l d l l . S ince A = ( a l l d l l - b l lCl l )(a22d22 - b22c22 ) we can rewr i te a~l = ( l /A )(a22d22 - b22c22 )d l l as a;l (al ldl l - b l l C l l ) = ] a l l l 2d l l - ] b l l l 2 d l l or Jail] 2 = ]Cll]2 -~ - l .

Now consider the equation W * J W = G * J G . From the (1,1) entry we obtain

latll 2 + la21l 2 = Iclll 2 + Iczl[ z + 1

2 -2d22 we now have tb2/[ ]d22]. But the (4,4) entry of which gives la21l = Ic21l. Since a21 = - ~ b 2 2 , C21 ~-- =

W * J W = G * J G gives 1b2212 = 1d2212 - 3 which is impossible.

306 A. Feintuch l Systems & Control Letters 25 (1995) 299-306

The J-spectral factorization problem is: given T * J T does there exist W invertible in C such that T * J T = W * J W ?

The answer in general is no! The following simple example was shown to me by my colleague A. Markus. Let

I -1 1

J = 1

0

T = 1 1

Then, 1 ] -1

T * J T = 1 •

If W C C such that W * J W = T * J T , then for

] W : WI 0 WI I

we have ~-]~=1 ] Wkl 12 = -1 which is impossible. As this and the above example show J-spectral factorizations for time-varying systems need a great deal more

thought.

R e f e r e n c e s

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