Some remarks on optimal interpolation

Systems & Control Letters 11 (1988) 259-264 North-Holland

259

Some remarks on optimal interpolation Ciprian F O I A S Departmont of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.

Allen T A N N E N B A U M Department of Electrical Engineering, University of Minnesota, 123 Church Street SE, Minneapolis, MI 55455, U.S.A., and Department of Mathematics, Ben-Gurion University of the Negev, Israel Received 29 March 1988 Revised 25 May 1988

Abstract: We give in this note very simple new formula for the optimal solution of a generalized interpolation problem which arises in the weighted sensitivity H~-minimization for distributed systems. This is based on our work on skew Toeplitz operators from [2].

Keywords: Weighted sensitivity minimization, Skew Toeplitz operator, Distributed parameter system, Generalized interpolation.

I. Introduction As is well known by n3w, a number of questions in systems design are reducible to a certain problem in generalized interpolation in H °° [10]. Namely, let w, O ~ H °° with w rational, and O nonconstant inner. Then the H°%optimal weighted sensitivity problem amounts to the computation of po'-mf{ IIw-Oqll~o: q ~ H ~ } and finding the corresponding qopt w}fich realizes this bound [8]. This problem has been considered from an operator theoretic point of view in a number of papers as well as the four block and multivariate generalizations. (See [2-7,12] and the references therein.) There has even been some work on solving such a problem for nonlinear

* This research was supported in part by grants from the Research Fund of Indiana University, NSF (ECS-8704047), and the Air Force Office of Scientific Research AFOSR-880020.

systems [1]. The point of the present paper is to give a very simple neat formula for the computation of Po and qopt which should be very easily implementable. The formuk, comes out of our work on more general multivariate distributed systems, and is embedded in our theory of skew Toeplitz operators. In order to explain our solution we first must recall the key fact that P0 is the norm of a certain operator (which is equivalent to the Hankel operator) and qopt may be computed from the corresponding singular vector. Namely, if S: H 2 ~ H 2 denotes the unilateral shift (all of our Hardy spaces will be defined on the unit disc D in the standard way),and P : H 2 --~ H E O O H 2 --: H denotes orthogonal projection, and if we set T ' = P S I H (T is the compressed shift), then po = [[ w(T)[[, w(T) may be defined as PAt,. [ H, where M w is the operator o n H 2 induced by multiplication by w.) In [3,7,12] procedures are given for the computation of P0. All of the~e amount to determining the invertibility of a 2n × 2n matrix where n .'= max {degree p, degree q } for relatively prime polynomials p and q such that w = p/q. Our contribution in this note is to give a procedure which reduces us to determining the invertibility of an n x n matrix and which can be written down almost immediately by inspection. (See Sections 3 and 4.) This could greatly speed up present computer implementations of our previous algorithms. Once again this comes from the theory of skew Toeplitz operators [2].

2. Background material In this section we will summarize some of the basic definitions of [2], and set up some necessary notation. We will first work in the general multivariate setting before specializing to the SISO

0167-6911/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

260

C Foias, A. Tannenbaum / Optimal interpolation

case. This will allow us to state how the matrix version of our algorithm should go through. Following [11], for A an r x s matrix-valued inner function, we set H ( A ) : = H 2 ( C r) ~ A H E ( C ' ) . Moreover for ~ ( a ) " HE(C,) --) H(A) orthogonal projection, we let

S(A):=PmA)SIH(A) where S is the canonical shift on H2(C'). (HJ(C ') for j - - 2 , c¢, denotes the j-th Hardy space of C~-valued functions on the unit disc D defined in the usual way.) Next let O denote a fixed N × N matrix-valued non-constant inner function. Then we set H,-H(O), T.'- S(O), and let S be the isometric dilation of T o n H2(C N) (i.e. S is the canonical shift on H2(CN)). Note that T ~ Co(Nl), i.e. the defect operators D r, Dr, have rank N 1 _< N, and

Tkh --) O,

T*kh ~ 0

for all h ~ H as k ~ ~ . (Recall that for a contraction B, i.e an operator such that [I B[I-< 1, the defect operator is given by D B := ( I - B ' B ) I/2. See [11] for details.) In particular I - T T * and ! - T * T are compact, and hence T and T* are Fredholm (i.e. essentially invertible). Now for Q ~ L(C N) (the^space of linear endomorphisms on C N), we set Q '= PHQI H, where Q is regarded as a multiplication operator on H2(CN). We are now at long last ready to define the main object of study in [2]. Set t!

j,k =o

where Cjk ~ L(C N) and Cj~. = Ckj. (Thus A - A*.) Notice that

may be regarded as the symbol of the skew Toeplitz operator A. Note however, that the correspondence between operator and symbol is not in general 1-1 in either direction. l a [2,4], it is shown how the solution of the most general H °° optimizatior~ problems (for multivariate distributed systems) may be reduced to the invertibility of an associated skew Toeplitz operator which is essentially invertible. We specialize here however to the SISO (numerical) one block case, where the formulae simplify considerably. Note that we are setting H j "= Hi(C) for j = 2, oo in our discussion below. So, using the above notation, we let N = 1. Let w ~ H °° be rational, and write w .'= p / q as a ratio of polynomials of degree _< n. For O > 0, set

A,'=q(r)(I-

1/p2w(T)w(T)*)q(T) *

= q ( T ) q ( T ) * - 1/p~p(T)p(T) *.

(1)

It is clear that Ap is not invertible iff p2~

a(w(T)w(T)*). (For B a bounded linear operator, o(B) denotes the spectrum of B; see [9].) Notice that if p > II w(T)lie (the essential norm of w(T)), then the non-invertibility of Ap is equivalent to p being a singular value of w(T). A~, will be the skew Toeplitz operator considered in this paper. Next we can clearly write n

a,= E Cj .,rJr

(2)

j,k=0

where the 'constants' Cjk,o depend on p. In other words, the determination of the singular values of w(T) reduces to the invertibility of a skew Toeplitz operator which is essentially invertible. We are thus led to consider the following kind of problem. Given a numerical skew Toeplitz operator n

A=

E

CjkTjT*k

(3)

j,k=O

where PH" H2(C N) ~ H denotes orthogonal projection. For obvious reasons we will call such an operator A skew Toeplitz. The matrix Laurent polynomial p!

E C:z j,t~ =0

with the property that 0 ~ o(A) iff 0 is an eigenvalue of A, give a procedure for the determination whether 0 is in point of fact an eigenvalue. Moreover (in order to find the optimal compensator), if 0 is indeed an eigenvalue, we must also compute an associated eigenvector. In Section 4, we will find a very simple formula for carrying all of this out.

C Foias, A. Tannenbaum / Optimal interpolation

3. Basic equalities In this section, we specialize some of the general results of [2] to the numerical case. We follow the notation of Section 2 above. Set

c,(~),= E g,,.,*."+'-"

(4)

j,k=O

We now make the following assumptions of gener-

icity:

261

where f ( z ) = f 0 + zfl + ' " is the Taylor expansion of f(z). Now from (5a), we have that there exist C (- 1), F ~ H °° such that

(7)

1 =CCt-1)-OF.

Multiplying (6b) by C t- l) and using (7) we see n--I

f(:.)- E

c'-"(~)c,(~)/,

I=0

{zeros of Co(z)} r i o ( T ) = ~ ,

(5a)

= O ( z ) ( z " E , ( z ) g ( z ) - F(z)f(z)) == O ( ~ ) h ( ~ ) .

and Co (z) has distinct roots all of which are non-zero.

(Sb) We will discuss below how to remove these assumptions. Note however that (5a) implies in particular that 0 ¢ o(Ao) iff 0 is an eigenvalue of A o. We should also add here that the genericity assumption (5a) is very easy to check in practice since from [11], we have that

(8)

If we apply the operator PH,O* to (8), we have n--l

h=

E Pu ' O - C ` - 1)Ctf,,

-

(9)

i=0

and so n--I

f = E (C(-°Ct-@PH'O*C(-"C~)ft •

(9a)

I=0

But from (6b), (9a), and (7), we get that

o ( T ) = {zeros of O in D }

n--I

U { essential singularities of O on aD } where aD denotes the unit circle. As we did at the end of the previous section, we will drop the explicit dependence of the various expressions on p, and thus set A = Ap, C = C0, etc. Finally z ~ C will denote a general complex variable, while ~' 0D. But now we clearly have that 0 ~ o(A) if and only if there exists a nonzero f ~ H ( O ) = H such that Af = O. This last condition is equivalent to the existence of f ~ H, g ~ H 2 such that f ~ 0 and n

E Cj, SJS**f = Üg.

I--0 n-I

n-I

= E c,/, + OF E c,/, 1=0

I=0 n-1

n-1

-oc E e..o*c'-"c,/,- E c./, I=0

I=0

n-I

= O E (FC,-CPn'O*Ct-"Ct)f, •

(10)

I=0

Now since 0(~') is unitary for almost ail ~"~ aD, we get that (10) holds iff

(6a) n--l

j,k = 0

E (Fc,- cP,,O*c,-',c,)(~)/,- ~"g(~).

Next set

c,(~)

~oo~ = c/- E c,/,

,=

i--0

E

(11)

c,~="'-~÷:-

O<_j
for 0 < l < n 1. Then from (6a) we get that 0 ~ o(A) iff there exist f, g as above such that

At:= ( F C , - CP,,O*C(-')C,)([)

n-I

C(z)f(z)- ~., C,(z)ft=z"O(z)g(z) i=0

But this brings us to the key point. Indeed since g ~ H 2, the Taylor series of z"g(z) has its first n coefficients equal to 0. Thus (11) is valid iff for the Taylor expansions (0 < 1 < n - 1)

(6b)

= Xto + ~;X,, + " "

(12)

C Foias, A. Tannenbaum / Optimal interpolation

262

sirrfibr. So multiplying both sides of (12) by @, and using (7), we have

we have n-1

E x,,/, = o

O X i = @FCI -

I=0

for0
=

OCPH20*C(-1)C!

( C C (-;) -

1)el- OCP,20*C(-1)Ci

= CPH(o)C(-I)cI

- Ct = C Y ! - C : ,

which completes the proof.

[]

Theorem 1 . 0 ~ o ( A ) if and only if

det

Lemma 2 means that XI~ H ~- is completely characterized as a solution of the equation

I x~ X~o "'" x~_,,o1 X°l

Xll

Xo.,,- i

Xl,,,- ~

X,,_ 1.1

"'"

ct=c~-ox,

=0.

X . - l.,,- i

(13) Proof. Immediate from the above discussion.

[]

Consequently, the key to gettmg, ~amplc procedure for computing the singular values of w ( T ) is finding the first n Taylor coefficients of the Xi. In the next section, we will see that this is quite easy.

(15)

for Yt~ H(O). But now we can find the requited sin~,ular values of w ( T ) , i.e. determine if A = Ap is invertible. Indeed degree C = 2n, and by our genericity assumption (5b) C has n distinct zeros in D (the unit disc), namely z t, z: . . . . . z,, and also the zeros 1/~q ..... 1/~,,. Note from (5a) that

e(~j),0

(16)

for j = 1. . . . . n. Thus from (15), we see that

X,( Zk ) = -- O( Zk ) - ' C,( zk ) 4. Main results

for k = l , . . . , n . Introduce the notation

In this section we give an easy way of computing the Taylor coefficient of the Xt and hence find our expression for the singular values and vectors of w t T ) . We use the notation of the previous sections here. Unless otherwise stated we will take 0 < ! _< n - 1 in what follows below. We begin with the following:

( - ~)Ct

= Pn(o)C(-1)Q,

(14)

and let X t be as in (12). Then

O X t = C Y t - C,.

,~(1/~ ) ,= ~/~ ',,a ( ~ ) for a ( z ) a polynomial of degree < 2n, and

G,(~) .= (~v,)(~), o,(f),= o-~,

~,(f).= x,(~)~ '~

where ~' ~ aD. Note that G t is analytic in ~ since Y~~ H(O). Now multiplying (15) by ~'2"O, we obtain

Lemma 2. (i) Set Y~ '= C ( - t)Ct - O P m O * C

(17)

(14a)

Note that X t ~ H 2, and Yt ~ H(O). (ii) Conversely if Yt ~ H ( O ) is such that there exists an X t ~ H 2 satisfying (14a), then Yt satisfies (14) and X I is given by (12).

Proof. We just prove (i). The proof of (ii) is

O'C, = ca,-

Xt .

(18)

This relation implies in particular that degree Xt _< 2 n - 1 since all the other functions are contained in L 2 0 H 2. Notice that although equation (18) was derived on OD, it extends analytically to the complement of D in the obvious way. We thus obtain

for all k = 1,..., n which yields

C Foias, A. Tannenbaum / Optimal interpolation

Xl(1/~k )

(19)

-- O( z, )Cl(1/~k )

=

for all k = 1,.. , n. Let now z,+j := i / f j for j = 1,..., n. We define the Lagrange polynomials, 2n

/ 2n

i----1

"--

/ i~j

i,j

for j = i , . . . , 2 n . Then n

x,(.,.) =

-

t.j(,,.)O(~.j)-'C,(,,.j)

E j=l n

- E Lj÷.(~)o-~c,O/~j).

(20)

j=l

263

tions to the above interpolation problem, i.e. we can explicitly parametrize the set

{q:

IIw-Oqll-
for p > P0. See [5] for all details. (iii) The genedcity condition (5b) can be eliminated as in [7]. (In this case we get a degenerate form of (22) exactly as in [7].) However, we do not see how to eliminate at this point, the genericity condition (5a). Hopefully this will not cause numerical problems in attempting to implement our algorithm, but we still must carefully investigate this issue. We should add that the previous algorithms from [3,6,7,12] seem to be numerically robust as evidenced by their implementation at the Systems Research Center of Honeywell and their application to actual design problems.

Now denoting 2n-1

5. Example

L,(,,.)= E Lj,~' i=0

In this section, v:e apply our above procedure to our favorite example and test case. The reader may compare the present method to others [3,6,7,12] that we have employed previously. We take

we have

x,,= -

n

E

.Lj,O(,,.j)-'C,(,,.j)

j=l Ft

- E L.+j.,O(z~)C~(1/~:j)

(21)

w(z)= 1-z 2

'

O

[z+l~

= exp[ z--Z--T ].

j=l

for all 1 = 0 , . . . , n - 1, i = 0 , . . . , n - 1. l~,at (21) can be written in the following matricial form: x;=

t x,,l

=

Then one may compute that here

(2)

!

c(,,.)=~z~+ 4- 7 z+p,

t Lki]

arid

- [ Ct (1/~.,)] [O--'(~6,k l t,-,,,+,,., ],

(1) 1

(22) where 8# denotes the Kronecker delta, and where 0 < 1, i _< n - 1, and 1 _
Corollary 3. With notation and hypotheses as above, p2 is an eigenvalue of w (T) w (T) * iff det X = 0. Remarks. (i) Notice that X is n x n. The eigenvector corresponding to 02 can be computed from (6a,b). (ii) Using our techniques from [5], we can also derive a parametrization of the suboptimal solu-

- 7

+ 7"

(Note that n = 1 in this case.) Moreover, the roots of C(z) are given by z,-(1-

202 ) + 2 i p 2 ¢ ( 1 / 0 2 ) - 1.

(Since II w II = = 1, w e may clearly take p ~ (0, 1).) But now from (22) and Corollary 3, we immediately get the following equation:

and

o-

22

~

--" ~r 1.

Z1

o(~)-'Co(~)

Z2 D Z1

+ ~ OZ (2 z , ) - ' C o ( z , ) . Z 1 -- Z 2

(23)

C Foias, A. Tannenbaum / Optimal interpolation

264

Then it is easy to compute that O ( z , ) = exp(-i~/(1/O 2) - i ) , and O(z 2) = 0---(~, and so plugging these expressions into (23) and simplifying, we get from Corollary 3 that the singular values of w(T) are precisely the roots of

(We should note that (13) extends to the multivariate setting; see [2].) Once again, the digital implementation of such an approach should somewhat improve our present methods in the area of H ~ optimization of multivariate distributed systems.

tan~/(1/O 2) - 1 + ~/(1/p 2) - 1 = 0 References

contained in the interval (0, 1). Thus in this case it seems that our present method is much more efficient than our previous procedures.

6. Conclusions

In this note we have given a very simple procedure for the computation of the singular values and vectors of operators of the form w (T). (Actually, it even gives a way of calculating the points of the discrete spectrum of w(T)w(T)*.) Formula (22) is so compact, that we believe it should even be of interest in the finite dimensional case, and perhaps even be more computationally efficient than present algorithms in this setting. In this regard in order to increase its value in computational operator theory, we would like to find a way of removing the genericity assumption (5a). Finally equation (15) generalizes to the multivariate case, and once again one must solve a linear equation of the form

C t ' = C Y t - O X i,

Y t ~ e H ( O ) forall ~ C

N,

where C, O are as in Section 1, and Ct is defined as in the numerical case. The solution of this last equation is discussed in great detail in [2], and of course we would like to develop a very simple expression for its solution along the lines of (22).

[1] J. Ball, C. Foias, J.W. Helton, and A. Tannenbaum, On a local nonlinear commutant lifting theorem, Indiana J. Math. 36 (1987) 693-709. [2] H. Bercovici, C. Foias, and A. Tannenbaum, On skew Toeplitz operators. I, Integral Equations Operator Theory, to appear. [3] C. Foias and A. Tannenbaum, On the Nehari problem for a certain class of L °~ functions appearing in control theory, J. Functional Anal. 74 (1987) 146-159. [4] C. Foias and A. Tannenbaum, On the four block problem. I, II, Integral Equations Operator Theory, to appear. [5] C. Foias and A. Tannenbaum, On the parametrization of the suboptimal solutions in generalized interpolation, submitred to Linear Algebra Appl. [6] C. Foias, A. Tannenbaum, and G. Zames, On the H°°-optimal sensitivity problem for systems with delays, SIAM J. Control Optim. 25 (1987) 686-706. [7] C. Foias, A. Tannenbaum, and G. Zames, Some explicit formulae for the singular values of certain Hankei operators with factorizable symbol, SIAM J. Math. Anal., to appear. [8] B.A. Francis, A Course in H °° Control Theory, Lecture Notes in Control and Information Science (Springer, New York, 1987). [9] N.K. Nikolskii, Treatise on the Shift Operator (Springer, New York, 1986). [10] D. Sarason, Generalized interpolation in H °°, Trans. Amer. Math. Soc. 127 (1967) 179-203. [11] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hiibert Space (North-Holland, Amsterdam, 1970). [12] G. Zames, A. Tannenbaum, and C. Foias, Optimal H ~ interpolation: a new approach, Proceedings of the CDC, Athens, Greece (1986) 350-355.