Numer. Math. 58, 641-660 (1991) Numerische Mathematik �9 Springer-Verlag 1991

Tracking poles and representing Hankel operators directly from data J.W. Helton 1, P.G. Spain 2, and N.J. Young 3' * 1 Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA 2 Department of Mathematics, University of Glasgow, Glasgow GI2 8QW, Scotland 3 Department of Mathematics, University of Lancaster LA1 4YF, UK

Received August 18, 1989/February 23, 1990

Summary. We propose and analyse a method of estimating the poles near the unit circle T of a function G whose values are given at a grid of points on T: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value of G on T, and then to estimate the modulus by seeking a best L 2 fit to G over a small arc by a first order rational function. These pole estimates lead to the construction of a basis of L 2 which is well suited to the numerical representation of the Hankel operator with symbol G and thereby to the numerical solution of the Nehari problem (computing the best H ~, analytic, approximation to G relative to the L ~ norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.

Subject classifications: AMS(MOS): 30-04; 30El0; 41A20; 65E05; 93-04; CR: G1.2.

1 Introduction

Some of the standard computat ional methods for the solution of filter design problems run into numerical difficulties when they are applied to systems having poles very close to the imaginary axis or unit circle. This is true for example for current algorithms for spectral factorization, solution of Lyapunov equations and the computat ion of Hankel singular values and vectors I-F]. At the same time there is serious engineering motivation for dealing with such systems. Stan- dard aircraft control problems often involve such poles. Even more technologi- cally pressing is the issue of the lightly damped beams occurring in satellite structures. There the frequency response functions have poles almost on the imaginary axis, so that design problems for satellites require the accurate numeri- cal manipulat ion of such functions.

* Partially supported by grants from the AFOSR and NSF

642 J.W. Helton et al.

What precautions can we take to alleviate the effects of these recalcitrant poles? Let us address this problem in the context of an important and representa- tive class of design problems: those that admit solution in terms of Hankel singular values and vectors. These are the singular values and vectors of the Hankel operator of the system [F, G], which may well be an operator of infinite rank since we are concerned with noisy data. We are obliged to work with a finite rank approximation to the operator, and the question we face is how best to obtain such an approximation. There are several possible approaches: which is the most appropriate depends on the form of the data of the system, as well as on the precise design problem under study. The most highly developed computational approach is the state space one: this presupposes a state space model of the system to be analysed. However, when there are system poles very close to the unit circle the identification process whereby one obtains a state space model is itself likely to be highly sensitive, so that in starting from a state space model one is ignoring a significant part of the difficulty. We prefer to enter the arena at an earlier stage of the analysis/design process by assuming data in the form of function values at a grid of points on the unit circle. These could be obtained, for example, by frequency response experiments or from specifications. As a concrete example we study the Nehari problem, though much of what we say adapts to other problems which are solvable via Hankel singular vectors. Accordingly we suppose we are given a bounded measurable function G on the unit circle T. For the purpose of analysis we assume G is known at every point of T; when it comes to computation, G enters via integration and so numerical integrals are evaluated with the aid of finitely many sample values. The Nehari problem is to find a function (~eH ~, the space of functions bounded and analytic in the open unit disc, such that the "e r ro r " G - ( ~ is minimised with respect to the essential supremum norm on T.

The fact that the Nehari problem and some of its generalizations can be solved numerically with the aid of Hankel operators lies at the heart of current H ~ control theory. Some basic design problems can be transformed to problems of Nehari type and hence solved - notably the H ~176 disc problem [HI, H2, HS] and the robust stabilization and the sensitivity minimization problems IF]. In particular, the H ~~ approach to robust controller design leads to precisely this problem IF] as does the predecessor to this subject, gain optimization of broad band impedance matching of passive circuits [H2].

A more general problem, termed the Carathrodory-Frjer problem by Tre- fethen, is that of approximating not by H ~ functions but by functions with fewer than a prescribed number of poles inside the unit disc. This occurs in in rational approximation IT], in gain optimization of active circuits [H3], in digital filter design [GST] and in reducing the order of a differential equation model I-G] (an area closely related to rational approximation). We surmise that the methods of this paper may apply to one of the steps of the Carathrodory- Fejrr problem, but we have not addressed the further step of projection onto H ~176 and we do not pursue that application here. Such problems do not occur directly in H ~ control, which is our main interest, so we restrict our attention to ordinary H ~176 approximation.

F rom a mathematical point of view the problem is solved by a theorem of Adamyan, Arov and Krein [AAK] expressing the desired function in terms

Tracking poles and representing Hankel operators 643

of singular vectors of a Hankel operator. If G is a bounded measurable function on the circle T with Fourier series

G(z)~ ~ G.z" n = - o ~

then the Hankel operator with symbol G is a certain operator H~ between infinite-dimensional Hilbert spaces. It can be represented by the infinite matrix

(1.1)

and can be thought of, in system-theoretic terms, as the mapping from past input sequences to future output sequences [G]. According to the AAK theorem the best analytic approximation problem for G is solved once the singular vectors of this operator are known.

Computationally the most straightforward approach to seeking the singular vectors of the infinite matrix (1.1) is to replace (1.1) by a large finite truncation and to compute the singular vectors of that. We shall call this method simple truncation. A similar idea has been studied and recommended in the context of model reduction by L. Trefethen et al. IT, GST]. If the operator (1.1) is compact and its largest singular value is simple then the singular vectors of the truncations will indeed converge to the singular vectors of (1.1) as the trunca- tions increase. However, if the Fourier coefficients of G decay slowly to zero, as will be the case if G is rational and has a pole very close to T, then one has to take extremely large truncations before the jettisoned entries are truly negligible. In practice simple truncation will often be good enough, but our purpose here is precisely to treat the cases where it fails. In [HY] it was shown that there is a superior form of truncation which allows more accurate approxi- mation of the Hankel operator and its singular vectors. We recapitulate the idea in w 4. However, to construct this "modified truncation" one needs estimates of the poles of G near T. In this paper we put forward a method of finding such estimates. We state a convergence theorem which supports the proposed algorithm and sketch its proof; the full proof appears in [HSY]. We give the results of tests which show how the conjunction of the present pole-tracking method and the modified truncation algorithm of [HY] are effective in solving the Nehari problem where simple truncation gives poor results.

We emphasize that our computed numerical representation of the Hankel operator H~ is derived directly from the data G, not from a high order rational approximation of G. There are algorithms for the Hankel singular values of an exactly known rational function g which entail no truncation error at all [PS, Y1]. It would therefore be feasible to pass from data G to an approximating g and then use one of these algorithms or a state space method. This is not what we do: rather, we use our pole estimates to pick finite-dimensional sub- spaces of the domain and codomain of H G and then compute the matrix of the compression of H~ to these subspaces by numerical quadrature using G itself. Thus we use the data twice. We believe that this is a significantly better approach. There is some discussion of the point in [HY].

644 J.W. Helton et al.

Our approach can be thought of as a way of finding a rough first order model for each "spike" or near-singularity of a frequency response function and of incorporating these models into the analysis. It would be interesting if analogous methods could be found for other important design algorithms such as spectral factorization. The preliminary extraction of poles by the present method would be a natural first step. We have not investigated this possibility.

The Nehari problem is customarily formulated for functions G which are essentially bounded and measurable on the unit circle, and the mathematical solution given by the Adamyan-Arov-Krein theorem applies in this generality. However, any numerical method which depends on finite matrix approximations of H~ only makes sense in the event that HG is a compact operator, which occurs if and only if G is the sum of a continuous function on T and an H ~ function [Po]. We therefore assume that G has this form. In this generality it does not make sense to speak of poles of G, and so there is a question as to what is meant by estimating poles in the case of non-rational G. Our analysis shows that if we find good estimates flj to the true poles ~j near the unit circle of some rational function which is close to G in the U~ then we can use the flj to get a good representation of H~ and hence to get a good approximation to the solution of the Nehari problem for G. Thus it is not essential to our method that G itself have well defined poles. This is important, since we envisage the case that G is a rational transfer function corrupted by additive non-rational noise which is small in the L ~ (T)-norm.

The plan of the paper is as follows. In w 2 we describe our strategy for tracking the poles near the unit circle of a rational function from its values given on a grid on the unit circle and in w 3 we state our convergence theorem and sketch its proof. We recapitulate the modified truncation algorithm, as proposed and justified in [HY], in w 4, then present the numerical evidence in w 5. We summarize our results in w 6.

2 The pole-tracking strategy

Our first aim is to estimate the poles near the unit circle T of the rational function G, presumed known in the form of function values on T. Briefly, our method is to inspect the graph of G to locate radii near which poles are to be found and then estimate these poles by means of a best L 2 fit over small intervals about these radii. More precisely, suppose G has a simple pole at

= I~le i~ where I~1 is close to 1, and suppose that we can find 7 (or an approxima- tion thereto) by inspecting the graph of G. We pick a small 0 > 0 and an arc I (~,, 0) of the unit circle: Fig. 1

(2.1) 1(~, 0)= {e": ~-O<t<~+O}.

We look for a best approximation to G in L2(I(?,, 0)) (with respect to Lebesgue measure) by functions of the form 2 (z - f l ) -1. That is, we define, for fixed ~ [ 0 , 2rt],

(2.2) D(fl, 0)= inf II a-2kp IIL2r162 ,,leC

+0)

ILO)

Tracking poles and representing Hankel operators

0 Fig. 1

645

where

1 (2.3) kt~ (z) = z - fl '

and for the chosen value of 0 we seek the value of fl for which D(fl, O) is a minimum. We expect that fl will be close to ~, and the effect of many dozens of numerical experiments is to confirm this expectation. In this section we give more details of the implementation, and in w 3 we sketch the proof of an inequali- ty which affords a measure of theoretical support for the method. We present some numerical results in w 5.

Our two heuristics are: A. A pole near T will typically give rise to a peak in the graph of ]GI or

of the derivative of arg G; B. If

G = a k ~ + h

where ~ = I~1 e i~, I~1 < 1, a ~ C, h ~ L ~ (T), and if we can f ind 7 by using Heuristic A, we may hope to recover ~t by choosing a suitable 0 and minimising D(fl, 0). O f course h must be small for this to yield a good estimate.

Description of the heuristics

Heuristic A. We scan the graph of [G[ in search of its peaks. We name the phases at which they occur candidate phases; and we later at tempt to find the corresponding candidate moduli, using our second Heuristic, so as to find candi- date poles.

The reciprocal Blaschke factor

G(z)-

has constant modulus on T for any pole ~. We cannot hope, therefore, to detect by examining the values of ]G]. We can, however, find the phase of such

646 J.W. Helton et al.

poles ~ by examining the derivative of arg(G). The determination of the modulus of ~ is more problematic (see Fig. 7).

Now poles very near Tcan mask other poles of smaller modulus: for instance, with

G = {(z - - 0.99 i)(z + 0.85 i)(z + 0.4 i)} -

the graph of IG[ is as follows:

60-

5o

~o ao

30

20

.-~ 10

.E

i i i

Poles at: 0.99i,-0.85,-tfi Fig. 2

i

7

We will therefore need to form G ~2) = GH(z - - ;6)

(where the fl are the candidate poles already found) and scan for peaks of [G~z) I in order to locate the (candidate) poles of smaller modulus. This process may have to be repeated several times; so to minimise the errors introduced by the use of candidate poles near, but not at, actual poles, we find it desirable to identify as many peaks as possible on each scan.

Instead, therefore, of looking for where [GI attains its maximum, we employ the following strategy.

We consider the set S(2)=[IG1>2] for 2~[0, maxlGI]. We hope to find a peak-isolating-level, a value of 2 such that these intervals form distinct bases for the peaks of G.

Clearly, if 2 is very small the structure of S(2) will reflect the local irregulari- ties in G; while if 2 is almost maxlG[ we may find only one peak, even though others may be quite apparent. For instance, with

G = {(z - 0.99 i)(z + 0.85 i)(z + 0.4 i)} - 1

(as above) we need to find a peak-isolating-level in the range [1.39, 4.39]; any lower level would conflate the two peaks, while a higher level would lose us the peak corresponding to -0 .85 i.

In practice we have considered values of 2 in a 10-step geometric progression from max[G] down to ]IGII~. We have then taken the peak-isolating-level to be the largest of these 10 values of 2 for which n(2), the number of disjoint intervals forming S(2), attains its maximum. This has worked very well.

Also, we have chosen to abort this peak-scan if max[G]<2 IIGI[1; this is to obviate useless searching for candidate moduli when G contains an amount of noise (this matter is discussed again later in w 5).

Tracking poles and representing Hankel operators 647

Heuristic B. By standard Hilbert space geometry we have

I(G, ka)l z D(//, 0)= I1G II 2

KI kp II 2 ,

the norms and inner product being those of L 2 (I (7, 0)). Thus finding the minimis- ing/~ is an unconstrained minimization problem in two real independent vari- ables. Note that it is easy to write down the gradient of D(., 0) analytically. We have tried two methods for this minimization problem: (1) a standard descent method, written in FORTRAN77 and using N A G routine E04GCF [NAG] ; and (2) the bisection method applied to the radius through eir and written, like the rest of the program, in MATLAB. The two implementations gave very similar results and so, for simplicity, we used (2) for most of our tests.

For each candidate-phase 7 we seek the value r, 0 < r < l , for which D(r) (=D(reie, O)) attains its minimum. If successful we obtain a candidate pole l = r e i~"

The shape of the graph of D varies considerably with the configuration of the poles of G, and in most cases hardly varies with changes in 0 (so long as 0 is small enough): but it is sensitive to the choice of 0 in the case of a reciprocal Blaschke factor (Fig. 7). We provide some examples.

Shape ofgraphofD

(i) G=(z-ro) -1 (O<ro< l ) :

4 4

" 2 . 2

" 5 m

~- 0 i i i i ~ l r

o 014 016 o8 ,0 f i r a p h o f D f o r pole e t 0.8

Fig. 3

The shape is similar for 7 near 0; but the minimum will not be attained at ro.

(ii) G=(z-ro)-1 (1 <ro): Fig. 4 The graph has a different shape: this allows us to distinguish between poles

inside and outside T. (iii) G=(z-ro) -2 ( 0 < r o < 1): Fig. 5 The graph of D has shape similar to that for a single pole: the minimum,

however, does not occur at ro.

(iv) G= z - r ~ with ro,r l close: Fig. 6 Z - - T 1

648 J.W. Hel ton et al.

..~ 11+0 t

,, 100 t

200 1

It

~- 50 L 0 0'2 . . . . . o '8 0./* 0.6

Graph of O for pole at 1.02 Fig. 4

110

1. ,, ~.0 /

, ,O

I I

m

t - . - :1 0 0.2 0.~, 0.6 0.8 1.0

Graph of 13 for double pole af 0.8

Fig. 5

..~ 8~ '~ /*0

0 i I

"~ 200

,, 100

0 1 2 ' ' ' o 1 6 ' ' ' ' 0 0./* 0.8 1.0

Graph of D for poles at 0.8, 0.8001

Fig. 6

Tracking poles and representing Hankel operators

(v) G - 1 - r ~ ( 0 < r o < l ) : Fig. 7 Z--Y 0

The pole-tracking algorithm

649

Scan I G(k) I for candidate phases

II Examine D to find

candidate modulus at each candidate phase

Form G (k+ 1)

Terminate III

I. Put G~I)= G and implement Heuristic A. The candidate phases are identified as those where [G(k) t attains its maximum

over eack peak-base. The candidate phases, together with peak-base lengths, are forwarded to

the next stage, dD II. We work with the explicit expression for the function ~ and seek its

dD d D < o zero by bisection. We abort if at any stage ~r-r > 0 at a left end point or dr

at a right end point. This procedure leads us to a genuine minimum when D has shape as in Fig. 3 while saving us from interpreting a maximum such as in Fig. 4 as a minimum.

The half angular interval 0 is taken to be the peak-base-length. Figure 6 shows that we cannot expect to determine the pole of a reciprocal

Blaschke factor by using Heuristic B directly: our dynamic choice of 0 may lead to either shape for D, and the stopping-rule may prevent us from finding the pole. If, however, we first subtract the 0 th Fourier coefficient (as calculated from the values presented) we obtain excellent results. In contrast, the more elaborate minimization routine E04GCF [NAG] does find the pole in all cases tried.

A genuine minimum for D provides us with a candidate modulus, hence with a candidate pole.

We run this stage of the algorithm for each candidate phase provided by I.

III. If some candidate poles have been found we form G (k+ 1)__ G(k)H(z_fl) where the fl are those candidate poles, and return to stage I.

If no candidate pole has been found at stage II the algorithm terminates. The modified truncation algorithm can then be performed, after choosing

a value for N, the number of extra candidate poles at 0.

650 J.W. Hel ton et al.

~-- o.15

.~ O.lO

0.05

~- 0.7 i i

0.6

0.5 0 0.2 O.t, ' 0'.6 0'.0 I;0

Graph of D for pole at 0.95 & zero at I/0.95

Fig. 7

Zeros of a rational function

Our procedures could also be used to estimate the zeros of a rational function near and inside T. We have not pursued this possibility.

3 A pole estimate

We can show that if G is made up of a pole 1/(z- cO and an additive remainder which is suitably dominated by the pole term, then a condition which is near to the condition that fl minimize D(fl, O) does indeed ensure that fl is close to ~. In terms of the notation (2.I)-{2.3) we have:

Theorem. Let ~ D have argument ? and let G=k~+h for same h~L~(T), where k~(z)= 1/(z-a) . Let e>0. Let O > 0 . Then there exists 6 > 0 (depending only on ~, e and O) such that, if

(i) IIh]lL~O(,(e,e))<6 and (ii) flissuchthatD(fl, q))<D(cqrp) forall qoe(O,O)

then If l-el < e.

The proof of this theorem can be found in full in [HSY]. Here we shall confine ourselves to an outline.

We introduce the notation

K(~,fl, 0)= Itk=ll 2 [(k~'k~)12 II k~ 1t2 ,

where the norms and inner products are those of L 2 (I (7, 0)), and we put

~ = 1 - 8 , ~ = l - f t .

Recall that [(G, k#)[ 2

D ( K 0 ) = II Gtl 2 IlkpH 2

Tracking poles and representing Hankel operators 651

In tegra t ion of the Tay lo r expansions leads to the est imate

o 0 ~ 1 1 1 2203 0 4 (3.1) K(~x,p, ) = ~ - ~ - ~ ~ - + O [ ~ ] .

G e o m e t r y shows that

(3.2) g(~,/3, 0 ) < 4 It hll 2

if D (/3, 0) < D (a, 0). Combin ing (3.1) and (3.2) leads to the est imate

(3.3) 1~-/312 < {12~41~12 IJh 1}2~,(~,o))} q~-2

3 M + ~ p , 0<~p < q~

where M depends only on a and 4~ only on a and O. Considera t ion of the value of r for which the right hand side of (3.3) at tains its m in imum demons t ra tes the existence of a fi as asserted by the theorem.

The theorem does not precisely analyse our a lgor i thm of w 2, but it comes close. In the a lgor i thm we choose a small 0 and then find/3 to minimize D(/3, 0), so that necessarily D(/3, 0) < D (a, 0). This is a weaker condi t ion than (ii) of the theorem, but in practice we have found that the point /3 obta ined is not very sensitive to the choice of 0 (so long as 0 is not too big), so that typically the chosen /3 will in fact satisfy (ii) for some O. The reason that we are obliged to assume the s t ronger p roper ty (ii) to obta in our inequali ty is that we allow a wide class of noises (that is, we have put only weak assumpt ions on h). Wi thou t a more restrictive noise model we must expect the analysis to be pessimistic.

4 The modified truncation algorithm

Suppose given a function GeL ~ the space of bounded Lebesgue measurab le functions on T with essential s u p r e m u m norm. Let H ~ denote the space of bounded analytic functions on the open unit disc D. Then H ~176 can be identified with a closed subspace of L ~ (see [ H o ] or [Y2]). Subject to mild restrictions on G (it suffices tha t G be cont inuous - [ A A K ] ) there exists a unique closest point d in H | to G with respect to the L ~ norm, that is, a unique d e l l ~ such tha t

1[ G - d II o~ = inf [l G - F [I o~. F E H ~

To find this G we make use of the Hankel operator

HG : H 2 ~ L 2 O H 2

defined by Hax=P_(Gx) , x e H 2.

Here L 2 denotes the space of measurab le functions on T square integrable with respect to Lebesgue measure with its na tura l inner product , H z is the H a r d y

652 J.W. Helton et al.

space consist ing of the functions in L 2 whose negative Four ie r coefficients vanish, Q denotes o r thogona l complemen t and P_ : L 2 ~ L 2 O H 2 is the o r thogona l pro- ject ion opera tor . We say tha t x is a maximising vector for H a if x + 0 and

II Ha x II : II H a ]1 It x II.

If G is cont inuous on T then H 6 is compac t and so H a does have maximis ing vectors. The theorem of A d a m y a n , Arov and Krein tells us that G is given by

(~ = G -- HG~v

where v is any maximis ing vector of H a . An exposi t ion of this mater ia l can be found in [Y2].

We have to compu te v and H o v. N o w maximis ing vectors are the same as s ingular vectors of the ope ra to r cor responding to its largest s ingular value, so we are faced with a singular value problem. This can be solved by l ibrary rout ines once a matr ix representa t ion of H a is found with respect to a pair of o r t h o n o r m a l bases in H 2 and L2~H z. Here we encounter a difficulty: as bo th these spaces are infinite-dimensional, an infinite mat r ix will result. In part ic- ular, if we take the obvious bases {1, z, z 2 . . . . } and {z- 1, z-Z, z -3 , ...} then the cor responding mat r ix is (1.1). Trunca t ing (1.1) is equivalent to considering the compress ion of H a act ing f rom span { 1, z, z 2 . . . . . z N} to span {z - 1, z - z, . . . , z - M} for some N, M e N , and since these spaces are not tuned in any way to the characterist ics of G they are unlikely to be the best choice. To see how to do bet ter observe tha t the singular vectors of H a cor responding to non-zero s ingular values lie in the space

Coke r H a = ( K e r Ha) •

so tha t it suffices to pe r fo rm the singular value decompos i t ion of the compress ion of H a acting f rom Coke r H a to Range H a. N o w Coker H a is determined by the poles of G. If G is ra t ional and its poles a l . . . . . a,. are all s imple then Coker H a is the linear span of the functions (1 - ~j z ) - 1, 1 < j < m. More generally, the conclusion still holds if (z-aO.. . (z-a, . )GeH ~, the a's being distinct. If we can use the k n o w n values of G to derive est imates fll . . . . . fir, of the true poles ax, -.-, a,. then we can work with a "modi f ied t runca t ion" of H a consisting of a compress ion of H a act ing on the linear span of the functions 1 - f i j z , 1 _<_j < m. This space is close to the true cokernel of H a when the flj are sufficiently close to the at : this is made precise and justified in [HY] . Range H o is also de termined (in the case of ra t ional G) by the poles of G, so we m a y also approxi - ma te Range H a using the fit's:

Range H a = s p a n { ( z - ctj)- 1 . 1 <j<m} - span {(z - flj)- 1 : 1 < j =< m}.

We call our es t imates i l l , . .- , 13,. of a l . . . . , am candidate poles. Once these candi- date poles are chosen the compress ion

H a : span {(1 - f i j z)-1} ~ span { (z - f l j ) -1}

Tracking poles and representing Hankel operators 653

is de termined. / /a is an operator between finite-dimensional spaces, so we may find a maximising vector f o f / /~ by standard linear algebra. We then form

/ /6 ~ = G

as our computed approximation to G. In practice we do not try to estimate all the poles aj. Both theory and

practical experience show that it is only the poles near the unit circle ("slow modes") which create difficulties for the numerical representation of Ha. The part of Coker H a corresponding to small I~jl (say I~jI <0.8) can be quite satisfac- torily approximated by the span of standard basis vectors z j. Fortunately the poles which are awkward for computation are the ones which can be detected comparatively easily by inspecting the values of G on T. This leads to the following high level procedure.

M o d i f i e d t r u n c a t i o n a l g o r i t h m

for computing an approximation G to the closest point G~H ~ to a given rational function G ~ L ~176

1. Find estimates fll . . . . . fir, of the poles of G in the annulus {z: 0.8 < [z I < 1}. 2. Choose a positive integer N~N. 3. Compute the matrix M (with respect to orthonormal bases) of the com-

pression//G of H6 acting from the linear span of the functions

1, z , z 2 . . . . . z ~ - 1 , (1--/~l z) -1 . . . . . (1-- fl,, z) -1

to the linear span of the functions

z - l , z - 2 . . . . . z -N , (z- /~ , ) -1 . . . . . ( z - / L 0 -1.

4. Perform the singular value decomposition of M and so obtain a maximis- ing vector f o f / / 6 and its image / /6 ~-

5. Form the desired approximation

~ = G / / 6 f f

We refer to Theorem 3 of [HY] for a theoretical justification of this algorithm. If a restriction of H a is used instead of the compression H G then (~ tends to

in the L ~ norm as flj--* e~, 1 < j < m, and N ~ ~ : furthermore, a rate of conver- gence is given. It is shown that if G is rational and the largest singular value of H6 is simple then there exist A, B > 0 such that

II (~ -- ~ II ~ < A (0.8) N/E + BS, Ifl ~ - otjl

for sufficiently large N and flj sufficiently close to ~j. It seems likely that the same conclusion remains true for the algorithm described here.

Our main purpose in this paper has been to put forward a method for stage 1 above and to describe our experience of it; however, it is worth pointing

654 J.W. Helton et al.

out what is involved in implementing stage 3, which at first sig.ht looks tricky. The basis functions 1, z . . . . . (1-f l , , ,z)-~ for the domain of H e are typically not or thonormal , so we actually use the basis

1, z, zU_t ,zN_ l (1-[/3~12)~,zN_ l z - f l , (1-1/3212) r .. . . i -~7~ z l,--/~-t z" I - /~2 z

z~_ * z - - f l , z-- f l2 ( 1 - Ifl,~[z) ~ ... . l__ffl z 1 - - f l 2 Z " " 1--flmZ

which is or thonormal and has the same span (note incidentally that with this choice of basis one need not insist that the fls be distinct). Call this basis e, . . . . ,eN+,,,. Similarly we need an or thonormal basis for the span of z - ' . . . . . (z - tim)- 1, and we take f l . . . . . fN + m where

m

j = l

The matrix of M of/:/G with respect to these or thonormal bases is now easily computed: the (i,j) entry of M is

m,j = (I:Ia e j, fi)

= (HG e s, fl)

=(Gej , f l )

1 2n = 2--n ~ G(ei~ e j ( e i ~ dO

6

and so M is computed with the aid of library quadrature routines. In the event that N is large a significant proport ion of the entries of M are simply Fourier coefficients of G, and so some saving can be made by the use of the fast Fourier transform.

Our algorithm includes the simple truncation method: if we choose no candi- date poles (m = 0) we obtain precisely the N x N truncation of the infinite Hankel matrix (1.1).

5 Numerical results

We present a small and representative sample of the many trials of our approxi- mation program. We have tried many different symbol functions G, mostly rational but some with added noise.

The programs were written in M A T L A B and run on the Glasgow Local Area Vax Cluster. The second author here records his sincere thanks to Dr. W.W. Stothers and Mrs. L.M.L. McCormick for their continued and unstinted support.

The examples presented here were (mostly) run using a grid of 2048 points equally spaced around T.

Tracking poles and representing Hankel operators 655

The function G was defined through its specified zeros ~j and specified poles ek ; i.e.

(z) = 1I (z - ~ ) / r l (z - ~ ) .

In the tables below we use the following symbols:

The specified pole (s), i.e. the true poles of G. Grid The number of grid-points on which G is known. NormH~ ]l HG {I calculated from the singular value decomposition of stage 4

of the modified truncation algorithm using the specified pole(s). // The candidate pole(s) obtained by the algorithm of w 3. NormH/~ ]I Ho II calculated as NormHa, but using the candidate pole(s). /~ This is an L ~176 measure of the error due to the use of the candidate

poles rather than the true poles. The modified truncation algorithm is run once using the true poles and once using the candidate poles. This produces two approximations (~, and G~ to the desired best ana- lytic approximant G. We define/~ to be H ( ~ - G~ I100.

When C=(z-~)-'

we can find tl HG tl and (~ explicitly:

II HG Ib = (1 -- balE) -1, •(z) = ~(1 --1~12) -1

We also have expressions for 11Ho 11 and G when

1 G=

( z - ~ , ) ( z - ~ 2 )

where ~1,0~2 eD\{0}, ~1 + ~2 # 0. We put

K = l l - a l ~21, o = ( 1 - 4 j ~2)/~,

m~=l--{~jl 2 ( j= 1, 2).

Let 2 be the larger root of

Then

Now with

we have [Y3]

~2 -- {m 2 +/,/,/2 _}_/,nl m21(z1 (D ..}_ (z 2 (,~12})~ j_ DI12 D12 = 0 .

tlHGII = V~ m 1 m 2 s:"

7 =2~1 oS+m~Z 4z co, 6 =41 ~5+42 o~,

G(z)= 4, y(1 --4 z z)+m~ 52(1 --41 ~2) 3 m, ,~ { ( ~ - m~)(1 -a2 z)+ml m2 6 ~ ( z - . , ) } "

656 J.W. Helton et al.

We thus can also calculate:

N o r m H tl Ho tl as calculated from the explicit expressions above. The supremum norm of the difference of G calculated using the candidate poles and the true G as given by the expressions above.

The quant i ty /~ is the proper indicator of the accuracy of the results; however, we have to make do with /~ except when G has one or two known poles. Nevertheless, the almost perfect agreement of/~ and E encourages us to believe that E is a good indicator for more general G.

I. Single specified pole: G(z)= 1/(z--ct)

The following results were computed with no extra candidate poles at 0 (i.e. N = 0), so working with a 1 x i compression of H~.

ct NormH Grid NormH~t fl NormHfl /~ /~

0.99 52.1256 64 311.063 0.99 311.069 6.5.10 -3 260 256 81.453 0.99 81.4545 1.86.10 - 3 31

2048 50.2513 0.99 50.2513 1.04.10 - 3 2.47 * 1 0 - 6

0.99 exp(ni2-11) 52.125 2048 50.2571 0.989806 49.9669 7.62 0.0794

0.999 500.2501 2048 996.002 0 .999 995 .63 0.441104 495 4096 587.474 0 .999 587.354 0.247744 87.224 8192 502.786 0 .999 502.779 0.196 2.535

2048 points were used in all the examples described below.

II. Single specified pole + random noise

We ran the programs on functions of the form

G = ( z - ~ ) -1 +0.1 * rand,

where rand is a random function uniformly distributed on [0, 1].

NormHct fl NormHfl /~ /~

0.99 50.2502 0.98991 50.2502 0.047 0.098

This tends to show that the procedures are numerically robust. However, the algorithm had to examine very many candidate phases on

the second phase-scan before rejecting them. We have therefore chosen to abort the scan if max lGI < 2 II all 1 in order to avoid searching in vain for candidate moduli when G contains noticeable noise. As a result, poles of modulus <0.55 may not be detected. The effect of ignoring such poles should be compensated by taking N sufficiently large (this is equivalent to taking a large number of candidate poles equal to zero).

Tracking poles and representing Hankel operators 657

III. Double specified pole:

NormH

0:~] 62.1984

0.99] 0.99] 6053.28

a (z) = I / ( z - ~ l ) ( z - ~2), ~, = ~

NormH~ fl N NormHfl /~ /~

62.1984 0.950043] 0.852387+0.06551ij 0 6 1 . 2 6 1 3 22.7263 22.7263

25 62.1605 5.2058 5.2058

6053.29 0.995381] 0.984084+0.00906ij

IV. Two specified poles, no specified zeros

0 5951.77 2213.37 2213.41 25 5970.82 1885.12 1885.13

The results are rather good, so long as the poles are angularly separated. The results improve as N increases (as we add extra candidate poles equal to 0).

We have not included the values of N o r m H a below because they agree exactly with those of N o r m H (to the accuracy of Matlab).

NormH fl

0.99 ] 0.99ij 36.2741 0.990082 ]

0.990086ij

0.99] 0.9iJ 37.5584

O0:~i] 4.4921

o%] 5.41666

0.9 ] 0.9+0.1i 53.2516

0.990099] 0.906422 +0.00556ij

0.904802+0.005552i] 0.005553 +0.905062i]

N NormHfl /~ /~

0 36.2465 0 .313254 0.313254 20 36.2467 0 .259118 0.259118

0 37.5575 0 .534379 0.534382 20 37.5578 0 .307758 0.307757

0 4.48649 0 .435076 0.435076 20 4.49201 0 .065127 0.065127

0.908274+0.00278i 0 5 .3902 3 . 0 7 2 8 9 3.07289 20 5.41647 0 . 0 9 1 1 5 0.09115

0.936513+0.05753i ] 0.866377-0.026588i

0.95] 0.5iJ 9.51565 0.951872

0.95] 0.3iJ 10.2266 0.952162

0 52.1276 23 .3042 23.3042 20 53.1965 6 . 2 8 8 4 3 6.28843

0 9.50529 4.3249 4.3249 20 9.51518 0.1626 0.1626

0 10.2163 3.2563 3.2563 20 10.2262 0.1556 0.1556

E Several specified poles, no specified zeros

NormH~ fl

0.991 0.9i /

-0.5 3 25.1994 0.990236 ] 0.916818ij

N NormHfl /~

0 25.1957 3.02232 10 25.1964 1.25958 20 25.197 0.516664

658 J.W. HeRon et al.

VI. With both specified poles and zeros

Reciprocal Blaschke factors: (1 - ~z) / (z - ~)

Using unmodified function:

N o r m H fl N NormHfi

0.95 1 No candidate poles found 20 0.871488

After subtracting 0 th Fourier Coefficient:

N o r m H fl

0.95 1 0.950001

0.99 1 0.99

N NormHfl

0 1 20 1

0 1 20 1

1.61057 0.681543

E E 1 .6 .1 0 -5 1 . 7 . 1 0 - 5 6 .6 . 10 - 6 6 . 0 . 1 0 - 6

2 . 5 . 1 0 - 5 2 . 6 . I 0 - 5 2 . 1 . 1 0 -s 2 . 1 . 1 0 -5

Four poles, two zeros

NormH~

0.98 ] 0.6i ]

- 0 . 7 - 0 . 8 9 i ]

0.6 0.99 i 5.3092

--0.00509 -0 .8979i | 0.979225[

--0.659811 -0 .00486 i ]

NormHfl /~

0 5.30687 1.26559 10 5.30803 0.389418 20 5.30849 0.181003

6 Conclusion

In this paper we give firm evidence in support of an algorithm proposed in [HY] for the numerical solution of the Nehari problem. The theoretical solution of this problem is in terms of Hankel operators, and the most straightforward approach to the representation of such operators is the simple truncation meth- od. There are important cases, however, where this method is unsatisfactory - when the symbol G is a function having poles close to the unit circle T. It was shown in [HY] that estimates of the poles of G near T can be used to obtain better representations of the Hankel operator HG and hence to derive better approximations to the solution of the Nehari problem. In the present paper we address the problem of estimating the relevant poles of G from sampled

Tracking poles and representing Hankel operators 659

data values on T. Our main result (in w 3) is a convergence theorem. We also present numerical evidence in favour of our method (w 5).

The upshot of our numerical trials is on the whole satisfactory: we improve markedly on the simple truncation method. We do not, however, claim to cope with all cases, and in particular we have not attempted to study the case of functions with multiple poles near T, though we think it likely that a straightfor- ward modification of our method would give good results.

In the first stage of our pole estimation process we look for the phases of the poles close to T. This we typically achieve to higher accuracy by seeking the peaks of hGl: even the phase of the pole c~ of the reciprocal Blaschke factor (1-~z)/(z--c~), which has constant modulus, can be determined precisely if we examine the derivative of arg(G) rather than the values of I G].

The second stage, finding the moduli, does not produce the same accuracy. Errors can accumulate on successive scans when dealing with a function that has several poles. The results are excellent for functions with a single pole and also for functions with two poles so long as they are not too close angulary: compare the results for (0.9, 0.9) and (0.9, 0.9 + 0.9 i). It is typical that we obtain two candidate moduli that straddle the correct value when we seek a double pole (w 5, III); we expect, however, that we could find the modulus of a suspected double pole by fitting a function of the form ()cz + #)/(z-~)2.

Our method may fail to find a candidate pole when we apply it to a reciprocal Blaschke product (see Fig. 7); if, however, we first subtract the 0 th Fourier coeffi- cient (calculated directly from the data) we obtain excellent results (w 5, VI); and the more elaborate minimization routine E04GCF [NAG] does find the pole in all cases tried. (It may be advantageous to perform this first step in all cases.)

For reasons explained in w 5 II we do not detect poles with modulus smaller than about 0.55. Luck is apparently on our side, though, for these poles are not too important if the number N of extra candidate poles at 0 is chosen large enough. It is instructive to compare the examples (0.9, 0.9 i) and (0.9, 0.3 i). In the first of these both poles are found and the result is quite good with N = 0 , improving by a factor of 7 when we increase to N =2 0 . In contrast, in the second, the smaller pole is not detected and the result with N = 0 is poor; but with N = 2 0 the accuracy increases 34-fold, giving almost the same relative accuracy.

Our results are significantly better than those of the simple truncation method when applied to the function 1/(z-0.999): with 8192 points we obtain the norm as 502.779 (the correct value being 500.250) and our computed best analytic approximation differs in norm from the true value by only 0.196, using just a one-dimensional compression. In contrast, a 55 x 55 truncation of the standard Hankel (1.1) yields a value of 52 for the norm.

In our implementation we have worked with an equally-spaced grid of points on T. This has obvious limitations: in particular it is unlikely to give good results for a pole whose distance from T is much less than the sampling interval. It is clear that there is scope for the refinement of the method by the use of an adaptive sampling technique. Where a pole has been detected close to T it is to be expected that faster sampling near this pole will yield both better pole tracking and more accurate computation of the matrix Ha. We recommend this as a promising try to anyone who requires greater accuracy than we have obtained by the methods described above.

660 J.W. Helton et al.

References

[AAK]

IF]

[G]

[GST]

[H1]

[H2]

[H3]

[HS]

[HSY]

[HY]

[Ho]

[NAG] [Po] [PSI

IT]

[Y1]

[Y2] [Y3]

Adamyan, V.M., Arov, D.Z., Krein, M.G.: Infinite Hankel matrices and generalized Carath6odory-Fej6r and Riesz problems. Funct. Anal. Appl. 2, 1-18 (1968) (English translation) Francis, B.: A course in H~ control theory. Lect. Notes Control Information Sci. Vol. 88, Berlin Heidelberg New York: Springer 1986 Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L~ bounds. Int. J. Control 39, 1115-1193 (1984) Gutknecht, M.H., Smith, J.O., Trefethen, L.O.: The Carath6odory-F6jer method for recursive digital filter design. IEEE Trans. Acoust. Speech Signal Process. 31, No. 6, 1417-1426 (1983) Helton, J.W.: Worst case analysis in the frequency domain: an H ~ approach to control. IEEE Trans Autom. Control 30, 1154-1170 (1985) Helton, J.W.: Broadband gain equalization directly from data, IEEE Trans. Circuits Syst. 28, 1125-1137 (1981) Helton, J.W.: The distance from a function to H ~ in the Poincar6 metric; electrical power transfer. J. Funct. Anal. 38, 273-314 (1980) Helton, J.W., Schwartz, D.F.: A primer on the H ~ disk method in frequency domain design: control. Department of Mathematics, University of California, San Diego 1985 Helton, J.W., Spain, P.G., Young, N.J.: Tracking poles and representing operators directly from data. University of Glasgow, Department of Mathematics Preprint Series 89/51, 1989 Helton, J.W., Young, N.J.: Approximation of Hankel operators: truncation errors in an H ~176 design method. In: Griinbaum, F.A., Helton, J.W., Khargonekar, P. (eds.) Signal Processing Part II: Control Theory and Applications, pp. 115-137, Berlin Hei- delberg New York: Springer 1990 Hoffmann, K.: Banach spaces of analytic functions, 1st Ed. Prentice Hall, Englewood Cliffs, NJ 1962 NAG Fortran Library Manual, NAG, Ltd., Oxford 1988 Power, S.C.: Hankel operators on Hilbert Space, 1st Ed. London: Pitman 1982 Pernebo, L., Silverman, L.M.: Model reduction via balanced state space representation. IEEE Trans. Autom. Control 27, 382-387 (1982) Trefethen, L.N.: Near circularity of the error curve in complex Chebyshev approxima- tion. J. Approx Theory 31, 344-367 (1981) Young, N.J.: The singular value decomposition of an infinite Hankel matrix. Linear Algebra Appl. riO, 639-656 (1983) Young, N.J.:An introduction to Hilbert space. Cambridge University Press, 1988 Young, N.J.: Best analytic approximation of rational functions of degree 2. Lancaster University, Department of Mathematics Technical Reports Series MA90/33, 1990