Approximation of delay systems by fourier-laguerre series

0005-1098191 $3.00 + 0.00 Pergamon Press plc ~) 1991 International Federation of Automatic Control

Automatica, Vol. 27, No. 3, pp. 569-572, 1991 Printed in Great Britain.

Brief

Paper

Approximation of Delay Systems by FourierLaguerre Series* JONATHAN

R. P A R T I N G T O N t

Key Words--Approximation theory; convergence analysis; delays; error analysis; Fourier analysis; Laplace transforms; model reduction; system order reduction.

Abstract--Error estimates are given for the approximation of stable recorded delay systems in the L 2 and H® norms, using two recently advocated techniques based on Laguerre series. In addition, some theoretical results on L~(0, oo) approximation are derived.

therefore helpful to have theoretical results indicating how fast they do converge. In many ways the most common, as well as the simplest infinite-dimensional systems to analyse are those arising from delay differential equations (delay systems), and in particular the retarded delay systems (defined in Section 3). We give a detailed analysis of convergence rates of the two methods based on the concept of the index of a delay system, a notion which is of importance in other methods of approximation. In Section 4 we give a discussion of a somewhat different norm, the time-domain L® norm, and give some theoretical results on approximation in this norm, with special reference to delay systems. Finally, Section 5 contains an example: we analyse one particular delay system in detail, and review all the approximation schemes for which error bounds are available.

1. Introduction CONSIDER A stable continuous-time linear time-invariant system, (which will usually be infinite-dimensional), with impulse response g(t) defined on (0, oo) and transfer function the Laplace transform of g, i.e.

C(s) = J; e-"g(t) dt. We restrict our discussion to single-input single-output systems, for simplicity of notation; however our results extend to multiple inputs and outputs. The problem of approximating g (or G) in various norms is one which has been much studied in recent years. In particular we shall be interested in the following norms:

2. Two techniques for approximating systems We shall analyse two techniques for approximating systems by systems with a single (repeated) pole. Both have appeared recently in the literature and have applications to delay systems. The first approximation technique that we shall discuss is given by Gu et al. (1989), and involves using the isometry between H®(C+) and the classical H® space on the disc, induced by the conformal map z =Ms = ( 4 - s ) / ( 4 + s ) , s = M-lz = 4 ( 1 - z)/(1 + z); here 4 is a fixed positive real number. In general we shall not be concerned with the actual value of 4, since the estimates hold whichever value we take; however, there is evidence (see e.g. Giover et al., 1990c) that best results may be obtained by taking 4 to be of the same order as n, the degree of the approximation. We are therefore led to consider the following expansion of a transfer function G(S):

(the L2 norm) and IIGII® = sup {llG(s)l[ : Re s > 0 } (the H~ norm). Note that we also have l

~

1/2

[see e.g. Partington (1988), Chapter 2], and we shall use the notation IIGII2 to denote Ilgll2. The H~ norm of a system has been shown to be of interest for problems of control system synthesis (Francis, 1987), whereas the L2 norm finds applications to questions of system identification (see e.g. Wahlberg and Ljung, 1986) as well as in LQG control; both norms have been considered from the point of view of model reduction (i.e. rational approximation) in a large number of papers. In Section 2, we discuss two recently suggested techniques for approximating systems, (Gu et al., 1989; M~ikil~i, 1990b), both based on Fourier-Laguerre series; these have advantages through being easier to calculate than more rapidly converging approximations (such as the Hankel-norm and truncated balanced realization methods), and it is

G(S)= ~=oCk(~--~+Ss)k,

where the (Ck) are constants (in fact Gu et al. consider the slightly more general matrix-valued case but this adds few extra difficulties). Moreover, if we write H(s)= V ~ / ( 4 + s)G(s) then we have (4 n(s) =

s) k

~o c~V~

so that we can identify the constants (Ok) a s the coefficients in the Fourier expansion of H(s) with respect to the orthonormal basis

~-- (4 - s) k

* Received 9 January 1990; revised 24 July 1990; received in final form 22 August 1990. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor H. Kimura under the direction of Editor H. Kwakernaak. t School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K. ~ro 27,3-x

(2.1)

(ek(s)) = V24 (~ ~

S~-'~I

of the Hardy space H 2 ( C + ) using the norm IIGII2 above, induced from L2(0, ~). (For simplicity of notation we suppress the dependence on 4.) Let LkCt)= e'/k! dk/dtk(t% -') be the classical Laguerre polynomial [see e.g. Szeg6 (1939) for details]. Then the 569

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functions (ek) are the Laplace transforms of the functions $ i ( t ) = ( - 1 ) " k V ~ e X'Lk(2At ), which form an orthonormal basis in the space L2(O, oo). A degree-n approximation to G(S) is given by

G"(s) = ~ o c k k ~ s }

"

by

3. Decomposition of retarded systems and error bounds The most general SISO retarded delay system has the transfer function

zk(I+z)

-~

o

and hence d o / V ~ = c o and (dk + d k _ O / V ~ = ck for k ~ 1. This technique was used in Glover et al. (1990c) with chosen to depend on n, and has been recently employed by Miikil~i in a n u m b e r of papers (including M~ikil~i, 1990a, b). Indeed Miikfl/i (1990b) gave an explicit error estimate for this technique applied to simple delay systems of the form G(s)=e-~rR(s) with R rational and strictly proper and T > 0. Namely, it showed that if r -> 1 is the relative degree of R (that is, the denominator degree is r more than the numerator degree), then (2.3)

d k = O ( k - 1/4- r/2)

(this estimate is tight) and so

ck = O(k

3/4--r/2).

(2.4)

Hence we obtain IIG - G, II2 = O(n 1/4-r/2) and can estimate IIG - a~ll® as follows:

(G - G,,)(M-1z) = ~+1 dkzk (17q~ + z) gk = n~+ l (dk + d k - 1) ~

-- dn Zn+ l I V e ,

and so

JIG - G.II: ~ ~ tc, I n+l

td.llV~

= O(n t/a-,n). (This is a similar argument to the one given in M~ikil~i, 1990b). Note that for any sequence (Ck) = O(k -~) and o¢ > 1, w e have E ~ = n + l ICkl = o ( n l - a ' ) , since

k-"<<-

n[ h 1 ( $ ) "~ Z pi(s) e - r ' ' , o

(3.2)

h2(s ) = ~ qi(s)e -ais,

Again we suppress the dependence on ~. in the notation. Note our notational convention that G~ and H~ are truncated Fourier series, whereas G ~ is not. The constants (ck) and (dk) are related as follows: we have that

o

(3.1)

n2

(X + s)

G(M-'z)=~ckzk:~dk

G ( s ) = h2(s)/hl(S),

and

i~, s'tk

k=O

(2.6)

for some constants A and B - - s e e Glover et al. (1990b, c) for details---so that there is a trade-off here between ease of calculation and accuracy of approximation.

to obtain an approximation n

t l a - GII~ -> Bn "

where

& ~ (~-s)* G(s ) = k~=odk V2X

a.(s) = X

(2.5)

and

(2.2)

Clearly H.(C+) convergence of the sequence (G ") to G is equivalent to H~ convergence of the power series expansion of G(Ms) on the disc. We cannot expect to obtain L 2 convergence of the series (G~), however, although we shall obtain L 2 convergence of the series H~(s) = v r ~ G~(s)/(;t + s) to H whenever H(s) is in HE(C+). The second technique consists in truncating a Fourier expansion

IIG - (~112~ An 1/2 ,

x-~'dx=nl-~'/(oc-1).

k=n+l

We shall use this estimation technique several times. In Section 3 we shall extend these results to general retarded delay systems; however we remark now that with G(s) = e-~rR(s), and r as above, the optimal convergence rate achievable with any rational approximation of G is given

(3.3)

0

with 0 = y0 < Yl < " " " < Yn~, 0-- 1 such that one may write P

G(s) = R(s) + ~ aie-~'iS/(s + 1) r + O(s r-l),

(3.4)

1

with R rational, P-> 1, the (ai) nonzero constants, and the (06) positive coefficients. Thus for example e-Z~/(s + e -s) = e z~/(s + 1) + O(s- 2), so it has index 1; whereas 1 / ( s + e - S ) = l / ( s + l ) + ( 1 e- s )/((s + 1)(s + e - s )) = 1/(s + 1) + 1/(s + 1) 2 - e - s /(s + 1) 2 + O(s-3), and has index 2. For the case G(s)= e-'rR(s) with R rational, (i.e. nl = 0), the index I(G) is just the relative degree of R. An equivalent formulation may be derived from Glover et al. (1990a, Section 5 and 1990b, Section 5), where it is shown that I(G) is such that the Hankel singular values of G and the minimum H~ error in approximating G by a system of degree n are bounded above and below by multiples of n-i(c). Similarly in Glover et al. (1990c) it is shown that the optimal L2 approximation error is of order n vz-~(a). The key to these discussions is that the dominant term which influences the approximation rate is the term Efa~e-~'~/(s+ 1)r occurring in (3.4). In the time domain this corresponds to analysing the impulse response g corresponding to G for which g t , - l ) is the first discontinuous derivative. It is possible to approximate unstable delay systems by writing them as G(s) = G~t,bl,(S ) + K(S), where K is rational, strictly proper and totally unstable, and then approximating G~t~b~,, but other approaches are often more satisfactory, e.g. the one based on coprime factorizations (Partington and Glover, 1990). Clearly the index of an unstable system is the same as the index of its stable part, but we shall assume in what follows that G is stable. As in Section 2, we write (ck) to denote the Fourier coefficients of the function H(s) = V ~ G(s)/(A + s) and we write (dk) for the Fourier coefficients of G(s) itself. Our first result is a consequence of the results of Miikilii and the decomposition technique of Glover et al. (1990a, b, c).

Lemma 1. Let G be a stable retarded delay system of index

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o0) we write Ek(g) to denote the minimum L 2 e r r o r of any degree-k rational approximation to g.

r -> 1. Then the coefficients (Ck) and (dk) satisfy

L2(0 '

ck -~ O(k -3/4-r/2) and

d k = O ( k -1/4-''2) as k - - ~ .

571

(3.5)

Both these estimates are tight in general.

Proof. Since H(s) is clearly a retarded delay system of index (r + 1) it is sufficient to prove the result for dk. We introduce the notation T , ( s ) = E ~ a ~ e - ~ ' / ( s + l y [cf. (3.4)], and obseve that it is possible to iterate the decomposition method and write, for any N-> r,

Theorem 1. Let g(t) be a function in L®(0, ~). Then, for any g(t) in L®(0, oo) whose Laplace transform is rational of degree n, we have IIg - gll® --- sup {En(g(t)e-"~)/V2-#~: # > 0).

Proof. It is a standard result sup (llfhll2/llhll2:h 6 L2, h #:0}. Hence IIg - g -,, ll®> - II(g

-

g)e-"ql2 = II(g

-

that

I1:11®=

g)e-'ql2/X/~ -

G(S) = RN(S ) dr Tr(S ) + T r + l ( s ) .Jr.....~_ TN(S ) + SN(S), where RN is rational, each T/is equal to a sum of delay terms divided by (s + 1)i, and SN(S) = O(S-N-i). Now it is easy to verify that dk(RN)~O exponentially for any stable rational function, (e.g. by transforming to the disc and examining the power series); moreover each T~ has the property that dk(Ti)=O(k-1/4-~:z), by (2.3). Finally, for sufficiently large N the coefficients dk(SN) are o(k-l:4-'n), by the results in Section 6 of Glover et al. (1990c), specifically since by Lemma 6.6 of that paper SN has an impulse response with ( N - 1 ) continuous derivatives, and exponential decay at ~: integration by parts shows that the Fourier-Laguerre coefficients of a function / of exponential decay will decrease faster than any given power of k if f is sufficiently smooth. It follows that dk(G)= O(k-U4-'/z), as required. To see that the estimate is tight, note that it is sufficient to verify this for dk(T,), since all the other terms are dominated by this. Apart from exceptional cases in which coefficients cancel (this cannot happen for P = 1 and may be impossible in general), this will be the case.

Corollary 1. Let G(s) be a stable retarded delay system of index r. Then the following bounds hold. IIG - G'II® = O(n u4-'r2)

(3.6)

O(n 1/4-r/2) O(n 1/4-':2)

(3.7)

IIG - Gnll® = IIa - G,,II2 =

(3.8)

Since ge - ~ also has a rational Laplace transform of degree n, the result now follows.

Corollary 2. Let G(s) be a retarded delay system of index r with impulse response g(t). Then there exists a constant A > 0 such that for any degree-n approximant 0 with impulse response g we have

/A

"g-g"®->[m nl/z-r

Proof. This follows from Theorem 1 (we may take # = 1) and the known L 2 result (2.5), given in Glover et al. (1990c) for r->2. In the case r = 1 the impulse response g is discontinuous and so it cannot be approximated uniformly by continuous impulse responses. [] This inequality is not tight, and there is some evidence to suggest that the "correct" inequality should be IIg-gll®-> An 1-" for all values of r. As regards achievable L®(0, ~) errors, little is known. One possible technique is to approximate g(t) in the L 1 norm. (in the case r->2, naturally), since I I g - g l l ® - < l l g - g l l l , provided that g(0o)= ~(0o)= 0. The methods of Giover and Partington (1987) can be used here and it can be shown that an achievable error bound is O((log n/n)'-l). Alternatively one can use the truncated series method of Sections 2 and 3, and, since IIqbkI1®-< 1 for all k (Szegt, 1939) we obtain the error bound

as n ---~ °°.

IIg - g, ll~ ~ ~

Proof. Since II((),-s)/(X+s))kl[®=l, for all k, it follows that IIG-G~ll®<-~..~llCkl and the first result follows because ck = Ok-a/4-rt2." Alternatively, for r->2 one can use the error estimate given in Gu et al. (1989, Theorem

5. Example Consider the linear system with transfer function G(s) =

case,

4, Approximation in the L®(O, ~) norm It is of interest in some applications to consider uniform approximation of the impulse response of a linear system, and accordingly we write Ilgll® = ess. sup Ig(t)l O~t~

for the norm in L®(O, ~). It is our first concern to derive a lower bound for the error in approximating a linear system in this norm; we then compare the various techniques available. Although the L®(0, oo) norm is an operator norm, being the norm of the Hankel operator

(ru)(t) = f o g ( t + r ) u ( r )

dr

as a map from L~(0, oo) to L®(0, oo), little appears to be little known about the approximation numbers of operators between these spaces, and we proceed differently. For g in

Idkl = O(n314-r12) •

n+l

2.17), namely

which gives a bound o f O ( n - l " 2 ( ~ n + l kl/2-r)l/2), which is O ( n 1/4-- r12). The other two estimates follow from (3.5) since the arguments given in Section 2 go through to this more general

if r->2; if r = l .

1 s + 1-exp(-2-s)"

This retarded delay system has been considered in several places (Partington et al., 1988; Gu et al., 1989; Glover et al., 1990b) for the purposes of assessing model reduction schemes and we collect together various results on this system. We begin with H® bounds. The decomposition G(s) = 1/(s + 1) + e-2-~/(s + 1)2 + O((s + 1) -3 shows that the index of G is 2, and it also follows that the Hankel singular values satisfy 1 n 2 c r , ( G ) - - - ) e ~2 as

n---)~,

(cf. Glover et al., 1990b): these provide an absolute lower bound for the approximation error by any scheme. In Partington et al. (1988), two approximation schemes were given: the partial fraction scheme, with error O(log n/n), and the Padt-like scheme of approximating e -s in the above expression by ((2n-s)/(2n+s))n: this has e r r o r 0(n-4/3). The best known theoretical bounds for Hankel-norm and truncated balanced realization approximations (Glover, 1984; Glover et al., 1988) yield errors of O(n -~) though in practice these methods appear to perform better than the currently known bounds would suggest. The techniques given in Glover et al. (1990b) (based on Pad6 approximation) achieve errors of O(n -z) which is the optimal rate. Next, the technique given in Gu et al. (1989),

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analysed in Sections 2 and 3, achieve errors of O(n ~/4) by (3.6), since the index of G is 2, and finally the technique given in M/ikil~i (1990b), also analysed above, also achieves errors of 0(n-314), by (3.7). Fewer results are available for the L 2 norm. The theoretical (and achievable) lower bound, given in Glover et al. (1990c), is O(n-3/2), since the singular values of the scaled Hankel operator are bounded above and below by multiples of n The partial fraction approach (Partington, 1990) yields an error of O(log n/Vnn). There is no error bound available for the optimal Hankel-norm approximation, and the error bound for the truncated balanced realization (Glover et al., 1988) is not explicit. Finally, the truncated Laguerre series gives an error bound of O(n 3/4), by (3.8).

6. Conclusions Fourier-Laguerre series do not achieve the optimal convergence rate for model reduction of delay systems in general but they are usually easily computable and (with k chosen appropriately) can provide a reasonably efficient approximation technique. Numerical results (see e.g. Glader et al., 1990) suggest that for lower order approximations several techniques perform satisfactorily, so that each may be useful in certain circumstances. For higher order approximations (e.g. n -> 5) the asymptotic behaviour starts to become more relevant and it is the case that a satisfactory approach in practice is to truncate a series to obtain a higher order approximation, and then perform a balanced or Hankelnorm reduction of the (large) finite-dimensional approximant. References Bellman, R. and K. L. Cooke (1963). Differential-Difference Equations. Academic Press, New York. Francis, B. A. (1987). A Course in H~ Control Theory. Springer, Berlin. Giader, C., G. H6gnas, P. M. M~ikila and H. T. Toivonen (1990). Approximation of delay systems--A case study . Int. J. Control (to appear). Glover, K. (1984). All optimal Hankei-norm approximations of linear multivariable systems and their L~ error bounds. Int. J. Control, 39, 1115-1193. Glover, K., R. F. Curtain and J. R. Partington (1988). Realisation and approximation of linear infinite dimen-

sional systems with error bounds. SIAM J. Control, 26, 863-898. Glover, K., J. Lam and J. R. Partington (1986). Balanced realisation and Hankel-norm approximation of systems involving delays. Proc. IEEE Conf. on Decision and Control, Athens, pp. 1810-1815). Giover, K., J. Lam and J. R. Partington (1990a). Rational approximation of a class of infinite-dimensional systems I: Singular values of Hankel operators. Math. Control Circ. Syst., 3, 325-344. Glover, K., J. Lam and J. R. Partington (1990b). Rational approximation of a class of infinite-dimensional systems II: Optimal convergence rates of L= approximants. Math. Control Circ. Syst. (to appear). Glover, K., J. Lam and J. R. Partington (1990c). Rational approximation of a class of infinite-dimensional systems: The Lz case. J. Approx. Theory (to appear). Glover, K. and J. R. Partington (1987). Bounds on the achievable accuracy in model reduction. In Modelling Robustness and Sensitivity Reduction in Control Systems, NATO ASI series F, 95-118. Gu, G., P. P. Khargonekar and E. B. Lee (1989). Approximation of infinite dimensional systems, IEEE Trans. Aut. Control, 34, 610-618. M~ikil~i, P. M. (1990a). Approximation of stable systems by Laguerre filters. Automatica, 7,6, 333-345. M~ikil~i, P. M. (1990b). Laguerre series approximation of infinite dimensional systems. Automatica, 2,6, 985-995. Partington, J. R. (1988). An Introduction to Hankel Operators. Cambridge University Press, Cambridge. Partington, J. R. (1990). The Lz approximation of infinite-dimensional systems. Proc. 5th I.M.A. Control Conf. (to appear). Partington, J. R. and K. Glover (1990). Robust stabilization of delay systems by approximation of coprime factors. Syst. Control Lett., 14, 325-331. Partington, J. R., K. Glover, H. J. Zwart and R. F. Curtain (1988). L= approximation and nuclearity of delay systems. Syst. Control Lett., 10, 59-65. Szeg6, G. (1939). Orthogonal Polynomials. American Mathematical Society, New York. Wahlberg, B. and L. Ljung (1986). Design variables for bias distribution in transfer function estimation, IEEE Trans. Aut. Control, 31, 131-144. Zwart, H. J., R. F. Curtain, J. R. Partington and K. Glover (1988). Partial fraction expansions for delay systems. Syst. Control Lett., 10, 235-244.