Math. Syst. 131. pp. 122-148. Springer-Ver- lag. Berlin. Fliess, H. (1977). Integrales iterees de K.T.. Chen. bruit blanc gaussien et filtrage non lineaire. C.R. Acad.
Copyright © IFAC Identification and System Parameter Estimation 1982 . Washington D.C .. USA 1982
ON THE APPROXIMATION OF NONLINEAR SYSTEMS BY SOME SIMPLE STATE-SPACE MODELS M. Fliess and D. Normand-Cyrot Laboratoire des Signaux et Systemes, C. N.R.S. - E.S.E. , Plateau du Mouion, 91190 Gij-sur-Yvetle, France
ABSTRACT. Interest in Volterra series depends to a great extent on their property to approximate non-linear systems. It is shown that continuous-time regular (or bilinear) systems or discrete-time state-affine systems have a similar property.
1_ INTRODUCTION The authors are pleased to thank the organizer of this session, Dr. S.A. Billings, for giving them the opportunity to present this survey written in English of works published in French and which have already found practical applications.
For practical identification purposes, there is a natural need to approximate general input-output behaviour through rather simple models. A classical approach to this topic has been, for a long time, the use of Volterra series (see, e.g., Barrett, 1963, Rugh, 1981). Following works due to Frechet and which were published at the beginning of the century (cf. Levy, 1951), it was possible to show that "continuous" non-linear systems could be arbitrarily well approximated by finite Volterra series. This has resulted in many papers which have tried to use this fact for various modelizations ranging from electronic circuits to biological organisms. But the computations of the Volterra kernels can be very tedious even today with digital computers. This is one of the reasons why this approach is still far from being satisfactory.
11. THE APPROXIMATION PROPERTY OF CONTINUOUS-TIME REGULAR (OR BILINEAR) SYSTEMS. a) Statement of the result. A regular (or bilinear) system is given by n
L
q(t)
(D
yet)
(A + ui(t)Ai)q(t) o i=1
=
Aq(t)
The state q belongs to a finite-dimensional R-vector space Q; the initial state q(o) is gIven. The operators Ao , AI' ••. , An : Q ~ Q,
Since the works of R.E. Kalman in the early sixties, the notion of state-space has now been widely popuralized in the control community. We survey here some results on the approximation of non-linear systems by simple state-space models, i.e., regular (or bilinear) systems in the continuous-time case, state-affine systems in the discrete-time case.
A : Q ~ Rare R-linear. The inputs (or controls) u1~ •.• ,un-: [o,oo[ ~ R are continuous. Let J be a compact subset of [o,oo[ and C a compact subset of CO(J)n, where CO(J) is the Banach space of continuous functions J ~ ~, with the topology of uniform convergence. Then the notion of continuous functionals JxC + R is obvious. vIi th respec t to the topology of uniform convergence, this space has a structure of Banach space (cf. Dieudonne, 1960, chap. VII).
The discrete-time systems were successfully applied to the identification of various parts of electrical power plants (Dang Van Mien and Normand-Cyrot, 1980, Bourdon, Dang Van Mien, Fliess and Normand-Cyrot, 1982). For regular (or bilinear systems), the approximation property certainly explains their flexibility to modelize various situations (see, e.g., Esparta and Landau, 1978, 1979).
A functional J xC + R is said to be causal, or non-anticipative, iff its value at time t £ J for (u , ... ,u ) £ C, does not depend on 1 n u 1(t' ) •..• ,un (t') for t t > t.
In the continuous-time case, the approximation property, was found independantly by Fliess (1975, 1976, 1981) and Sussmann (1976)The two versions do not coincide and we give here the one due to the first author. For discrete-time systems, the result is due to Fliess and Normand-Cyrot (1980).
Theorem. Any continuous causal functional JxC + R can be arbitrarily well approximated by regular (or bilinear) systems. Remarks. (i) The approximation theorem does not remain true with an unbounded time interval (cf. Brockett, 1977). 511
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(ii) The theorem remains true with restricted classes of regular systems, for example, where the matr ices Ao' AI' ... , An are nil-
( J. = 0 , I , ... , n ) , where I.( 2- ) : Q(n ~ Q(2-) J
l'S
potent and simultaneously triangular (cf. Fliess, 1976, 1981).
the identity mapping. The verification can be done by direct computations or, better, by the use of non-commutative generating power series ( cf. Fliess, 1976, 1981).
(iii) It is possible to generalize the appr~ ximation theorem to systems driven by white Gaussian noises (cf. Fliess, 1977).
8) Constants. Any constant functional c can be realiZ;ci-as one-dimensional system q(t)=o (q(o)=c), y(t)=q(t).
b) Proof.
y) ~~
