An approximation of discrete-time nonlinear systems using stochastic inputs

SYSTEMS & CONTROL LETTERS

Volume I. Number 6

May 1982

An approximation of discrete-time nonlinear systems using stochastic inputs Patrick BOURDON Eleclricile de France, DireClian des EllIdes el Recherches, 1, Avenlle dll General de Galllle, 92141 C!a/llar!. Frallce

Received 31 January 1982 Slale-affine systems approximate most nonlinear discrete-time input/output systems using Gaussian while noise process inputs. Keywords: Nonlinear stochastic approximation. Discrete-time state-affine systems. Gaussian white noise.

Introduction

The interest of the study with a state space of linear systems gave rise to a similar approach of nonlinear systems. Fliess [1,3J and Sussmann [6] have already shown that most of nonlinear systems could be approximated by regular (or bilinear) systems, when time is continuous. This result has been extended in the stochastic case when inputs are Gaussian white noise processes [2J. In the discrete-time case, Fliess and Normand-Cyrot [4] have shown that the state-affine systems have to take the place of regular systems to keep the approximation theorem true. The aim of this note is to extend this result when the inputs are white Gaussian noises.

Notations

The state-affine systems introduced by Sontag [5] are defined by the equations

q(t+I)==[Aa+.~ Uiu(t)'''Ui,(t)AiO ... i,]q(t),

y(t)=Aq(t).

[mile

- the state q(t) belongs to IR N (q(O) is given). - AiD" 'i~ ,are N X N square matrices; i a, ... , ik E {I, ... ,11}. - The inputs ui(t) (i= 1, ... ,n) are real-valued. - A is a row matrix of order N. The inputs are supposed to be Gaussian white noise processes, i.e.

E(ult») =0. E(u}{t») = I, E[ui(t)uj(t')] =0

i=I .... ,I1.

(=O,I, ... ,

ift*t'ori*j.

If the noises are not zero-mean or without unit variance, a simple change of variables bring us back.

Nonlinear approximation

We define first the form of the systems to be approached and the topology we use, before we state and prove the approximation theorem. Let 1= {O, I, ... , T}, T< 00. or 1= N. A finite measure on I is given by a set {mt)tEI where every m t is 382

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Volume 1. Number 6

May 1982

SYSTEMS & CONTROL LETIERS

real-valued and nonnegative and l.lm, < DO. Denote by :1/.", the set of functions f; I -> ~ satisfying l.,I11,FU) < I and the family {m,}rEI are chosen to verify these twO hypotheses: (i) polynomials P:l->~,

t--+

~

tXAt

00.

A

finite

are everywhere dense in :\/1/' (ii) Any state-affine system (with output

~m,E[a2(t)] <

a( I)

verifies

DO.

I

If 1 is finite these two conditions are fulfilled.

Denote by H I . 1It the set of nonlinear discrete-time processes, with inputs ui' ... ,u", which satisfy

l.lm,E[y2(t)] < 00 modulo the equivalence l.lm,E[(y'(t) -y"(t»2] = O. HI . which contains the state-affine systems is a Hilbert space with the inner product LIm, E[y'( I)y"(t )]. 1It

In the induced topology, the notion of dense subset in H I . 1It is obvious. Theorem. The state-affine system set is dense ill HI. lit'

=

Proof. Let us recall that the system with output y(t) a l (l)a 2 (1) where al(l) and a 2(1) are the outputs of two state-affine systems is a state-affine system [4]. Let yet) be the output of an HI.IIt-process in the orthogonal complement of the state-affine system set. For any state-affine system (output aCt»~ ~m,E[y(t)a(t)] =0. I

With any polynomial p(t) = l.r.nilotXktA, a(t)p(t) is the output of a state-affine system. In fact we nolice that the system with outpul y(t) = t can be written

q(t+I)=[[6

~]+~Ui(t)[~ ~]]q(t), q(O)=[~],

y(t)=[1 O]q(t).

So ~ m,E[a(t )p(t)y(t)] I

== 0 =: 2: m,p(t )E[a(t)y(t )]. I

The hypothesis (i) allows

E[a(t )y(t)] = 0 with any state-affine syslem and any 1 in J. According to the causality principle,y(t) and a(t) only depend on U;('1') (i= 1, ... ,11) for'1'
y(t) =g,[ul(O), ... ,u,,(O), ... ,uj(t- l), ... ,u,,(c- I)]. This function is polynomial for the state-affine systems

a(t)

= p,[ ul(O), ... ,ulI(O), ... ,u,(t -

l), ... ,ulI(t - I)J

with p,E~<{U;('1')li= 1, ... ,/1, O~'1'
1, .... 11, O~r
<

at time t:

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SYSTEMS & CONTROL LETTERS

May 1982

for any PI E ~<01'" Oq) and q= nt. The polynomials with q indeterminates are dense in L 1(1R 'I, m), the set of the square-summable real-valued functions, defined over IRq, with the Gaussian measure dm = exp[ -

Ha~ + ... +0,;]] dOl' .. da".

So g, == 0 for any t. This yields the theorem. This result is still valid for any measure which gives completeness properties to the polynomials L 1 [1R", m].

In

References [1] M. Fliess. Un Dutil algebrique: les series formelles non commutatives, in: MlIIhematical Systems TheOJ}' (G. Marchesini and S.K. Miller, Eds.), Lect. Notes Econom. Math. Syst. 13, (Springer, Berlin, 1976) pp. 122-148. [2] M. Fliess, Integrales iterees de K.T. Chen, bruit blanc gaussien et filtrage non-lineaire, C.R. A cad. Sci. Paris Ser. A 284 (1977) 459-462. [3] M. Fliess, Fonctionnelles causales non-lineaires et indeterminees 110n commutatives, Bull. Soc. Math. France 109 (1981) 3-40. [4] M. Fliess and D. Normand-Cyrot, Vers une approche algebrique des systemes non-lineaires en temps discret. in: Analysis und Optimization of Systems (A. Bensoussan and J.L. Lions, Eds.) Versailles, Dec. 1980, Lect. Notes Control Informat. Sci. 28 (Springer, Berlin, 1980). [5] E.D. Sontag, Realization theory of discrete-time nonlinear systems, [ - the bounded case, IEEE TrailS. Circuits und Systems 26 (1979) 342-356. [6] H,J. Sussmann, Semi-group representations, bilinear approximation of input-output maps, and generalized inputs, in: Mathematical Systems Theory (G. Marchesini and S.K. Miller, Eds), Lect. Notes Econom. Math. 13 (Springer, Berlin. 1976) pp. 172-191.

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