SE REARS ON THE INVERTIBILITY OF NONLINEAR DISCRETE-TIE SYSTEMS W P7 - 3:30 S. Monaco* and D. Normand-Cyrot** * Istituto di Automatica, Universit& di Rama, Via Endossiana 18, 00184 R0X4A, ITALY. ** Laboratoire des Signaux et Syst&mes, CNRS-ESE, Plateau du Moulon, 91190 Gif-sur-Yvette,FRANCE. ABSTRACT Conditions of invertibility of nonlinear discrete time systems where the controls appear linearly are dis- cussed. I. INTRODUCTION Many problems which are usual in nonlinear con- trol theory like for example the inversion, the decou- pling, the imnersion problems are only studied in the continuous time case [1,3,4,5,8,10,111 by means of dif- ferential geometric tools or of input/output develop- ments in terms of generating series. Until now such stu- dies were impossible in the discrete time case. From a recent work (9] on nonlinear discrete-time systems where convenient input/output representations are introduced, these problems can now be studied and solved in the dis- crete-time case. This paper concerns the invertibility of nonli- near discrete time systems, it follows a pecent work of the authors 16,7) on the imnersion and feedback im- mersion of a nonlinear discrete-time system into a li- near system where the same tools and techniques of pro- ofs are used. Many related problems can be solved in the same way. Let us consider a nonlinear discrete-time system where the controls appear linearly: p (x(t+l) - x(t) + f(x(t)) + I ui(t)gi(x(t)) fy(t) - (h1,...,hq)(X(t)) The state x(t) E RN, gj,...,gp: RN > RN and hi,. ..,hq: RN .* R are analytic functions, the controls u a (u.1,.,,up) are real-valued. A control system is said invertibie when the cor- responding input/output map is injective. The invertibility of linear systems has been trea- ted extensively in the literature and results have been recently extended to noulinear continuous-time systems, [3,4,8,10,11]. Here with the same definitions, we give in theo- rem 3 a sufficient condition of invertibility of the system I in terms of a certain matrix rank. Moreover the left inverse system is constructed. Let us quote that recently a very sirnfiar rank condition of invertibility has been obtained (2] for several-input, several-output nonlinear continuous- time systems. II. PRELIMINARIES Review on input-output development Given a system 1, initialized at-x0 of R,N the outputs y (a), iEjl,..,sq}, a > 0 are computed by means of successive compositions of analytic functions, one has: y (o) - hi(xo ) p Yi(l) - h(I+f+ I u.(o)g.)(x0 ) i j1 I p p Y (a) i- h(I'f+ I uCc(a-1)gj).....(I+f. I u.(o)gj)(x0 ) j-1 3 - In order to extend yi(a) in powers of the con- trols uj(T), T k 0, j E (1,...p,), let us recall the following notations (6,7,91 : Given an analytic function f RN + RN, we de- note: N Lf = f a wher fi i-I ih ax. where fj s the ith component of f. Besides the usual composition of differential operators denoted o we introduce the following preduet: If f and g : RN -_ R are two analytic functions, we de- note: N Lf O Lg I i,j-1 figj as;2 f ig ~ axax Now, let us denote exp0Lf or briefly Af %he dif- ferential series such that, expLf a Af I+L f + 02 fL + 4*. GP +L .. 0 f f f -2.:T'Lf PI f where L fP . Lf*L* O.*SL (p times) The following equality is at the basis of all the next computations related to the input-output development of the system 1. For any analytic function f : RN RN and h : RN + R, the composition h.(I+f) verifies: V x E RN, ho(I+f) (x) - exp0Lf(h) 1 Af(h) (the bar indicates the evaluation at x). Consider again the system Ex where for simpl- city's sake we assume that p a q - , the output y(a) is given by: y(a) - h A ..fu().g.* Af+u(a)g h) I xoa>O xc, Because of the equality, A A + I :;Af@L f+Ug f n>l n f g yQa) can be easily expressed as a series in powers of the variables u(T), 0 c T S a-1. We obtain a "Idiscrete Volterra" extension of y(a) ; the first kernels can be easily computed, one has : a-i y(a) - a (h)j + I u(T)AfOL A'. C (h)1 aaIIUT r2 r2 -t1-T + u 2(1A AfOL-goAf f f (h) l + I> U(T )U(T l ) o6 OL 0 O L OA 2 (h 324 Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on April 23,2010 at 13:40:24 UTC from IEEE Xplore. Restrictions apply.