Functional expansions for nonlinear discrete-time systems

Given an input-output map associated with a nonlinear discrete-time state equationx(t + 1) =f(x(t);u(t)) and a nonlinear outputy(t) =h(x(t)), we present a method for obtaining a “discrete Volterra series” representation of the outputy(t) in terms of the controlsu(0), ...,u(t − 1). The proof is based on Taylor-type expansions of the iterated composition of analytic functions. It allows us to make an explicit construction of each kernel, that is, each coefficient of the series expansion ofy(t) in powers of the controls. This is achieved by making use of successive directional derivatives associated with a family of vector fields which are deduced from the discrete state equations. We discuss the use of these vector fields for the analysis and control of nonlinear discrete-time systems. This work was carried out while D. Normand-Cyrot was working at the I.A.S.I. (from March to October 1984) and with the financial support of the Italian C.N.R. (Consiglio Nazionale delle Ricerche).