Approximation of nonlinear dynamic systems by rational series

Abstract

Given an analytic system, we compute a bilinear system of minimal dimension which approximates it up to order k (i.e. the outputs of these two systems have the same Taylor expansion up to order k). The algorithm is based on noncommutative series computation: let s be the generating series of the analytic system; then a rational series g is constructed, whose coefficients are equal to those of s, for all words of length smaller than or equal to k. These words are digitally encoded, in order to simplify the computations of the Hankel matrices of s and g. We then associate with g, a bilinear system, which is a solution to our problem. Another method may be used for computing a bilinear system which approximates a given analytic system ( S). We associate with ( S) an R-automaton of vector fields and build the truncated automaton by cancelling all the states which have the following property: the length of the shortest successful path labelled by a word that gets through this state is strictly greater than k. Then, the number of states of this truncated automaton yields the dimension (not necessarily minimal) of the state-space.