Left inversion of analytic nonlinear SISO systems via formal power series methods

Given a single-input, single-output (SISO) system, FF, and a function yy in the range of FF, the left inversion problem is to determine an input uu such that y=F[u]y=F[u]. The goal of this paper is to provide an exact and explicit analytical solution to this problem in the case where FF is an analytic mapping in the sense that it has a convergent Chen–Fliess functional expansion, and yy is a real analytic function. In particular, it will be shown that given a certain condition on the generating series cc of FF, a corresponding unique analytic uu can always be determined via operations on formal power series. The condition on cc turns out to be equivalent to having a well-defined relative degree when FF has an input-affine analytic state space realization with finite dimension. But the method is applicable even when FF does not have such a realization. The technique is demonstrated on four examples, including a continuous stirred chemical reactor.