Stable inversion for nonlinear systems

There have been two recent major developments in output tracking for nonlinear systems, and our first main contribution is to relate these. Under appropriate assumptions, we show that the bounded solution of the partial differential equation of Isidori and Byrnes for each trajectory of an exosystem must be given by an integral representation formula of Devasia, Chen and Paden. Under restrictive Chen and Paden develop a Picard process that converges to the solution of the integral equation. This solution to the integral equation is also a bounded solution a dynamical equation driven by the desired outputs. In aircraft applications our nonlinear are perturbations of 'pure-feedback systems' with outputs, and we find a solution to the stable inversion problem, i.e. finding bounded controls and bounded state trajectories in response to bounded output signals, in two steps. The first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory. The second step computes the remaining control and states by finding a noncausal and stable solution to an 'error-driven dynamical equation'. In other words, the method of Devasia, Chen and Paden is applied to an 'error system' and not to the full system. This two-step procedure is the second main contribution of this paper. Throughout this paper it is assumed that our nonlinear systems have vector relative degree. (C) 1997 Elsevier Science Ltd.