Singular PDEs and the single-step formulation of feedback linearization with pole placement

Abstract

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.