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

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 linearizatio n, where restrictive conditions arise from a two-step design method (transf ormation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employ ment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In p articular, the mathematical formulation of the problem is realized via a sy stem of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient Conditions for solvability is derived, by using Ly apunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution meth od, that is easily programmable with the aid of a symbolic software package . Under a simultaneous implementation of a nonlinear coordinate transformat ion and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization arid pole-placement d esign objectives may be accomplished in a single step, effectively overcomi ng the restrictions of the other approaches by bypassing the intermediate s tep of transforming the original system into a linear controllable one with an external reference input. (C) 2000 Elsevier Science B.V. All rights res erved.