Stabilization via Polynomial Lyapunov Function

Abstract

In this paper we summarize some recent results on local state feedback stabilization of a nonlinear system via a center manifold approach. A systematic design procedure for stabilizing nonlinear systems of nonminimum phase, i.e., with unstable zero dynamics, was presented recently in [11]. The basic idea can be described as follows. We first propose some sufficient conditions to assure the approximate stability of a dynamic system. Using these conditions and assuming the zero dynamics has stable and center linear parts, a method is proposed to design controls such that the dynamics on the designed center manifold of the closed-loop system are approximately stable. It is proved that using this method, the first variables in each of the integral chains of the linearized part of the system do not a.ect the approximation degree of the dynamics on the center manifold. So the first variables are considered as nominating controls, which can be designed to produce a suitable center manifold. Based on this fact, the concept of injection degree is proposed. According to different kinds of injection degrees certain sufficient conditions are obtained for the stabilizability of systems of non-minimum phase.KeywordsNormal FormCenter ManifoldNonlinear Control SystemApproximate SystemNonminimum PhaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.