Stabilization of nonlinear systems in compound critical cases

Authors:
Der-Cherng Liaw;Chun-Hone Chen
Affiliations:
Department of Electrical and Control Engineering, National Chiao Tung University, Hisnchu, 30010 Taiwan, ROC;Department of Electrical and Control Engineering, National Chiao Tung University, Hisnchu, 30010 Taiwan, ROC
Venue:
Applied Mathematics and Computation
Year:
2002

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Abstract

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local non-linear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.