A New Approach to Searching Lyapunov Function Candidates for Autonomous Nonlinear Systems

Stability of nonlinear systems is a problem of fundamental importance in system engineering. Specifically, the computation of a Lyapunov Function presents one of the tools enabling the study of the stability of nonlinear systems. The aim of this work is to study the Lyapunov approaches for polynomial systems. These approaches have been investigated in order to develop numerical algorithms based on the synthesis of Polynomial Lyapunov Functions. We proceed in two steps: Firstly, we implement a Threshold Accepting Algorithm technique to determine a candidate Lyapunov function. Secondly, we use an optimization strategy based on a Linear Matrix Inequality (LMI) to compute the Region of Attraction (RA). The parameters of the Lyapunov Function are computed by combining Threshold Accepting Algorithms (TAA) and LMI. The proposed approach yields a larger stability region for polynomial systems than an existing method does. Examples are given to illustrate the efficiency of the proposed approach.