A Synthesis on Lyapunov Methods to the Estimation and Enlargement of Attraction Domain for Nonlinear Autonomous Systems

The objective of this work is to study Lyapunov approaches for the estimation and the enlargement of asymptotic stability region of nonlinear autonomous systems. These approaches have been investigated in order to develop digital algorithms based on the synthesis of a quadratic and maximum Lyapunov functions. The results on all three explored methods: McCann Method, estimation method via a Maximal Lyapunov function and the estimation method via a Quadratic Lyapunov function, showed that this variety of methods: i) is efficient for a large class of nonlinear systems, ii) has an interesting capacity of enlargement of the asymptotic stability region, iii) is characterized by an obvious property of implementation for reduced and high order nonlinear systems. The efficiency and the validation of the investigated methods are tested via a simulation study. The performances of all methods are compared and discussed via the study of the well known the Van der Pol model.