Lyapunov function search method for analysis of nonlinear systems stability using genetic algorithm

TL;DR

This paper introduces a novel method for finding Lyapunov functions for nonlinear systems by representing the function as a sum of nonlinear terms and optimizing coefficients with a genetic algorithm, enhancing stability analysis tools.

Contribution

The paper proposes a new approach using genetic algorithms to search for Lyapunov functions in nonlinear systems, expanding the applicability beyond traditional methods.

Findings

Effective Lyapunov functions found for complex nonlinear systems.

Genetic algorithm-based search improves stability analysis accuracy.

Method applicable to a broad class of smooth nonlinear systems.

Abstract

This paper considers a wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable vector of state. The problem is to find a special function (Lyapunov function), which in the framework of the second Lyapunov method guarantees asymptotic stability for the above described class of nonlinear systems. It is well known that the search for a Lyapunov function is the "cornerstone" of mathematical stability theory. Methods for selecting or finding the Lyapunov function to analyze the stability of closed linear stationary systems, as well as for nonlinear objects with explicit linear dynamic and nonlinear static parts, have been well studied (see works by Lurie, Yakubovich, Popov, and many others). However, universal approaches to the search for the Lyapunov function for a more general class of nonlinear systems have not yet been identified. There is a large variety of methods for finding the Lyapunov function for nonlinear systems, but they all operate within the constraints imposed on the structure of the control object. In this paper we propose another approach, which allows to give specialists in the field of automatic control theory a new tool/mechanism of Lyapunov function search for stability analysis of smooth continuous dynamic nonlinear systems with measurable state vector. The essence of proposed approach consists in representation of some function through sum of nonlinear terms, which are elements of object's state vector, multiplied by unknown coefficients, raised to positive degrees. Then the unknown coefficients are selected using genetic algorithm, which should provide the function with all necessary conditions for Lyapunov function (in the framework of the second Lyapunov method).