Study on Lyapunov Functions for Liénard-type Nonlinear Systems

In this paper, a generalized Lyapunov function for the Lienard-type nonlinear system which is important as a representative system expressing LRC electric circuits and mechanical spring-mass systems etc., is con-structed using the Lagrange-Charpit method. The Lyapunov function includes particular nonlinear terms as arbitrary functions, by which the quadratic term appearing in the Luré-type Lyapunov function can be extended. The result yields all the conventional Lyapunov functions as special cases, changing the forms of the arbitrary functions. To investigate the relation between the arbitrary function in the Lyapunov function and the stability region obtained, the stability boundaries for various types of the arbitrary functions are illustrated in the application to a simple system. In addition, numerical values of the time derivative of the Lyapunov function along the stability boundary are calculated to study the relation between the stability region and the values of the time derivative.