Crasovskii approach to construct Lyapunov function and its derivative function for analyzing stability of non-linear systems

Abstract

This paper aims at constructing Lyapunov function and its derivative function to analyze and judge the stability of non-linear time-invariant systems by Crasovskii approach. A canditate Lyapunov function is selected first according to the system function. Then a replacement matrix function, which is used to judge the negative definteness of derivative function of candidate Lyapunov function, is derived by solving Jacobi matrix of the system function. When this replacement matrix function is determined to be negative definite, in light of Lyapunov stability theory, thereby the non-linear time-invariant system is asymptotically stable. Examples indicate that this approach is valid for judging the stability of some non-linear time-invariant systems, and it still can be used to determine the unknown parameters for some non-linear time-invariant systems. This approach provides another way to analyze the stability of non-linear time-invariant systems.