Methods for Constructing Lyapunov Functions for A Class of Nonlinear Systems

In automatic control theory, the stability of differential equations can be determined with Lyapunov's second method, which is simple and concise. However, an approach to constructing Lyapunov functions for special scenarios is an issue that remains to be addressed. Lyapunov function s that are identified for special scenarios can confer many conveniences to engineering practice. In this study, we proved that, for linear time-invariant systems, Lyapunov functions can be obtained by analyzing the relationships among coefficients to enable a rapid evaluation of the system stability. We then generalized the conclusion on linear time-invariant systems to nonlinear systems by investigating the Lyapunov function for four types of nonlinear systems. It was determined that the methods developed in this study can be used to quickly find Lyapunov function for assessing the system stability in differential forms when undifferentiated variables vary in terms of odd powers. Lastly, we analyzed Lyapunov functions for nonlinear time-invariant and nonlinear time-variant systems-both types of systems containing function-type coefficients instead of constant coefficients-and used the Lyapunov functions to assess the system stability.