Extension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems

The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov function derivative. We introduce new sufficient conditions for asymptotic stability of equilibrium states of nonlinear systems considering some inequalities for the higher order time derivatives of Lyapunov function. If the above-mentioned inequalities are found, then the stability analysis of an equilibrium state is reduced to check the characteristic equation for a controllable canonical form LTI co-system. The poles of co-system are required to be negative real ones. Some examples are presented to demonstrate the approach.