Stability Analysis of Nonlinear Dynamic Systems Using the Lyapunov Method Approach

Abstract

A new approach to the stability analysis of homogeneous nonlinear systems is proposed, based on the concept of candidate Lyapunovfunctions, where the focus is not on the positive definiteness of the candidate Lyapunovfunctions, but on the negative definiteness of their derivatives. After having ensured the negative definiteness of the derivative function, on the basis of the sign assignment of the primitive function, the stability of the equilibrium is analyzed, where the necessary and sufficient conditions are declared at the same time. The selection of the tendency of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions. The unknown coefficients in the structure of the candidate function are computed based on the negative definiteness of the derivative function. Then, using these coefficients in Lyapunov function, sign of primitive function in state space is argued. Thus, the stability/instability of the equilibrium point can be deduced from the triple sign settings of the candidate function. Furthermore, in the process of negative definiteness of the derivative function, the coefficients are obtained using two independent methods. The proposed theoretical results are supported and their effectiveness is demonstrated by numerical simulations.