Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions

摘要

In this paper, based on Lyapunov functions candidates, a new approach in the stability analysis of homogeneous nonlinearsystems is proposed in which instead of concentrating on the positive definiteness of the Lyapunov candidatefunctions, we stress on the negative definiteness of its derivative. Having ensured of negative definiteness of the derivativefunction, based on sign assignment of the primitive function, the stability of the equilibrium is analyzed wherein thenecessary and sufficient conditions are declared simultaneously. Selecting the trend of the Lyapunov candidate functionis primarily performed in the form of a linear combination of some simple functions whose unknown coefficientsin the candidate function structure are computed based on negative definiteness of the derivative function. Afterward,using these determined coefficients in the Lyapunov function, the sign of the primitive function in the state space isargued. Therefore, the triple sign attitudes of the candidate function can be used to deduce the stability/instability ofthe equilibrium point. Moreover, in the process of the negative definiteness of the derivative function, the coefficientsare obtained using two independent methods. Numerical simulations support the proposed theoretical results andshow their effectiveness.