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

In this paper, based on Lyapunov functions candidates, a new approach in the stability analysis of homogeneous nonlinear systems is proposed in which instead of concentrating on the positive definiteness of the Lyapunov candidate functions, we stress on the negative definiteness of its derivative. Having ensured of negative definiteness of the derivative function, based on sign assignment of the primitive function, the stability of the equilibrium is analyzed wherein the necessary and sufficient conditions are declared simultaneously. Selecting the trend of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions whose unknown coefficients in 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 is argued. Therefore, the triple sign attitudes of the candidate function can be used to deduce the stability/instability of the equilibrium point. Moreover, in the process of the negative definiteness of the derivative function, the coefficients are obtained using two independent methods. Numerical simulations support the proposed theoretical results and show their effectiveness.