Novel stability and passivity analysis for three types of nonlinear LRC circuits

Abstract

In this paper, the global asymptotic stability and strong passivity of three types of nonlinear LRC circuits are investigated by utilizing the Lyapunov's direct method. The stability conditions are obtained by constructing appropriate energy (or Lyapunov) function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. Many specialists construct Lyapunov functions by using some properties of the functions with much trial and errors or for a system they choose candidate Lyapunov functions. So, for a given system the Lyapunov function is not unique. But we insist that the Lyapunov (energy) function is unique for a given physical system. Thus, this study clarifies Lyapunov stability with suitable tools and also improves some previous studies. Our approach is constructing energy function for a given nonlinear system that based on the power-energy relationship of the system. Hence for a dynamical system, the derivative of the Lyapunov function is equal to the negative value of the dissipative power in the system. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The provided results are central importance for the stability analysis of nonlinear systems. Some simulation results are also given successfully that verify the theoretical predictions.