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

Abstract

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

On the global uniform stability analysis of non-autonomous dynamical systems: A survey

“…For time-varying systems, expect for some special cases, the Lasalle invariant principle is either not valid or difficult to use [49]. Hence, some researchers attempt to use the available Lyapunov function, whose time-derivative is not strictly negative definite, to construct a new Lyapunov function whose time-derivative is negative definite ( [1,3,4,34,39]). One could hope that a method for proving the existence of a Lyapunov function might carry with it a constructive method for obtaining this function.…”