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.
Highlights
In history, modeling and stability analysis of nonlinear systems are the most important and popular problems in control theory
Since almost all systems are nonlinear in nature [1], a number of promising studies have been analyzed in the literature
The matter under discussion is the stability of the origin (0, 0) and the passivity of the following nonlinear resistive, inductive, and capacitive LRC circuits for one input variable (m = 1) and two state variables (n = 2)
Summary
In history, modeling and stability analysis of nonlinear systems are the most important and popular problems in control theory. The Lyapunov’s direct method is still recognized as an effective tool to study the stability theory of dynamical systems such as: the global asymptotic stability of the electrical RLC circuit [8], neural networks with time varying delays [9,10], power systems analysis [11], robot manipulators [12], dissipativity analysis of discretetime neural networks [13], global robust passivity analysis [14], dissipativity and passivity analysis of neural networks [15]. The direct method provides the opportunity to examine the stability of the equilibrium points with minimum energy This meaning (diminishing of energy) tends us to the passivity of the systems. The interested knows how to construct the energy (Lyapunov) function and checks the result of the time derivative (directional) of the energy function with (1)
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