An International Journal of Optimization and Control: Theories & Applications ISSN:2146-0957 eISSN:2146-5703 Vol.11, No.2, pp.227-237 (2021) http://doi.org/10.11121/ijocta.01.2021.001073 RESEARCH ARTICLE Novel stability and passivity analysis for three types of nonlinear LRC circuits Muzaffer Ate¸ s a* and Nezir Kadah b a Departments of Electrical-Electronics Engineering, University of Van Yuzuncu Yil,Turkey b Departments of Electrical-Electronics Engineering, Adana Alparslan Turkes Science and Technology University, Turkey mates@yyu.edu.tr, nkadah@atu.edu.tr ARTICLE INFO ABSTRACT Article History: Received 08 January 2021 Accepted 09 May 2021 Available 31 July 2021 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. Keywords: Lyapunov stability Nonlinear systems Nonlinear LRC circuits Passivity Gronwall’s inequality AMS Classification 2010: 34D23; 34D20; 34C23 1. Introduction In history, modeling and stability analysis of non- linear systems are the most important and pop- ular problems in control theory. Since almost all systems are nonlinear in nature [1], a number of promising studies have been analyzed in the liter- ature. Many researchers as Lagrange, Hamilton, Poincare and Lyapunov are focused on the mod- elling problem to analyze the dynamic behavior of systems [1, 2]. The aforementioned methods are based on the energy utilization of the related systems. However, since all systems are not in linear forms, certain mathematical solutions are not available to solve these issues. Furthermore, closed-form expressions for the solutions of the linear systems are not possible to solve nonlinear systems. Nevertheless, it is important to be able to make some assumptions about the conduct of a nonlinear system called qualitative analysis. The stability of the equilibrium point was first ex- amined by Lagrange; however, the Lagrange prin- ciple was only suitable for the Lagrange systems (conservative systems) [1], but engineering sys- tems usually have damping [3]. Then, the sta- bility theory of motion derived from the concepts of Lagrange’s principle and Poincare’s regular so- lution (Lyapunov stable motion) was developed by Lyapunov [2]. Hamiltonian and Lagrangian systems comply with conservative systems (ex- act differential equations), but Lyapunov stability theory can be applicable to arbitrary differential equations. Thus, the Lyapunov direct method is *Corresponding Author 227