Qualitative analysis of second-order nonlinear dynamical systems with RLC circuit systems

In this thesis, Lyapunov stability and passivity problems of some second-order nonlinear and time-varying dynamical systems are discussed. This study is mainly based on the circuit theory approach and Lyapunov's direct method. It is to develop stability and passivity analysis by attributing physical meaning to some second-order systems that can be made or expressed with RLC circuit elements. That is, to create electrical circuit models using RLC circuit elements and then determine the qualitative behavior of the systems with Lyapunov's direct method. It has been proven in the literature that Lyapunov's direct method is irreplaceable for the stability analysis of nonlinear systems. For nonlinear systems, the construction of the Lyapunov function is difficult because there is no precise method and algorithm for constructing the candidate Lyapunov or energy function. That is, there may not be a Lyapunov function for a system with positive definiteness and negative derivative. The Lyapunov function is constructed by using the power-energy relationship in the modeled electrical circuit. RLC Circuits are energy-consuming systems and it has been shown by examples that the construction of energy storage (candidate Lyapunov) functions of such circuits based on the power-energy relationship of circuit theory is more applicable and concrete than the selected candidate Lyapunov functions. It is advantageous to treat a given system as an electrical circuit, because the energy and derivative of the system determine the dynamic behavior of the system. The modeled electrical circuit model may contain three nonlinear elements, which can generalize the second-order differential equations in the literature. In addition, using the circuit theory approach, Lyapunov stabilities of some well-known second-order differential equations in the literature are both developed and combined. The power-energy relationship is used to construct the Lyapunov or energy function with the circuit theory approach of each system considered, and the directional derivative of the Lyapunov function gives the negative value of the power consumed in the system. These two techniques are rarely encountered in the literature. Thus, the proposed approach simplifies the derivative of the respective Lyapunov functions. The proposed approach to determine the construction of the Lyapunov function in terms of the physical meaning of the system can combine the present results.