Energy decay estimates for certain class of nonlinear systems arising in models of power systems

This dissertation is concerned with stabilization and explicit energy decay estimates for a class of bilinear system and nonlinear ordinary differential equations, used to model oscillatory phenomena in electrical or civil engineering. We also focus specifically on the so-called critical homogeneous bilinear systems (HBLS) with viscous quadratic feedback for whom we compute the energy decay estimates.; Chapter 1 is an introduction, which provides a brief review of the mathematical model of an electric power system, motivating the present work and a overview of the main problem.; In Chapter 2 we consider a well-known Lienard's system of ordinary differential equations, modeling oscillatory phenomena in various applications in engineering. It is known that such a system is asymptotically stable when a linear viscous damping with constant gain is engaged. However, in many applications it seems more realistic that the aforementioned damping is not constant and does depend upon on the deviation from the equilibrium. In this Chapter we consider a nonlinear feedback, introduced in [10], which is proportional to the square of such deviation and develop a methodology using geometric "visualization" technique to derive explicit energy decay estimates for solutions of the corresponding "damped" Lienard's system.; In Chapter 3 we consider a model of a power system which consists of a single generator connected to an infinite bus by a transmission line. This model is widely used to study the effects of various controllers on power system stability. In this Chapter we introduce a smooth nonlinear stabilizing feedback, motivated by so called facts FACTS (Flexible a.c. Transmission Systems) devices, which changes the admittance of the transmission line. We apply the methodology developed in Chapter 2 to derive explicit energy decay estimates for solutions of the corresponding system of nonlinear ordinary differential equations.; Finally, in Chapter 4 we consider a Homogeneous Bilinear System (HBLS) such that the drift term A is n x n skew-symmetric matrix and provide a detailed proof of global asymptotic stability for the given system. This result is a special case of a Theorem obtained by Jurdjevic and Quinn in [10]. We also obtain explicit energy decay estimates for the given HBLS by assuming that the input matrix B is symmetric positive definite.