Stability region of nonlinear dynamical systems and applications

Abstract

The determination of stability region (or basin of attraction) of nonlinear dynamical systems is of great importance in many applications in engineering and sciences. In particular, the theory of stability region encounters important applications in the stability analysis of electrical power systems, in the development of global optimization techniques and in the theory of recurrent neural networks.Motivated by problems of stability analysis of electrical power systems, a complete characterization of the stability boundary was developed for a large class of nonlinear continuos and autonomous dynamical systems. Exploring this characterization, algorithms to obtain optimal estimates of the stability region were developed. In spite of the incredible development of this theory, there are still many open problems in this area whose solution is very desirable from both the scientific and technological point of view. This research project has the main aim of investigating some of these problems realted to the characterization of the stability region of nonlinear dynamical systems and developing algorithms to obtain optimal estimates of the stability region via energy functions.The existing characterization of the stability region was developed for a class of nonlinear dynamical systems that admit only hyperbolic equilibrium points in their limit sets. Understanding the characterization fro other classes of systems that admit more complex behavior, such as closed and chaotic orbits, in their limit sets is one of the goals of this research project. Other objective of this project is the investigation of the behavior of the stability region of attractors under the influence of parameter variation. This is an open problem in the literature that has a large potential for applications. In particular, the theory of stability region of singular perturbed systems will be studied.More precisely, the following research topics will be considered in this research project:1. Development of the characterization of the stability region of nonlinear dynamical systems that admit complex behavior, such as closed and chaotic orbits, in their limit sets.2. Studies of the behavior of the stability region of nonlinear dynamical systems under the influence of parameter variation.3. Development of the charaterization of the stability region of singularly perturbed systems.All these topics are related with open problems n the literature. Applications in electrical power sytems and general nonlinear systems will be investigated. In particular, we will investigate the possibility of applying the results of this research project in the problem of transient stability analsis of electrical power systems via direct methods. (AU)