Publication Type: B. Thesis
Source: PhD. Thesis, University of California, Santa Barbara (2007)
Keywords: hybrid systems
This thesis deals with systems exhibiting both continuous and discrete dynamics, perhaps due to intrinsic behavior or to the interaction of continuous-time and discrete-time dynamics emerging from its components and/or their interconnection. Such systems are called hybrid systems and permit the modeling of a wide range of engineering systems and scientific processes. In this thesis, hybrid systems are treated as dynamical systems: the interplay between continuous and discrete behavior is captured in a mathematical model given by differential equations/inclusions and difference equations/inclusions, which we simply call hybrid equations. We develop tools for systematic analysis and robust design of hybrid systems, with an emphasis on systems that involve control algorithms, that is, hybrid control systems. To this effect, we identify mild conditions that hybrid equations need to satisfy so that their behavior captures the effect of arbitrarily small perturbations. This leads to novel concepts of generalized solutions that impart a deep understanding not only on the robustness properties of hybrid systems but also on the structural properties of their solutions. In turn, these conditions enable us to generate various tools for hybrid systems that resemble those in the stability theory of classical dynamical systems. These include general versions of Lyapunov and Krasovskii stability theorems, and LaSalle-type invariance principles. Additionally, we establish results on robustness of stability of hybrid control for general nonlinear systems. We also present a novel mathematical framework for numerical simulation of hybrid systems and its asymptotic stability properties. The contributions of this thesis are not limited to the theory of hybrid systems as they have implications in the analysis and design of practically relevant engineering control systems. In this regard, we develop general control strategies for dynamical systems that are applicable, for example, to autonomous vehicles, multi-link pendulums, and juggling systems.