UC Santa Cruz

This dissertation focuses on developing tools to study the robust forward invariance of sets for systems with unknown disturbances and hybrid dynamics. In particular, the notions of robust forward invariance properties are proposed for hybrid dynamical systems modeled by differential and difference inclusions with state-depending conditions enabling flows and jumps. A set is said to enjoy robust forward invariance for a system when its solutions start within the set always stay in the set regardless of disturbances. These proposed notions allow for a diverse type of solutions (with and without disturbances), including solutions that have persistent flow and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Moreover, sufficient conditions for sets to enjoy such properties are presented. The proposed conditions involve the system data and the set to be rendered robust forward invariant.

Furthermore, such conditions are exploited to derive conditions guaranteeing that sublevel sets of Lyapunov-like functions are robust forward invariant and, in turn, inspired a constructive way to design invariance-based control algorithms for a class of hybrid systems with control inputs and disturbances. More precisely, when a hybrid system have a Lyapunov-like function V satisfying a set of specific conditions, existence of feedback laws that render sublevel sets of V robustly forward invariant for the closed-loop hybrid systems are presented. In addition, two selection theorems are proposed to design invariance-based controllers for the class of hybrid systems considered.

Applications and academic examples are given to illustrate the results. In particular, the presented forward invariance analysis and design tools are applied to the design and validate of hybrid controllers for power conversion systems, specifically, a single-phase DC/AC inverter and a DC/DC boost converter. Moreover, results are applied to the estimation of weakly forward invariant sets, which is an invariance property of interest when employing invariance principles to study convergence properties of solutions. Finally, the developed algorithms are tested on the control of a constrained bouncing ball system.