Path-Complete Methods and analysis of constrained switching systems

Switching systems are dynamical systems having several operating modes. For example, consider a controlled system where the controller may take several configurations, e.g. failure or active. Other examples include multi-agent systems where varying communication patterns dictate the dynamics of agents. They also appear as models of cyber-physical systems, which result from the interaction of algorithms and physical devices. These applications are key in motivating researchers to tackle the deep mathematical challenges arising from the study of switching systems. In this thesis, we study a general class of switching systems where the sequences of modes in time are generated by an automaton. They are known as constrained switching systems. We provide theory and algorithms for their stability analysis, discussing the notions of uniform, exponential and dead-beat stability, and including I/O stability and feedback stabilization. In particular, we provide tools to approximate the exponential decay rate of constrained switching systems arbitrarily accurately in finite time. Our techniques revolve around and expand Path-Complete Methods. These are mathematical tools merging automata theory with multiple Lyapunov functions within a general stability analysis framework. These techniques show potential for tackling further challenges, which pushes us to further analyze them. First, in the context of arbitrary switching systems, we show how these multiple Lyapunov functions can be represented as common Lyapunov functions. Second, we provide algorithms to compare their performances as stability certificates. This work contributes to a global effort in the study of complex dynamical systems where computer science and systems theory meet.