Learning stability guarantees for black-box hybrid systems : from arbitrary to constrained switching linear systems, a step towards complexity

Hybrid systems are dynamical systems whose dynamics is characterized by continuous and discrete behaviours. In particular, switching linear systems are an important family of hybrid systems in which a switching rule plays a critical role. Consider a linear system whose controller is physically separated from its plant, and where the feedback has to go via a network. If the network fails, this linear system switches from an active to a failing status. Although this hybrid behaviour makes their range of application extensive, switching linear systems are notoriously hard to analyze. However, they often generate rich data (harvested from devices such as cameras, lidars, etc.), to which the engineer has access in large quantities. This motivates a global research effort in developing data-driven tools to analyze such complex systems. Recently, data-driven stability analysis techniques were developed for a subcategory of switching linear systems, where the switching rule is arbitrary. In this thesis, we take a step towards complexity by generalizing these results to a more general framework, where we consider that the switching rule is subject to logical rules. These systems are called constrained switching linear systems. We leverage existing data-driven approaches and combine them with specific tools for analyzing the stability of these systems, such as lifting techniques and multiple Lyapunov functions. Using these concepts, we first demonstrate that we can derive probabilistic guarantees for the stability of constrained switching linear systems. We investigate two different approaches, and compare them with multiple examples. Second, we provide algorithmic tools to compute these guarantees.