Generalized Cactus and Structural Controllability of Switched Linear Continuous-Time Systems

This paper explores the structural controllability of switched linear continuous-time systems. It first identifies a gap in the proof for a pivotal criterion for the structural controllability of switched linear systems in the literature. To address this void, we develop novel graph-theoretic concepts, such as multi-layer dynamic graphs, generalized stems/buds, and generalized cacti, and based on them, provide a comprehensive proof for this criterion. Our approach also induces a new, generalized cactus based graph-theoretic criterion for structural controllability. This not only extends Lin's cactus-based graph-theoretic condition to switched systems for the first time, but also provides a lower bound for the generic dimension of controllable subspaces.