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

In this article, we explore the structural controllability of switched linear continuous-time systems. We first identify a gap in the proof of a fundamental criterion for the structural controllability of these systems. To solve this problem, we develop several novel graph-theoretic concepts, such as multilayer dynamic graphs, generalized stems/buds, and generalized cacti. Using these concepts, we provide a comprehensive proof for the criterion. Our approach also induces a new, generalized cactus-based graph-theoretic criterion for the 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 of switched systems.