Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem

Structural Controllability of Switched Continuous and Discrete Time Linear Systems

This paper explores the structural controllability of switched continuous and discrete time linear systems. It identifies a gap in the proof for a pivotal criterion for structural controllability of switched continuous time 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 cactus configurations, 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 (which is conjectured to be exact). Finally, we present extensions to reversible switched discrete-time systems, which lead to not only a simplified necessary and sufficient condition for structural controllability, but also the determination of the generic dimension of controllable subspaces.Comment: Submitted to a journa