New Gramians for Linear Switched Systems: Reachability, Observability, and Model Reduction
For control theorists and engineers working with linear switched systems, this provides a novel theoretical framework and practical method for model reduction, though the results are incremental as they extend existing Gramian-based concepts to a specific system class.
The paper introduces new algebraic Gramians for continuous-time linear switched systems that satisfy generalized Lyapunov equations, showing their ranges encode reachability and observability spaces, and proposes a balancing-based model reduction technique with stability preservation and error bounds.
In this paper, we propose new algebraic Gramians for continuous-time linear switched systems, which satisfy generalized Lyapunov equations. The main contribution of this work is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a linear switched system. As a consequence, a simple Gramian-based criterion for reachability and observability is established. Second, a balancing-based model order reduction technique is proposed and, under some sufficient conditions, stability preservation and an error bound are shown. Finally, the efficiency of the proposed method is illustrated by means of numerical examples.