Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from Shaker and Wisniewski (2011, 2009)  [1] and [2]. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems (Wood et al., 1996)  [3]. Specifically in this paper, we provide a bound on the approximation error in the L2L2 norm for continuous-time and the l2l2 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 balanced 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 L2L2 and l2l2 gains depend only on the input–output map.