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

Model Reduction by Moment Matching for Linear Switched Systems

Two moment-matching methods for model reduction of linear switched systems (LSSs) are presented. The methods are similar to the Krylov subspace methods used for moment matching for linear systems. The more general one of the two methods, is based on the so called "nice selection" of some vectors in the reachability or observability space of the LSS. The underlying theory is closely related to the (partial) realization theory of LSSs. In this paper, the connection of the methods to the realization theory of LSSs is provided, and algorithms are developed for the purpose of model reduction. Conditions for applicability of the methods for model reduction are stated and finally the results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201

Midpoint based balanced truncation for switched linear systems with known switching signal

We propose a novel model reduction approachfor switched linear systems with known switching signal. Theclass of considered systems encompasses switched systems withmode-dependent state-dimension as well as impulsive systems.Our method is based on a suitable definition of (time-varying)reachability and observability Gramians and we show that theseGramians satisfy precise interpretations in terms of input andoutput energy. Based on balancing the midpoint Gramians, wepropose a piecewise-constant projection based model reductionresulting in a switched linear system of smaller size