Generalised gramian framework for model/controller order reduction of switched systems

Abstract

In this article, a general method for model/controller order reduction of switched linear dynamical systems is presented. The proposed technique is based on the generalised gramian framework for model reduction. It is shown that different classical reduction methods can be developed into a generalised gramian framework. Balanced reduction within a specified frequency bound is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, generalised gramian-based Petrov-Galerkin projection is constructed instead of the similarity transform approach for reduction. The framework is developed for switched controller reduction. To the best of our knowledge, there is no other reported result on switched controller reduction in the literature. The method preserves the stability under an arbitrary switching signal for both model and controller reduction. Furthermore, it is applicable to both continuous and discrete time systems for different classical gramian-based reduction methods. The performance of the proposed method is illustrated by numerical examples.

Balanced truncation for linear switched systems

“…In particular, balanced truncation was explored in [1,2,14,13,15,12,16]. The procedure dealt with in this paper was already described in [1,2].…”

“…Invariance of the grammians on state-space representation Under which conditions the controllability and observability inequalities (2) and (3) and their discrete counterparts have solutions ? Does the existence of a solution to these inequalities depend on the choice of the state-space representation ?…”

“…To describe the contribution of this paper in more details, we will first present an informal overview of balanced truncation for linear switched system which appeared in [1,2]. In fact, this method has already been used for linear time varying systems in [3].…”

“…The performance of this procedure has been extensively tested by means of numerical examples in [1,2]. However, many theoretical questions remain open.…”

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AbstractIn this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from [1,2]. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems [3].

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