The one-step-map for switched singular systems in discrete-time

Abstract

We study switched singular systems in discrete time and first highlight that in contrast to continuous time regularity of the corresponding matrix pairs is not sufficient to ensure a solution behavior which is causal with respect to the switching signal. With a suitable index-1 assumption for the whole switched system, we are able to define a one-stepmap which can be used to provide explicit solution formulas for general switching signals.

Stability analysis for switched discrete-time linear singular systems

“…The main relevance of regularity and index-1 is the following statement about existence and uniqueness of solutions of (3), which is a straightforward consequence from the QWF (4) and was already stated as Lem. 2.5 in Anh et al (2019).…”

“…This nice solvability characterization is lost in the discrete time case (see e.g. Example 1.1 in Anh et al (2019)); in particular, causality with respect to the switching signal may be lost. It is therefore necessary to impose more strict assumptions on the matrix pairs (E i , A i ) to be able to conclude suitable solution properties.…”

“…It is therefore necessary to impose more strict assumptions on the matrix pairs (E i , A i ) to be able to conclude suitable solution properties. Inspired by our previous works on discretetime singular systems (Anh and Yen, 2006;Anh et al, 2007;Loi et al, 2002;Anh et al, 2019), we introduce the following index-1 notion for the overall switched system.…”

“…Since condition (6) also needs to hold for i = j and in view of Lemma 2.9 it follows that each individual pair (E i , A i ) needs to be regular and of index-1, i.e. the definition is indeed a generalization of the index-1 property for the non-switched case; however, Example 1.1 in Anh et al (2019) shows that regularity and index-1 (causality) of each individual mode is not sufficient in general for the whole SDLS (1) to be of index-1 in the above sense. Note furthermore, that we do not assume that the "mixed" matrix pairs (E j , A i ) are regular and index-1 (because i is defined in terms of E i and not E j ).…”

“…The references Darouach and Chadli (2013); Koenig and Marx (2009); Meng and Zhang (2006) which actually study the general SDLS system (1) seem to have overlooked the fact that even if each mode is causal (i.e. regular and index-1, see Section 2.2) the corresponding switched system is not well-posed in general, see the recent publication Anh et al (2019); in particular, these references lack a proper solution theory capable to handle arbitrary switching. In fact, to study the SDLS system (1) under arbitrary switching it seems reasonable to impose an additional causality assumption with respect to the switching signal, i.e.…”