Observability of Singular Linear Switched Systems in Discrete Time: Single Switch Case

Abstract

In this paper, we investigate the observability of singular linear switched systems in discrete time. As a preliminary study, we restrict ourselves to systems with a single switch switching signal, i.e. the system switches from one mode to another mode at a certain switching time. We provide two necessary and sufficient conditions for the observability characterization. The first condition is applied for arbitrary switching time and the second one is for switching times that are far enough from the initial time and the final time of observation. These two conditions explicitly contain the switching time variable that indicates that in general, the observability is dependent on the switching time. However, under some sufficient conditions we provide, the observability will not depend on the switching time anymore. Furthermore, the observability of systems with two-dimensional states is independent of the switching time. In addition, from the example we discussed, an observable switched system can be built from two unobservable modes and different mode sequences may produce different observability property; in particular, swapping the mode sequence may destroy observability.

Observability and Determinability Characterizations for Linear Switched Systems in Discrete Time

“…Utilizing Lemma 3.6 proves that it is not possible to have different observability properties with different switching times, which is omitted due to space limitation. This result is indeed stronger than the result presented in [16,Corr. 3.4] where we have now confirmed that the observability of LSSs, with any dimensional states and under a single switch switching signal, is always constant as long as each mode is long enough active.…”

“…III. OBSERVABILITY Under a single switch switching signal, the observability characterization for (2) has been covered in [16]. We present in the following the results regarding the observability characterizations for general switching signals.…”

“…Inspired by the single-switch result for the continuous time case, one may conjecture that also in the discrete time case the dependence of the observability property on the switching time in the single-switch case can only come from not obtaining enough information from the individual modes. We have already investigated this question in [16] in the context of singular systems, but we were only able to show the constant observability property under some additional subspace assumptions. Here we are now able to state the constant observability without any further assumptions; the result is based on the following technical lemma.…”

“…However, this is not enough to preserve the observability if the switching time(s) is changed and thus it becomes dependent on the switching time(s). Apart from our own study of the single-switch case [16], the dependency of observability on (multiple) switching times seems not to be discussed yet in any reference and therefore is a novel aspect we study in this paper.…”