Observability and Determinability Characterizations for Linear Switched Systems in Discrete Time

Abstract

In this article, we study the observability and determinability for discrete-time linear switched systems. Studies for the observability for this system class are already available in literature, however, we use assume here that the switching signal is known. This leads to less conservative observability conditions (e.g. observability of each individual mode is not necessary for the overall switched system to be observable); in particular, the dependencies of observability on the switching times and the mode sequences are derived; these results are currently not available in the literature on discrete-time switched systems. In addition to observability (which is concerned with recovering the state from the initial time onwards), we also investigate the determinability which is concerned with the ability to reconstruct the state value at the end of the observation interval. We provide several simple examples to illustrate novel features not seen in the continuous time case or for unswitched systems.

Reachability and Controllability Characterizations for Linear Switched Systems in Discrete Time: A Geometric Approach

“…Indeed, this will make powers to A i matrices in the reachability and controllability conditions do not affect the characterization results. The second situation is when the state is one-dimensional as proved in the following proposition, which is a property also seen for observability/determinability, see [16].…”

An automata theoretic approach to observer design for switched linear systems

“…One can then adapt the approach described in Subsection 4.4 with the DBA Bk instead of B k . In that case, LMIs similar to (25) and (26) need to hold for all transitions to non-accepting states and to accepting states, respectively.…”

“…Let us consider Example 1 again and let B k be the DBA constructed using Algorithm 1 for k = 3 as shown in Figure 1. Then, we solve the LMIs ( 24), ( 25), (26) for ρ = 1.5, λ = 0.1, to synthesize the following observer gains 1 :…”

“…In our work, we assume that the switching signal is known. In that context, there have been several works for characterizing observable or reconstructible switching sequences [15,26], i.e. finite switching sequences that make it possible to estimate the initial or the current state of the system, respectively.…”