Observability of Switched Linear Systems

Abstract

The observability of deterministic, discrete-time, switched, linear systems is considered. Depending on whether or not the modes are observed, and on whether the continuous state or the mode sequence is to be recovered, several observability concepts are defined, characterized through linear algebraic tests, and their decidability assessed.

Identification of PWARX hybrid models with unknown and possibly different orders

“…The error E λ between the estimated discrete stateλ t and the true discrete state λ t was computed as the percentage of times in whicĥ λ t = λ t , averaged over the number of trials. 2 The number of zeros was estimated as described in Remark 2, with the weight of the complexity term chosen as µ = 2 × 10 −8 .…”

“…Work on identification/filtering of hybrid systems first appeared in the seventies (see [19] for a review). More recent works consider variations of Problem 1 in which the model parameters, the discrete state and/or the switching mechanism are known, and concentrate on the analysis of the observability of the hybrid state [2], [4], [9], [11], [18], [21], [22] and the design of hybrid observers [1], [3], [7], [8], [10], [12], [14], [16], [17], [20].…”

Active mode observability of switching linear systems

“…More recent advances on this topic have been developed in [2], where arbitrary switching sequences were considered. In [1], such results have been extended to comply with the presence of bounded disturbances that corrupt the dynamics and the measures.…”

“…In fact, even in the best-case scenario, it is not possible to determine uniquely the switching sequence when the initial continuous state is null (or, in the noisy case, when it belongs to a non-empty neighborhood of the origin). In [2], it has been shown that, for noise-free systems, such a drawback may be overcome by means of a suitable choice of the controls, thus making it possible to determine uniquely the switching sequence over a finite horizon on the basis of the observations, regardless of the initial continuous state. Such controls are called discerning controls.…”

“…The original contributions of this paper can be summarized as follows: i) in the finite horizon setting, the results of [2] are extended to the noisy case and a characterization of discerning control sequence in the presence of unknown but bounded noises is derived; ii) it is proved that, when the mode observability property holds over a finite horizon, then it is possible to derive a persistently discerning control sequence over an infinite horizon that, at any time, allows one to reconstruct the discrete state up to the current time; iii) a sequential scheme is proposed that allows to derive a persistently discerning control sequence by choosing the control vectors on line, one at each time stage.…”