Active mode observability of switching linear systems

Abstract

In this paper active mode observability is addressed for a class of discrete-time linear systems that may switch in an unknown and unpredictable way among different modes taken from a finite set. The active mode observation problem consists in determining control sequences (discerning control sequences) that allow to identify the switching sequence on the basis of the observations. The presence of unknown but bounded noises affecting both the system and measurement equations is taken into account. In particular, a general condition is derived that characterizes discerning controls in a finite-horizon setting. Such a result is extended to the infinite-horizon case in order to derive "persistently discerning" control sequences. Numerical examples are reported to clarify the approach.

Rank tests for the observability of discrete-time jump linear systems with inputs

“…Notice that our definition of mode observability differs from the definition of mode observability of [1], [2] in the way the quantifiers are placed. 1 There, it is required that there exists a universal input, i.e., an input independent of the switching sequences, such that for any two distinct switching sequences, the outputs corresponding to the inputs are different.…”

“…In contrast, in Definition 1, the input U T may depend on the sequences w andw. Therefore, a system might be mode observable according to Definition 1, but not according to the definition of [1], [2]. Moreover, the sufficient conditions of [1], [2] are also sufficient conditions for mode observability in our setting.…”

“…Related work. Among the existing works on observability of discrete-time JLSs, the ones that are more closely related to our approach are [1], [3], [4], [2]. In comparison to the work of [3], the main contribution of this paper is that the switching sequence is viewed as an unknown parameter which needs to be estimated, rather than as a measured input.…”

“…With respect to the work of [1], [2], which also considered JLSs with inputs, there are two fundamental differences. First, we consider several different concepts of observability.…”

“…In §II we present several definitions of observability for JLSs. The first two definitions are mode observability [1], [2] and strong mode observability, which refer to the ability of uniquely reconstructing the discrete state for some or for all inputs, respectively. The other two definitions are observability and strong observability, which refer to the ability of uniquely reconstructing both continuous and discrete states for some or for all inputs, respectively.…”