On Detectabilities of Fuzzy Discrete Event Systems

Many biomedical systems and some engineering systems can be modeled as fuzzy discrete event systems. In this article, fuzzy discrete event systems with constraints (FDESwC) are introduced and detectabilities of FDESwC are investigated. While detectabilities of conventional crisp discrete event systems (DES) have been investigated before, detectabilities of FDESwC are much more complex because the state space of FDESwC is infinite, unlike that of crisp DES. To overcome this difficulty, trajectories of FDESwC bounded by
$N$
observations are considered and
$N$
-detectabilities are introduced.
$N$
-detectabilities of crisp DES are first defined and proved to be equivalent to detectabilities if
$N$
is sufficiently large. Fuzzy
$N$
-detectabilities of FDESwC are then defined. An algorithm is derived to check fuzzy
$N$
-detectabilities. While fuzzy detectabilities require to determine which state the system is in, this requirement can be relaxed in some applications. To do so, fuzzy
$D$
-detectabilities are introduced and investigated, where the requirement is to distinguish certain pairs of states.