Detectability of Discrete-Event Systems Under Nondeterministic Observations

In practical systems, due to reasons such as sensor limitations, sensor faults, and packet losses in networks, the observation of events becomes nondeterministic. In this article, we extend strong detectability and weak detectability to the case of nondeterministic observations and denote them as A-(
$k_{1}$
,
$k_{2}$
)-detectability and O-(
$k_{1}$
,
$k_{2}$
)-detectability, respectively. A-(
$k_{1}$
,
$k_{2}$
)-detectability says that, for any string, we can distinguish state pairs in the specification for all possible observations of the string. O-(
$k_{1}$
,
$k_{2}$
)-detectability says that, for at least one string, we can distinguish state pairs in the specification for all possible observations of the string. For A-(
$k_{1}$
,
$k_{2}$
)-detectability, we construct a transformed automaton and then translate the A-(
$k_{1}$
,
$k_{2}$
)-detectability problem into the traditional detectability problem that has been solved. We show that A-(
$k_{1}$
,
$k_{2}$
)-detectability can be used to solve the deterministic supervisory control problem. For O-(
$k_{1}$
,
$k_{2}$
)-detectability, we construct an augmented automaton that includes all the information of the given automaton and its state estimates. Based on the augmented automaton, we propose a depth-first search (DFS)-based algorithm to check O-(
$k_{1}$
,
$k_{2}$
)-detectability.
Note to Practitioners
—Nowadays, practical engineering systems become more and more complex. In these systems, the observation of events often becomes nondeterministic due to reasons such as sensor limitations, sensor faults, and packet losses in networks. Consider a mobile robot as an example. The availability of a sensor output may depend on the current location of the mobile robot. If the mobile robot is in an area where the wireless network is unreliable, the sensor output may not be received by the supervisor. In this article, we investigate the state estimation problem for practical engineering systems under nondeterministic observations within a discrete-event system framework. The results in this article provide not only insights for engineers in the automatic control field to understand nondeterministic observations in practical systems but also the methodology to estimate the current discrete state that is always an important issue. Therefore, we believe that the engineers in the automatic control field should be interested in this article and can benefit from it.