Fault Diagnosis with Dynamic Observers

arXiv:1004.2810v1 [cs.FL] 16 Apr 2010 Fault Diagnosis with Dynamic Observers∗ Franck Cassez† CNRS, IRCCyN Laboratory 1 rue de la No¨e BP 92101 44321 Nantes Cedex 3 France Email: franck.cassez@cnrs.irccyn.fr. Stavros Tripakis Cadence Research Laboratories 2150 Shattuck Avenue, 10th floor Berkeley, CA, 94704 USA and CNRS, Verimag Laboratory Centre Equation 2, avenue de Vignate, 38610 Gi`eres France Email: tripakis@cadence.com. Abstract— In this paper, we review some recent results about the use of dynamic observers for fault diagnosis of discrete event systems. Fault diagnosis consists in synthesizing a diagnoser that observes a given plant and identifies faults in the plant as soon as possible after their occurrence. Existing literature on this problem has considered the case of fixed static observers, where the set of observable events is fixed and does not change during execution of the system. In this paper, we consider dynamic observers: an observer can “switch” sensors on or off, thus dynamically changing the set of events it wishes to observe. It is known that checking diagnosability (i.e., whether a given observer is capable of identifying faults) can be solved in polynomial time for static observers, and we show that the same is true for dynamic ones. We also solve the problem of dynamic observers’ synthesis and prove that a most permissive observer can be computed in doubly exponential time, using a game-theoretic approach. We further investigate optimization problems for dynamic observers and define a notion of cost of an observer. I. INTRODUCTION A. Monitoring, Testing, Fault Diagnosis and Control Many problems concerning the monitoring, testing, fault diagnosis and control of discrete event systems (DES) can be formalized using finite automata over a set of observable events Σ, plus a set of unobservable events [3], [4]. The invisible actions can often be represented by a single unob- servable event ε. Given a finite automaton over Σ∪{ε} which is a model of a plant (to be monitored, tested, diagnosed or controlled) and an objective (good behaviours, what to test for, faulty behaviours, control objective) we want to check if a monitor/tester/diagnoser/controller exists that achieves the objective, and if possible to synthesize one automatically. The usual assumption in this setting is that the set of observable events is fixed (and this in turn, determines the set of unobservable events as well). Observing an event usually requires some detection mechanism, i.e., a sensor of some sort. Which sensors to use, how many of them, and where to ∗Preliminary versions of parts of this paper appeared in [1] and [2]. † Work suported by the French government under grant ANR-06-SETI. place them are some of the design questions that are often difficult to answer, especially without knowing what these sensors are to be used for. In this paper we review some recent results about sensor minimization. These results are interesting since observing an event can be costly in terms of time or energy: computation time must be spent to read and process the information provided by the sensor, and power is required to operate the sensor (as well as perform the computations). It is then essential that the sensors used really provide useful information. It is also important for the computer to discard any information given by a sensor that is not really needed. In the case of a fixed set of observable events, it is not the case that all sensors always provide useful information and sometimes energy (used for sensor operation and computer treatment) is spent for nothing. For example, to detect a fault f in the system described by the automaton B, Figure 1, page 3, an observer needs to watch only for event a initially, and watch for event b only after a has occurred. If the sequence a.b occurs, for sure f has occurred and the observer can raise an alarm. If, on the other hand, event b is not observed after a, then f has not occurred. It is then not useful to switch on sensor b before observing event a. B. Sensor Minimization and Fault Diagnosis We focus our attention on sensor minimization, without looking at problems related to sensor placement, choosing between different types of sensors, and so on. We also focus on a particular observation problem, that of fault diagnosis. We believe, however, that the results we obtain are applicable to other contexts as well. Fault diagnosis consists in observing a plant and detecting whether a fault has occurred or not. We follow the discrete- event system (DES) setting of [5] where the behavior of the plant is known and a model of it is available as a finite-state automaton over Σ ∪{ε, f} where Σ is the set of potentially observable events, ε represents the unobservable events, and f is a special unobservable event that corresponds to the faults1. Checking diagnosability (whether a fault can be detected) for a given plant and a fixed set of observable events can be done in polynomial time [5], [6

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