Diagnosis and Degradation Control for Probabilistic Systems
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Diagnosis and Degradation Control for Probabilistic Systems Nathalie Bertrand1
· Serge Haddad2 · Engel Lefaucheux1,3
Received: 22 May 2018 / Accepted: 30 April 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Systems prone to faults are often equipped with a controller whose aim consists in restricting the behaviour of the system in order to perform a diagnosis. Such a task is called active diagnosis. However to avoid that the controller degrades the system in view of diagnosis, a second objective in terms of quality of service is usually assigned to the controller. In the framework of stochastic systems, a possible specification, called safe active diagnosis requires that the probability of correctness of the infinite (random) run is non null. We introduce and study here two alternative specifications that are in many contexts more realistic. The notion of (γ , v)-fault freeness associates with each run a value depending on the discounted length of its correct prefix where the discounting factor is γ . The controller has to ensure that the average of this value is above the threshold v. The notion of α-resiliency requires that asymptotically, at every time step, a proportion greater than α of correct runs remain correct. From a semantic point of view, we determine the equivalences and (non) implications between the three notions of degradations both for finite and infinite systems. From an algorithmic point of view, we establish the border between decidability and undecidability of the diagnosability problems. Furthermore in the positive case, we exhibit their precise complexity and propose a synthesis of the controller which may require an infinite memory. Keywords Stochastic systems · Partial observation · Fault tolerance · Diagnosis The work of Serge Haddad was supported by the project ERC EQualIS (FP7-308087). Nathalie Bertrand
[email protected] Serge Haddad [email protected] Engel Lefaucheux [email protected] 1
Inria, CNRS, IRISA, Universit´e de Rennes, Rennes, France
2
LSV, ENS Paris-Saclay, CNRS, Inria, Universit´e Paris-Saclay, Saint-Aubin, France
3
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbr¨ucken, Germany
Discrete Event Dynamic Systems
1 Introduction Diagnosis The designer of a system aims at eliminating faults that could trigger unwanted behaviours. However, for embedded systems interacting with an unpredictable environment, the absence of faults is not a reasonable hypothesis. Thus diagnosis, whose goal consists to detect faults from the observations of the run of the system, is a crucial task. One of the approach frequently used to analyse diagnosability (i.e. the existence of a diagnoser) consists in modelling the system by a transition system whose states (depending on the internal part of the system) are unobservable and events may, depending on their nature, be observable or not. A diagnoser must fulfill two requirements: correctness and reactivity. A diagnoser is correct if it never erroneously claims a f
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