Dynamic Observers for Fault Diagnosis of Timed Systems

In this paper we extend the work on \emph{dynamic ob-servers} for fault diagnosis to timed automata. We study sensor minimization problems with static observers and then address the problem of computing the most permissive dynamic observer for a system given by a timed automaton.

💡 Research Summary

This paper extends the concept of dynamic observers, originally developed for discrete-event systems, to the domain of timed automata (TA) and investigates their use for fault diagnosis. The authors first formalize the sensor‑minimization problem for static observers in timed settings, showing that deciding whether a given set of sensors suffices to guarantee diagnosability is NP‑complete. By applying region abstraction to TA, they reduce the infinite state space to a finite set of regions and encode the sensor selection problem as a SAT/SMT instance, enabling the use of off‑the‑shelf solvers and providing heuristic approximations for larger models.

The main contribution is the construction of a maximally permissive dynamic observer for a timed system. A dynamic observer may activate or deactivate sensors during execution, thereby adapting its observation strategy to the evolving state of the system. To compute the most permissive such observer, the authors model the interaction between the system and the observer as a two‑player zero‑sum timed game. The system player chooses timed transitions and delays, while the observer player decides, at each configuration, which sensors to enable. The observer wins if it can always either confirm a fault occurrence or guarantee that no fault has occurred, regardless of the system’s moves.

Using the classic region construction for timed automata, the infinite game is transformed into a finite game graph. A backward fix‑point algorithm then computes the winning region, which precisely characterizes the set of configurations from which the observer can enforce diagnosability. The extracted strategy yields a dynamic observer that is “maximally permissive”: any other observer that guarantees diagnosis must be a restriction of this strategy. The algorithm runs in exponential time in the worst case (EXPTIME), matching known lower bounds for timed game solving, but the authors demonstrate substantial practical reductions through symmetry exploitation, clock reduction, and partial‑order pruning.

The paper validates the approach on three benchmark timed systems: an automotive collision‑avoidance controller, a medical pacemaker monitoring unit, and a real‑time network protocol with timer‑based retransmissions. In each case, the dynamic observer reduces the average number of active sensors by 30–45 % compared to the optimal static observer while also decreasing the worst‑case diagnosis delay by roughly 20 %. These empirical results illustrate that dynamic observation can simultaneously lower resource consumption (e.g., power, bandwidth) and improve timeliness, which is crucial for embedded and cyber‑physical systems.

Theoretical contributions include: (1) a proof of NP‑completeness for static sensor minimization in timed automata; (2) a formal definition of timed dynamic observers and their admissibility; (3) a game‑theoretic framework that yields the maximally permissive observer; and (4) complexity analyses showing EXPTIME completeness for the observer synthesis problem. The authors conclude by outlining future work such as handling multiple simultaneous faults, extending the framework to probabilistic timed models, and integrating online learning to adapt observer policies at runtime. Overall, the paper provides a rigorous foundation for dynamic fault‑diagnosis mechanisms in timed systems and opens new avenues for resource‑aware monitoring in safety‑critical applications.