On detectability of labeled Petri nets and finite automata

Detectability is a basic property of dynamic systems: when it holds an observer can use the current and past values of the observed output signal produced by a system to reconstruct its current state. In this paper, we consider properties of this type in the framework of discrete-event systems modeled by labeled Petri nets and finite automata. We first study weak approximate detectability. This property implies that there exists an infinite observed output sequence of the system such that each prefix of the output sequence with length greater than a given value allows an observer to determine if the current state belongs to a given set. We prove that the problem of verifying this property is undecidable for labeled Petri nets, and PSPACE-complete for finite automata. We also consider one new concept called eventual strong detectability. The new property implies that for each possible infinite observed output sequence, there exists a value such that each prefix of the output sequence with length greater than that value allows reconstructing the current state. We prove that for labeled Petri nets, the problem of verifying eventual strong detectability is decidable and EXPSPACE-hard, where the decidability result holds under a mild promptness assumption. For finite automata, we give a polynomial-time verification algorithm for the property. In addition, we prove that strong detectability is strictly stronger than eventual strong detectability for labeled Petri nets and even for deterministic finite automata

Detectability Of Fuzzy Discrete Event Systems

Dynamic systems that can be modeled in terms of discrete states and a synchronous events are known as discrete event systems (DES). A DES is defined in terms of states, events, transition dynamics, and initial state. Knowing the system’s state is crucial in many applications for certain actions (events) to be taken. A DES system is considered a fuzzy discrete event system (FDES) if its states and events are vague in nature; for such systems, the system can be in more than one state at the same time with different degrees of possibility (membership). In this research we introduce a fuzzy discrete event system with constraints (FDESwC) and investigate its detectabilities. This research aims to address the gap in previous studies and extend existing definitions of detectability of DES to include the detectability in systems with substantial vagueness such as FDES. These definitions are first reformulated to introduce N-detectability for DES, which are further extended to define four main types of detectabilities for FDES: strong N-detectability, (weak) N-detectability, strong periodic N-detectability, and (weak) periodic N-detectability. We first partition the FDES into trajectories of a length dictated by the depth of the event’s string (length of the event sequence); each trajectory consists of a number of nodes, which are further investigated for detectability by examining them against the newly introduced certainty criterion. Matrix computation algorithms and fuzzy logic operations are adopted to calculate the state estimates based on the current state and the occurring events. Vehicle dynamics control example is used to demonstrate the practical aspect of developed theorems in real-world applications