Abstract
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.
Highlights
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 (Giua and Seatzu 2002; Shu et al 2007; Shu and Lin 2011, 2013a; Fornasini and Valcher 2013; Xu and Hong 2013; Zhang et al 2016; Ru and Hadjicostis 2010; Yin and Lafortune 2017b; Masopust 2018; Keroglou and Hadjicostis 2015; Sasi and Lin 2018)
We propose some new notions of detectability in the context of ω-languages, and characterize the related decision problems for both finite automata and labeled Petri nets
For labeled Petri nets, we show that the property is decidable and the corresponding decision problem is EXPSPACE-hard: note that this decidability result holds under the promptness assumption (collected in (ii) of Assumption 2) that is equivalent to condition (ii) of Assumption 1 for labeled Petri nets
Summary
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 (Giua and Seatzu 2002; Shu et al 2007; Shu and Lin 2011, 2013a; Fornasini and Valcher 2013; Xu and Hong 2013; Zhang et al 2016; Ru and Hadjicostis 2010; Yin and Lafortune 2017b; Masopust 2018; Keroglou and Hadjicostis 2015; Sasi and Lin 2018). We uniformly call this property “detectability”, and call another similar property “observability” implying that the initial state can be determined by the observed output signal produced by a system (e.g., Yin 2017; Shu and Lin 2013b; Zhang et al 2018; Zhang and Zhang 2016)
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