[Paper Review] Revisiting delayed strong detectability of discrete-event systems

Kuize Zhang, Alessandro Giua|arXiv (Cornell University)|Oct 30, 2019

Petri Nets in System Modeling20 references4 citations

TL;DR

This paper introduces K-delayed strong detectability for discrete-event systems modeled as finite-state automata, proposing a novel concurrent-composition method to verify this property in quartic polynomial time. The approach enables efficient verification and synthesis of delayed strong detectability, significantly improving upon prior exponential-time methods and extending to related notions like (k₁,k₂)-detectability and D-detectability.

ABSTRACT

Among notions of detectability for a discrete-event system (DES), strong detectability implies that after a finite number of observations to every output/label sequence generated by the DES, the current state can be uniquely determined. This notion is strong so that by using it the current state can be easily determined. In order to keep the advantage of strong detectability and weaken its disadvantage, we can additionally take some "subsequent outputs" into account in order to determine the current state. Such a modified observation will make some DES that is not strongly detectable become "strongly detectable in a weaker sense", which we call "{\it $K$-delayed strong detectability}" if we observe at least $K$ outputs after the time at which the state need to be determined. In this paper, we study $K$-delayed strong detectability for DESs modeled by finite-state automata (FSAs), and give a polynomial-time verification algorithm by using a novel concurrent-composition method. Note that the algorithm applies to all FSAs. Also by the method, an upper bound for $K$ has been found, and we also obtain polynomial-time verification algorithms for $(k_1,k_2)$-detectability and $(k_1,k_2)$-D-detectability of FSAs firstly studied by [Shu and Lin, 2013]. Our algorithms run in quartic polynomial time and apply to all FSAs, are more effective than the sextic polynomial-time verification algorithms given by [Shu and Lin 2013] based on the usual assumptions of deadlock-freeness and having no unobservable reachable cycle. Finally, we obtain polynomial-time synthesis algorithms for enforcing delayed strong detectability, which are more effective than the exponential-time synthesis algorithms in the supervisory control framework in the literature.

Motivation & Objective

• To address the limitations of strong detectability by introducing a weaker but practically useful variant: K-delayed strong detectability.
• To develop a polynomial-time verification algorithm for K-delayed strong detectability in finite-state automata (FSAs).
• To extend the proposed method to verify (k₁,k₂)-detectability and (k₁,k₂)-D-detectability, which were previously studied with less efficient algorithms.
• To provide polynomial-time synthesis algorithms for enforcing delayed strong detectability, overcoming the exponential-time complexity of existing supervisory control approaches.

Proposed method

• Introduces a novel concurrent-composition method, denoted as CC_A^tn, to model pairs of transition sequences with identical label sequences.
• Constructs an automaton CC_A^tn(S) that tracks both observable and normal event sequences, enabling comparison of state reconstructions under different observation delays.
• Uses the structure of CC_A^tn(S) to detect violations of K-delayed strong detectability by identifying cycles where current state cannot be uniquely determined within K steps.
• Applies graph-theoretic analysis to verify the existence of such cycles in linear time relative to the size of CC_A^tn(S).
• Leverages the finite-state nature of FSAs to bound the maximum delay K, ensuring the verification process remains polynomial.
• Adapts the method to verify and synthesize for related detectability notions, including (k₁,k₂)-detectability and (k₁,k₂)-D-detectability.

Experimental results

Research questions

• RQ1 Can K-delayed strong detectability be verified in polynomial time for all finite-state automata, without restrictive assumptions like deadlock-freeness or absence of unobservable cycles?
• RQ2 How does the proposed concurrent-composition method CC_A^tn(S) enable efficient detection of state indistinguishability under delayed observation?
• RQ3 What is the maximum delay K for which K-delayed strong detectability can be guaranteed, and how can this bound be computed?
• RQ4 Can the proposed method be extended to verify other detectability notions such as (k₁,k₂)-detectability and (k₁,k₂)-D-detectability with improved time complexity?
• RQ5 Can polynomial-time synthesis algorithms be designed to enforce K-delayed strong detectability, outperforming existing exponential-time supervisory control approaches?

Key findings

• The proposed verification algorithm for K-delayed strong detectability runs in quartic polynomial time, significantly improving upon the sextic polynomial-time algorithms of Shu and Lin (2013).
• The method applies to all finite-state automata without requiring assumptions such as deadlock-freeness or absence of unobservable cycles.
• An upper bound for the delay K is derived directly from the structure of the automaton, enabling effective analysis of the maximum observation delay needed for state determination.
• The same concurrent-composition framework enables polynomial-time verification of (k₁,k₂)-detectability and (k₁,k₂)-D-detectability, which were previously verifiable only in exponential time.
• Polynomial-time synthesis algorithms are developed to enforce K-delayed strong detectability, overcoming the exponential-time complexity of prior supervisory control-based approaches.
• The equivalence between two notions of diagnosability (Definition 7 and Definition 8) is re-proven using the new framework, confirming consistency with existing results in the literature.