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Coordination control of discrete-event systems revisited

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Abstract

In this paper, we revise and further investigate the coordination control approach proposed for supervisory control of distributed discrete-event systems with synchronous communication based on the Ramadge-Wonham automata framework. The notions of conditional decomposability, conditional controllability, and conditional closedness ensuring the existence of a solution are carefully revised and simplified. The approach is generalized to non-prefix-closed languages, that is, supremal conditionally controllable sublanguages of not necessary prefix-closed languages are discussed. Non-prefix-closed languages introduce the blocking issue into coordination control, hence a procedure to compute a coordinator for nonblockingness is included. The optimization problem concerning the size of a coordinator is under investigation. We prove that to find the minimal extension of the coordinator event set for which a given specification language is conditionally decomposable is NP-hard. In other words, unless P=NP, it is not possible to find a polynomial algorithm to compute the minimal coordinator with respect to the number of events.

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Acknowledgments

The research of two first authors was supported by RVO: 67985840. In addition, the first author was supported by the Grant Agency of the Czech Republic under grant P103/11/0517 and the second author by the Grant Agency of the Czech Republic under grant P202/11/P028. The authors gratefully acknowledge very useful suggestions and comments of the anonymous referees.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 616 62, Brno, Czech Republic

    Jan Komenda & Tomáš Masopust

  2. Van Schuppen Control Research, Gouden Leeuw 143, 1103 KB, Amsterdam, The Netherlands

    Jan H. van Schuppen

Authors
  1. Jan Komenda
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  2. Tomáš Masopust
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  3. Jan H. van Schuppen
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Corresponding author

Correspondence to Tomáš Masopust.

Additional information

A preliminary version was presented at the 11th International Workshop on Discrete Event Systems (WODES 2012) held in Guadalajara, Mexico (Komenda et al. 2012a).

Appendix Auxiliary results

Appendix Auxiliary results

In this section, we list auxiliary results required in the paper.

Lemma 7 (Proposition 4.6, (Feng 2007))

Let L i over Σ i , for i = 1, 2, be prefix-closed languages, and let K i be a controllable sublanguage of L i with respect to L i and Σ i, u . Let Σ = Σ1 ∪ Σ2. If K 1 and K 2 are synchronously nonconflicting, then K 1 ∥ K 2 is controllable with respect to L 1 ∥ L 2 and Σ u .

Lemma 8 (Komenda et al. 2012c)

Let K be a subset of a language L, and L be a subset of a language M over Σ such that K is controllable with respect to \(\overline {L}\) and Σ u , and L is controllable with respect to \(\overline {M}\) and Σ u . Then K is controllable with respect to \(\overline {M}\) and Σ u .

Lemma 9 (Wonhan 2012)

Let \(P_{k} : \Sigma ^{*}\to \Sigma _{k}^{*}\) be a projection, and let L i be a language over Σ i , where Σ i is a subset of Σ, for i = 1, 2, and Σ1 ∩ Σ2is a subset of Σ k . Then P k (L 1 ∥ L 2) = P k (L 1) ∥ P k (L 2).

Lemma 10 (Komenda et al. 2012c)

Let L i be a language over Σ i , for i = 1, 2, and let \(P_{i} : (\Sigma _{1}\cup \Sigma _2)^{*} \to \Sigma _{i}^{*}\) be a projection. Let A be a language over Σ1 ∪ Σ2 such that P 1(A) is a subset of L 1 and P 2(A) is a subset of L 2. Then A is a subset of L 1 ∥ L 2.

Lemma 11 (Pena et al. 2009)

Let L i be a language over Σ i , for i ∈ J, and let \(\cup _{k,\ell \in J}^{k\neq \ell } (\Sigma _{k}\cap \Sigma _{\ell })\subseteq \Sigma _{0}\). If \(P_{i,0}:\Sigma _{i}^{*} \to (\Sigma _{i}\cap \Sigma _0)^{*}\) is an L i -observer, for i ∈ J, then \(\overline {\|_{i\in J} L_{i}} = \|_{i\in J} \overline {L_{i}}\) if and only if \(\overline {\|_{i\in J} P_{i,0}(L_i)} = \|_{i\in J} \overline {P_{i,0}(L_i)}\).

Lemma 12 (Komenda et al. 2011b)

A language K ⊆ (Σ1 ∪ Σ2 ∪ … ∪ Σ n )∗is conditionally decomposable with respect to event sets Σ1, Σ2,…, Σ n , Σ k if and only if there exist languages \(M_{i+k}\subseteq \Sigma _{i+k}^{*}\), i = 1, 2, … , n, such that \(K=\parallel _{i=1}^{n} M_{i+k}\).

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Komenda, J., Masopust, T. & van Schuppen, J.H. Coordination control of discrete-event systems revisited. Discrete Event Dyn Syst 25, 65–94 (2015). https://doi.org/10.1007/s10626-013-0179-x

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