Delay-coobservability and its algebraic properties for the decentralized supervisory control of discrete event systems with communication delays

doi:10.1016/j.automatica.2008.12.006

Automatica

Volume 45, Issue 5, May 2009, Pages 1252-1259

https://doi.org/10.1016/j.automatica.2008.12.006 Get rights and content

Abstract

In networked control systems, uncontrollable events may unexpectedly occur at a plant before a proper control command is applied to it due to communication delays. In this paper, we address the problem of decentralized supervisory control under such communication delays based on the C&P (conjunctive and permissive) and D&A (disjunctive and antipermissive) decision architecture. In particular, for the existence of a decentralized supervisor, we present the notion of delay-coobservability of a given language specification and a polynomial-time algorithm for verifying it. In addition, algebraic properties of the delay-coobservability are investigated. We further present a synthesis method of the decentralized supervisor for practical usefulness.

Introduction

The architecture of decentralized supervisory control of discrete event systems (DESs) is composed of local control actions and the fusion decision of such local actions. In this respect, the C&P (conjunctive and permissive) and D&A (disjunctive and antipermissive) decision rules were introduced as a general architecture for decentralized supervisory control (Yoo & Lafortune, 2002). In that work, local supervisors decide their control actions using P&A (permissive and antipermissive) rules and the C&D (conjunctive and disjunctive) fusion decision is made by integrating the local decisions for its final application to a plant. In most of the previous works on decentralized supervisory control (Rudie and Wonham, 1992, Yoo and Lafortune, 2002, Yoo and Lafortune, 2004), it was assumed that there is no event occurrence in a plant before a fusion control decision is applied to it. However, this assumption does not hold in practice for many decentralized supervisory control systems connected by communication networks. During the computation of a control command in a supervisor or during the transmission of the command message through a network, some uncontrollable events (e.g., faults, completion of certain tasks, etc.) may unexpectedly occur in the plant. Hence, in the presence of such delays, the general C&P and D&A architecture for decentralized supervisory control and the notion of coobservability (Yoo & Lafortune, 2002) need to be reconsidered.

Recently, some studies on the supervisory control under communication delays were reported (Balemi, 1994, Debouk et al., 2003, Park and Cho, 2006, Park and Cho, 2007, Tripakis, 2004). In particular, a centralized supervisory control problem to achieve a given language specification under delays was addressed in Park and Cho (2006). The existence condition of a decentralized supervisor that can achieve a given language specification under delays was presented in Park and Cho (2007) within the C&P decision architecture. In this paper, we address the decentralized supervisory control problem for a given language specification in consideration of communication delays based on the C&P and D&A decision architecture. For this purpose, we assume that every controllable event is permitted to occur only if it is enabled by a decentralized supervisor and that some uncontrollable events may unexpectedly occur in a plant before a proper control command is applied to it. Based on the first assumption, we present the notion of enabling-required strings with which a local supervisor can make its decision on issuing control commands. Based on the second assumption, we present a design method for local supervisors which can achieve a given language specification according to permissive and antipermissive local decision rules under delays. A design method for the decentralized supervisory control action based on the local decisions is provided as well. As a core condition for the existence of a decentralized supervisor to achieve a given specification, we present the notion of delay-coobservability based on C&P delay-coobservability and D&A delay-coobservability. Moreover, we provide a polynomial-time algorithm to verify the delay-coobservability based on construction of nondeterministic automata. Furthermore, we present the algebraic properties of the delay-coobservability.

Although the presented delay-coobservability is more restrictive than the coobservability of Yoo and Lafortune (2002), it is more useful for the decentralized supervisory control problems within the C&P and D&A framework under communication delays in the following respects. In the framework of Yoo and Lafortune (2002), a decentralized control command should be computed upon the observation of a string with the assumption that there will be no event occurrence in a plant while the control command is actually transmitted to it. However, during the transmission of the control command through a communication channel, other data packet arrival events or some breakdown events might occur in the plant. In such cases, the control command could be transmitted to the plant after the arrival of other data packets or some component failures if we follow the framework of Yoo and Lafortune (2002). So, we might then have unexpected behaviors of the controlled system. The framework presented in this paper provides a more systematic way of computing decentralized control actions in such a situation. Moreover, if a given language specification is not delay-coobservable, the synthesis method we presented for a decentralized supervisor is practically more useful.

We shall assume general knowledge of supervisory control and its most common notations. For a reference, the reader is directed to Park and Cho (2007).

Section snippets

Problem formulation

Let us consider the problem of decentralized supervisory control of a DES $G$ with communication delays as shown in Fig. 1 where $a_{1}, \dots, a_{D}$ denote uncontrollable events (in this paper, we assume that a plant $G$ is not necessarily trim, i.e. $p r (L_{m} (G)) \neq L (G)$ ). In Fig. 1, we note that a proper decentralized supervisory control action $S_{d e c} (t)$ for a string $t$ is actually applied to a plant after the occurrences of $a_{i}$ , $i = 1, \dots, D$ subsequent to $t$ .

Each local supervisor $S_{i}$ can observe the set of locally observable

Supervisor existence: Delay-coobservability for a fixed partition of $Σ_{c}$

Let us present a language property called delay-coobservability as a principal condition for the existence of a decentralized supervisor that can achieve a given language specification under communication delays.

Definition 5

A language $K \subseteq L_{m} (G)$ is C&P delay-coobservable w.r.t. ${(Σ_{o, i}, Σ_{c, i})}_{i \in I}$ if $\forall s u \in p r (K)$ , $σ \in Σ_{c}$ , and $u \in Σ_{u c}^{*}$ s.t. $s u σ \in L (G) ∖ p r (K)$ and $s \in e r (K)$ , $(\exists i \in I) [[σ \in Σ_{c, i}] \land [E_{i} (s) Σ_{u c}^{*} σ \cap p r (K) = 0̸]] .$

Definition 6

A language $K \subseteq L_{m} (G)$ is D&A delay-coobservable w.r.t. ${(Σ_{o, i}, Σ_{c, i})}_{i \in I}$ if $\forall s u \in p r (K)$ , $σ \in Σ_{c}$ , and $u \in Σ_{u c}^{*}$ s.t. $s u σ \in p r (K)$ and $s$

Verification of delay-coobservability

To verify the delay-coobservability of a given language specification, we need to construct two different nondeterministic automata $M_{c} (Σ_{c})$ and $M_{d} (Σ_{c})$ for C&P delay-coobservability and D&A delay-coobservability, respectively, based on the approach of Yoo and Lafortune (2002). For illustration, we consider two local supervisors here, but the result can be applied to any finite number of supervisors through a straightforward extension. We assume that $K$ is generated by the trim finite-state

Algebraic properties of delay-coobservability

In this section, we present the algebraic properties of delay-coobservability that are essential in order to solve supervisor synthesis problems under communication delays. For this purpose, we consider the closure of C&P delay-coobservability and D&A delay-coobservability under union and intersection. Note that the closure of these delay-coobservabilities is checked w.r.t. not a specific partition $Σ_{c, e}$ and $Σ_{c, d}$ of $Σ_{c}$ but the set $Σ_{c}$ for the sake of generality. If a certain delay-coobservability

Conclusions

In this paper, we have developed a decentralized supervisory control framework under communication delays. Specifically, we have addressed the problems of supervisor existence and delay-coobservability verification. We have further investigated the algebraic properties of delay-coobservability and presented the synthesis method of a decentralized supervisor.

Acknowledgements

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-311-D00652), by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea Ministry of Education, Science & Technology through the Systems Biology grant (M10503010001-07N030100112), the Nuclear Research grant (M20708000001-07B0800-00110), and the 21C Frontier Microbial Genomics and Application Center Program (Grant MG08-0205-4-0). We thank anonymous reviewers for

Seong-Jin Park received the B.S., M.S., and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1994, 1997, and 2001, respectively. From September 2001 to February 2004, he was a Senior Engineer in the Telecommunication Division, Samsung Electronics, Suwon, Korea. In March 2004, he joined the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea, where he is currently an Associate Professor. His

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Kwang-Hyun Cho received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 1993, 1995, and 1998, respectively. From March 1999 to August 2004, he was a Lecturer and an Assistant Professor in the School of Electrical Engineering at the University of Ulsan, Korea. From June 2002 to August 2003, he worked at the Control Systems Centre in Department of Electrical Engineering & Electronics at the University of Manchester Institute of Science and Technology (UMIST), UK as a Visiting Professor. From January to March in 2004 he also worked at the Automatic Control Group in Department of Signals, Sensors and Systems at the Royal Institute of Technology, Sweden as a Visiting Research Fellow. He was invited to the Hamilton Institute in Ireland as a Senior Research Fellow for the systems biology research between June and August in 2004. From September 2004 to August 2007, he was an Associate Professor at the College of Medicine, Seoul National University, Korea and worked at the Bio-MAX Institute, Seoul National University, Korea as a Director of Systems Biology Laboratory. He is currently a tenured Associate Professor in the Department of Bio and Brain Engineering at the Korea Advanced Institute of Science and Technology (KAIST), also holds a joint position at the KAIST Institute for the BioCentury, and is a director of Laboratory for Systems Biology and Bio-Inspired Engineering (http://sbie.kaist.ac.kr). He has been guest editor of SIMULATION: Transactions of the Society for Modeling and Simulation International for the special issue of systems biology (vol. 79, no. 12, 2003) and IEEE Control Systems Magazine for the special issue of systems biology (vol. 24, no. 4, 2004). He co-founded Systems Biology as an Editors-in-Chief which is the world-first international journal in systems biology launched by IEE (changed to IET in 2007) in London from 2004. He is currently an Editorial Board Member of Systems and Synthetic Biology (Springer, Netherlands, from 2006), BMC Systems Biology (BMC, London, UK, from 2007), Gene Regulation and Systems Biology (Libertas Academica, New Zealand, from 2007), and Bulletin of Mathematical Biology (Springer, New York, from 2008), and an Editorial Advisory Board Member of Molecular BioSystems (The Royal Society of Chemistry, UK ). He is also an Associate Editor of BMC Systems Biology and BMC Research Notes (BMC, London, UK, from 2008), and Bulletin of Mathematical Biology (Springer, New York, from 2008). He is a Senior Member of the IEEE Engineering in Medicine and Biology Society (EMBS). He is one of the Editors-in-Chief of ‘Encyclopedia of Systems Biology’ to be published by Springer and a Research Grant Review Committee Member of Human Frontier Science Program (HFSP) in 2008–2009. He was the recipient of the IEEE/IEEK Joint Award for Young IT Engineer of the Year 2008 and other numerous awards from various institutions as well as from Korean government.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Bart De Schutter under the direction of Editor Ian R. Petersen.

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Automatica

Brief paperDelay-coobservability and its algebraic properties for the decentralized supervisory control of discrete event systems with communication delays☆

Abstract

Introduction

Section snippets

Problem formulation

Supervisor existence: Delay-coobservability for a fixed partition of Σc

Verification of delay-coobservability

Algebraic properties of delay-coobservability

Conclusions

Acknowledgements

Decentralized supervisory control of discrete event systems with communication delays based on conjunctive and permissive decision structures

Automatica

Input/output discrete event processes and communication delays

Discrete Event Dynamic Systems: Theory and Applications

On the effect of communication delays in failure diagnosis of decentralized discrete event systems

Discrete Event Dynamic Systems: Theory and Applications

Delay-robust supervisory control of discrete event systems with bounded communication delays

Institute of Electrical and Electronics Engineers Transactions on Automatic Control

A framework for synthesis of communicating decentralised supervisors for discrete-event systems

Reliable decentralized supervisors for discrete-event systems under communication delays: Existence and verification

Brief paper
Delay-coobservability and its algebraic properties for the decentralized supervisory control of discrete event systems with communication delays☆

Supervisor existence: Delay-coobservability for a fixed partition of $Σ_{c}$