Elsevier

Automatica

Volume 142, August 2022, 110431
Automatica

Brief paper
Modular control of discrete-event systems using similarity

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Abstract

We investigate modular supervisory control of discrete-event systems composed of several groups of components, where each group consists of similar modules. Because of the similar structures of the modules, such systems can be represented as a set of (group) templates. Supervisory control can then be performed on these templates, resulting in a set of template supervisors. We propose a modular approach to construct the template supervisors based on the local computation of supremal symmetric sublanguages and on the concept of conditional decomposability. The supremal symmetric sublanguage of a decomposable language turns out to be decomposable, and can thus be computed locally. It is proven that the local supervisors of the components of a group are similar and can thus be obtained by a symmetry map from the template supervisor of the group.

Introduction

In large-scale engineering systems, cf. Agarwal et al., 2019, Ding et al., 2019 and Du et al. (2020), multiple agents are often divided into groups of subsystems according to their roles as in Amini et al. (2020). Within each group, the agents (such as robots or AGVs in Luo et al. (2015) and Wu and Zhou (2007b)) handle similar jobs, and have therefore similar behaviors, cf. also Su and Lennartsson, 2017, Wu and Zhou, 2007a and Wurman et al. (2008). Such multi-agent systems are referred to as discrete-event systems (DES) with similar components, see e.g. Rohloff and Lafortune, 2006, Su and Lennartsson, 2017, Wu and Zhou, 2007a and Wurman et al. (2008), where components in the same group have similar state transition structures.

The control problems of DES with similar components have been discussed in the literature.  Eyzell and Cury (2001) investigated the symmetry of systems to reduce the complexity of supervisory control by constructing a reduced automaton. A quotienting technique has been recently presented in Basu and Kumar (2021) for simplification of non deterministic automata.  Wang et al. (2019) studied blocking and deadlocking for systems with isomorphic modules.  Rohloff and Lafortune (2006) explored the control and verification problems of DES with similar components, and introduced the concept of symmetry. They focused on existential results and identified the necessary and sufficient conditions for the existence of a set of similar local supervisors that enforce a given specification. These conditions include, among others, separability (also known as decomposability) and symmetry.

Since the monolithic synthesis becomes computationally expensive and infeasible for a large number of agents, modular methods have been proposed for systems with a similar structure to avoid the synthesis of a monolithic system.  Jiao et al. (2017) considered systems consisting of groups of machines having an isomorphic structure and extracted the control information with the help of the state tree structures.  Liu et al. (2019) investigated modular control of DES with similar components under some restrictive assumptions, such as the agents do not share events and all events are observable.

In this paper, we use the symmetry map of Rohloff and Lafortune (2006), which is better suited for the case, where agents share some (global) events and for the case of partial observation of events. Our work extends the results of Rohloff and Lafortune (2006) by dropping the assumptions that the specification is symmetric, decomposable, controllable, and normal. If the specification language fails to be symmetric, we show that the supremal symmetric sublanguage always exists, and we show how to compute it in a modular way (Theorem 7). We use a more general concept of conditional decomposability, which relaxes the assumption that the given specification is decomposable according to the alphabets of local agents. Concerning controllability and normality, we make use of supremal controllable and relatively observable sublanguages, and of the fact that supremal controllable and relatively observable sublanguages of a set of similar (local) specifications remain similar. This observation allows us to base our template supervisors on these notions. Moreover, Rohloff and Lafortune (2006) assume that all local agents are similar, that is, they considered only a single group of similar agents. We consider a modular system that consists of several groups of similar agents, that is, the agents from different groups need not be similar. The relaxation of these four assumptions and the similarity of all local plants lead to Theorem 10 where a template supervisor is designed for each group. Local supervisors of the subsystems within a group are obtained from the template supervisor of the group with the help of a symmetry map that maps the events of the template supervisor to the relevant global and private events of each subsystem.

Section snippets

Preliminaries

A generator G=(Z,Σ,δ,z0,Zm) is a quintuple, where Σ is a finite set of events, Z is a finite set of states, z0Z is an initial state, ZmZ is a set of marker states, and δ:Z×ΣZ is a (partial) transition function. As usual, δ can be extended to δ:Z×ΣZ, where Σ is the set of all finite-length strings, including the empty string ɛ. The closed behavior of G is the language L(G)={sΣδ(z0,s)Z}, and the marked behavior of G is the set Lm(G)={sL(G)δ(z0,s)Zm}. The length of a string s is

Modular control of DES with similar components

We investigate modular supervisory control for DES with 1 groups of similar agents, {G1,,G}, i.e. for i=1,,, Gi={Gi1,,Gini}, where Gij,j=1,,ni, are similar as defined below. For all i, Gij are over Σij of the same cardinality that are decomposed as Σij=ΣpijΣgi, where Σgi is the global (shared) alphabet of the group Gi, and Σpij is the private alphabet of Gij. Since Σpij are of the same cardinality for the group Gi, we denote the “rewriting” bijection by ψjji:ΣpijΣpij. The set of all

Illustrative example

Consider a small factory system, adapted from Wonham and Cai (2019). As displayed in Fig. 2 small factory consists of three input machines G1, G2, G3 and two output machines G4 and G5 linked by a buffer with capacities three in the middle. The generators are shown in Fig. 2. Let Σ=ΣċΣuc={l1,l3,3}̇{l2} for l[1,5], Σ=ΣȯΣuo={l1,l3,3}̇{l2}, and Σg={3}. The alphabet of agents in the input group is Σ1=j=13Σ1j=j=13{j1,j2,j3,3} and the alphabets of agents in the output group are Σ2=j=45Σ2j=j=

Conclusion

An efficient modular approach to compute a set of template supervisors for plants composed of systems similar by group is proposed based on local computations of supremal symmetric sublanguages and on decomposing specifications. It is shown that local supervisors for the components of a group are similar, and can be computed by a symmetry map from a template supervisor. In a future we plan to extend our approach to real-time systems.

Yingying Liu was born in Xi’an, China, in 1989. She received the B.Eng. degree from the Taiyuan University of Technology, China, in 2013 and the Ph.D. degree from the Xidian University, Xi’an, in 2020, all in electrical engineering. Since 2020, she is currently a Post-doctoral with the Department of Automation, Shanghai Jiao Tong University.

Her research interests include supervisory control of discrete event systems, distributed control of multi-agent systems, and system reconfiguration, game

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Yingying Liu was born in Xi’an, China, in 1989. She received the B.Eng. degree from the Taiyuan University of Technology, China, in 2013 and the Ph.D. degree from the Xidian University, Xi’an, in 2020, all in electrical engineering. Since 2020, she is currently a Post-doctoral with the Department of Automation, Shanghai Jiao Tong University.

Her research interests include supervisory control of discrete event systems, distributed control of multi-agent systems, and system reconfiguration, game theory, model-based fault diagnosis, security of cyber and cyber–physical systems. Dr. Liu received the Workshop on Discrete-Event Systems Best Student Paper Award in 2018.

Jan Komenda received a M.Sc. degree in mathematics from the Masaryk University (Brno), Czech Republic, in 1994, and was awarded a Ph.D. degree in control and computer science from Université de Franche-Comté (Besançon), France in 1999. He worked at CWI Amsterdam (The Netherlands) in 2001–2003 in the group of J.H. van Schuppen. He is researcher at the Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic. His current research interests are in supervisory control of logical and timed discrete event systems using methods of coalgebra and idempotent algebra. He serves as Associate Editor of IEEE Trans. on Automatic Control and as Department Editor of Discrete Event Dynamic Systems and has served as Associate Editor of Automatica in the period 2009–2015.

Tomáš Masopust is an associate professor at the Faculty of Science, Palacky University in Olomouc, Czechia. He received M.Sc. in computer science from the Masaryk University, Brno, Czechia in 2004, Ph.D. in computer science from the Brno University of Technology, Brno, Czechia in 2007, and was appointed a Research Professor in Informatics and Cybernetics by the Czech Academy of Sciences, Prague, Czechia in 2019. He worked at CWI Amsterdam, the Netherlands, in the Systems and Control Group, at the University of Bayreuth, Germany, in the Theoretical Computer Science Group, at TU Dresden, Germany, in the Knowledge-Based Systems Group, and at the Institute of Mathematics of the Czech Academy of Sciences, Prague, Czechia, where he holds a part-time appointment. His research interest includes verification and control of discrete-event systems and theoretical computer science. He is an associate editor of journals Kybernetika and JDEDS.

Zhiwu Li received the B.S., M.S., and Ph.D. degrees in mechanical engineering, automatic control, and manufacturing engineering, respectively, all from Xidian University, Xi’an, China, in 1989, 1992, and 1995, respectively. He joined Xidian University in 1992 and now he is also with the Institute of Systems Engi- neering, Macau University of Science and Technology, Taipa, Macau. Over the past decade, he was a Visiting Professor at the University of Toronto, Technion (Israel Institute of Technology), Martin-Luther University of Halle-Wittenburg, Conservatoire National des Arts et Metiers (CNAM), Meliksah Universitesi, and King Saud University.

Dr. Li’s research interests include Petri net theory and application, supervisory control of discrete event systems, workflow modeling and analysis, system reconfiguration, game theory, and data and process mining. He is a member of Discrete Event Systems Technical Committee of the IEEE Systems, Man, and Cybernetics Society, and a member of IFAC Technical Committee on Discrete Event and Hybrid Systems from 2011 to 2014. He serves as a frequent reviewer for 90+ international journals including Automatica and a number of the IEEE Transactions as well as many international conferences. He is listed in Marquis Who’s Who in the world, 27th Edition, 2010. Dr. Li is a recipient of an Alexander von Humboldt Research Grant, Alexander von Humboldt Foundation, Germany, and Research in Paris, France. He is the founding chair of Xi’an Chapter of IEEE Systems, Man, and Cybernetics Society.

Partially supported by the MŠMT INTER-EXCELLENCE project, Czechia LTAUSA19098, the GAČR, Czechia grant 19-06175J, the National Key Research and Development Program of China (2018AAA0101700) and the National Natural Science Foundation of China (62061136004, 61803259, 61833012), Shanghai Jiao Tong University, China Scientific and Technological Innovation Funds and by RVO 67985840. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Christoforos Hadjicostis under the direction of Editor Christos G. Cassandras.

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