Elsevier

Automatica

Volume 47, Issue 5, May 2011, Pages 1028-1034
Automatica

Brief paper
Design of a maximally permissive liveness-enforcing supervisor with a compressed supervisory structure for flexible manufacturing systems

https://doi.org/10.1016/j.automatica.2011.01.070Get rights and content

Abstract

In this paper, a deadlock prevention policy for flexible manufacturing systems (FMS) is proposed, which can obtain a maximally permissive liveness-enforcing Petri net supervisor while the number of control places is compressed. By using a vector covering approach, the sets of legal markings and first-met bad markings (FBM) are reduced to two small ones, i.e., the minimal covering set of legal markings and the minimal covered set of FBM. A maximally permissive control purpose can be achieved by designing control places such that all markings in the minimal covered set of FBM are forbidden and no marking in the minimal covering set of legal markings is forbidden. An integer linear programming problem is designed to minimize the number of control places under an assumption that a control place is associated with a P-semiflow. The resulting net has the minimal number of control places on the premise that the assumption holds, and possesses all permissive states of a plant. The only problem of the proposed method is its computational complexity that makes it inapplicable to large-scale Petri net models. An FMS example from the literature is presented to illustrate the proposed method.

Introduction

Flexible manufacturing systems (FMS) Wikipedia (2011) are usually used to automatically finish different kinds of jobs by using a limited number of shared resources such as machines, robots, automated guided vehicles, and buffers. Deadlocks always occur in FMS when some processes keep waiting indefinitely for the other processes to release resources, which often offset the advantages of these systems.

Petri nets Murata (1989) are widely used to model and control FMS since they are suitable to detect deadlocks of a system and develop a policy to deal with them Ezpeleta, Colom, and Martinez (1995), Ghaffari, Rezg, and Xie (2003), Huang, Jeng, Xie, and Chung (2001), Li and Zhou (2009) and Reveliotis (2007). Deadlock prevention is one of the most important approaches whose goal is to impose constrains on a system to prevent it from reaching deadlock states Ezpeleta et al. (1995), Fanti and Zhou (2004), Fanti and Zhou (2005), Jeng and Xie (2005), Piroddi, Cordone, and Fumagalli (2008) and Piroddi, Cordone, and Fumagalli (2009). In this case, the computation is carried out offline and once the control policy is established and applied, deadlocks can no longer occur.

There are three very important criteria in evaluating and designing a liveness-enforcing supervisor for a system to be controlled: behavioral permissiveness, structural complexity and computational complexity. A maximally permissive supervisor always leads to high utilization of system resources. A supervisor with the minimal number of control places can decrease the hardware and software costs in the stage of control validation and implementation. A deadlock control policy with low computational complexity means that it can be applied to complex systems.

Our previous work Chen, Li, Khalgui, and Mosbahi (in press) develops a new method that can definitely derive a maximally permissive liveness-enforcing supervisor for Petri net models of FMS if such a supervisor exists. In Chen et al. (in press), a maximally permissive control place is designed by a place invariant (PI) that forbids one of the first-met bad markings (FBM) and none of legal markings is forbidden. The PI is computed by solving an integer linear programming problem. In order to overcome the complexity of this method, a vector covering approach is developed to reduce the sets of legal markings and FBM to be small, i.e., the minimal covering set of legal markings and the minimal covered set of FBM. Then, only the two reduced sets are considered in the design of a supervisor. In this case, the computational burden is greatly reduced. However, the previous work does not ensure that the supervisor has a compact structure and hence suffers from the structural complexity problem.

In this paper, a non-iterative approach is proposed, aiming to overcome the structural complexity problem and ensures that the controlled system is still maximally permissive. First, a vector covering approach is used to reduce the sets of legal marking and FBM to be small. Then, an integer linear programming problem is designed to satisfy the following three conditions:

  • (1)

    Each FBM in the minimal covered set of FBM is forbidden by at least one control place. Therefore, all the FBM are forbidden, implying that the controlled system is live.

  • (2)

    No markings in the minimal covering set of legal markings are forbidden. Therefore, no legal markings are excluded, implying that the final supervisor is maximally permissive.

  • (3)

    The objective function minimizes the number of control places to be added under an assumption that a control place is associated with a P-semiflow.

If the integer linear programming problem has a solution, we can obtain a maximally permissive supervisor with a minimal structure within the class of supervisors where each control place is associated with a P-semiflow. Then, the controlled system can be implemented with less hardware and software costs, and the resources of the system are highly utilized as well. This method is motivated by the previous work Chen et al. (in press). Thus, it has the same application scope with the method in Chen et al. (in press). That is to say, it can be used to all classes of Petri net models of FMS in the literature such as PPN Hsieh and Chang (1994) and Xing, Hu, and Chen (1995), S3PR Ezpeleta et al. (1995), ES3PR Tricas, Garcia-Valles, Colom, and Ezpelata (1998), S4PR Tricas, Garcia-Valles, Colom, and Ezpelata (2000), S*PR Ezpeleta, Tricas, Garcia-Valles, and Colom (2002), S2LSPR Park and Reveliotis (2000), S3PGR2 Park and Reveliotis (2001), and S3PMR Huang, Jeng, Xie, and Chung (2006). A drawback of the proposed method is its computational complexity that makes it inapplicable to large-scale Petri net models. Also, it cannot be applied to Petri net instances from the above models that have no maximally permissive liveness-enforcing supervisor expressed by monitors.

The rest of the paper is organized as follows. Section 2 reviews some basics of Petri nets used throughout this paper. The computation of a control place by a PI in Yamalidou, Moody, Lemmon, and Antsaklis (1996) and the method to obtain a maximally permissive control place in Chen et al. (in press) are briefly recalled in Section 3. Section 4 develops an approach to design a maximally permissive supervisor with the minimal number of control places where each such control place is associated with a P-semiflow. A deadlock prevention policy is formulated in Section 5. Section 6 provides some examples and compares the approach with other methods existing in the literature. Finally, some conclusions are provided in Section 7.

Section snippets

Petri Nets

We assume that the readers are familiar with the basics of Petri nets. Only some key concepts of Petri nets are provided. More details can be found in Murata (1989).

A Petri net is a four-tuple N=(P,T,F,W) where P and T are finite and non-empty sets. P is a set of places and T is a set of transitions with PT and PT=. F(P×T)(T×P) is called a flow relation of the net, represented by arcs with arrows from places to transitions or from transitions to places. W:(P×T)(T×P)N is a mapping

Control place computation by a place invariant

This section only reviews the main results of the control place computation method proposed in Yamalidou et al. (1996).

Let [Np] be the incidence matrix of a plant net with n places and m transitions. The control places can be represented by a matrix [Nc] that shows the incidence relationship between the control places and transitions of the plant. The controlled net with incidence matrix [N] consists of both the original net and control places, i.e., [N]=[NpNc] Suppose that the control goal is

Minimal number of control places synthesis

This section presents an approach to minimize the number of the considered control places. In the following, NFBM denotes {i|MiMFBM}.

A PI may forbid more than one FBM. Given PIj for an FBM MjMFBM, any FBM MkMFBM(kj) is forbidden if it satisfies the following constraint: iNAlj,iMk(pi)iNAlj,iMj(pi) where lj,i(iNA) are the coefficients of PIj. By simplifying Constraint (12), we have iNAlj,i(Mk(pi)Mj(pi))0. For PIj, we introduce a set of variables fj,k (kNFBM and kj) to

Deadlock prevention policy

This section presents a deadlock prevention policy to obtain a maximally permissive liveness-enforcing supervisor with a minimal number of the considered control places if such a supervisor exists.

Algorithm 1

Deadlock Prevention Policy.

This algorithm can obtain a maximally permissive liveness-enforcing supervisor with the minimal number of control places on condition that each control place is associated with a P-semiflow. We first compute the reachability graph of a Petri net model, and then the sets of

Experimental results

In this section, an FMS example available in the literature is considered to show the experimental results of the proposed approach. It can also be applied to other examples that can be found in Chen and Li (2009). As stated previously, only operation places are considered for markings in ML and MFBM. For the sake of an expedient description, a compact multiset formalism iNAM(pi)pi is used to represent marking M in ML and MFBM.

The Petri net model of an FMS given in Uzam (2002) is shown in

Conclusions

This paper presents a deadlock prevention policy to obtain a maximally permissive supervisor with a compressed number of control places if such a supervisor exists. The performance analysis of a deadlock prevention policy is always carried out by considering the following three criteria: behavioral permissiveness, structural complexity, and computational complexity. For the proposed method, in terms of behavioral permissiveness, the derived supervisor is maximally permissive and in terms of

YuFeng Chen received the B.S. degree from Xidian University, Xi’an, China, in 2006. He is currently a Ph.D. student at the School of Electro-Mechanical Engineering, Xidian University. His research interests include Petri net theory and applications, and supervisory control of discrete event systems.

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    YuFeng Chen received the B.S. degree from Xidian University, Xi’an, China, in 2006. He is currently a Ph.D. student at the School of Electro-Mechanical Engineering, Xidian University. His research interests include Petri net theory and applications, and supervisory control of discrete event systems.

    ZhiWu Li (M’06, S’07) received the B.S., M.S., and Ph.D. degrees in mechanical engineering, automatic control, and manufacturing engineering, respectively, from Xidian University, Xi’an, China, in 1989, 1992, and 1995, respectively. He joined Xidian University, in 1992, where he is currently a Professor of School of Electro-Mechanical Engineering and the director of Systems Control and Automation Group. From June 2002 to July 2003, he was a visiting professor at the Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada. From February 2007 to February 2008, he was a visiting scientist at the Laboratory for Computer-Aided Design (CAD) & Lifecycle Engineering, Department of Mechanical Engineering, Technion–Israel Institute of Technology, Technion City, Haifa, Israel. Since November 2008, he has been a visiting professor in Automation Technology Laboratory, Institute of Computer Science, Martin-Luther University of Halle-Wittenburg, Halle (Saale), Germany. He serves as a host professor of Research Fellowship for International Young Scientists, National Natural Science of Foundation of China.

    He is the author or coauthor of over 120 publications including a book chapter in Deadlock Resolution in Computer-Integrated Systems (Marcel Dekker, 2005). He is a coauthor with MengChu Zhou, Deadlock Resolution in Automated Manufacturing Systems: A Novel Petri Net Approach, (Springer, 2009) and Modeling, Analysis and Deadlock Control in Automated Manufacturing Systems (Beijing, 2009, in Chinese). His current research interests include Petri net theory and application, supervisory control of discrete event systems, workflow modeling and analysis, and systems integration. He is the General Co-Chair of the IEEE International Conference on Automation Science and Engineering, August 23–26, Washington, DC, 2008. He is a financial Co-Chair of the IEEE International Conference on Networking, Sensing, and Control, March 26–29, 2009, a member of International Advisory Committee, 10th International Conference on Automation Technology, June 27–29, 2009, a Co-Chair of the program committee of the IEEE International Conference on Mechatronics and Automation, August 24–27, 2010, members of the program committees of many international conferences. He serves an associate editor of the IEEE Transactions on Automation Science and Engineering, International Journal of Embedded Control Systems, IST Transactions of Robotics, Automation & Mechatronics — Theory & Applications, and IST Transactions of Control Engineering-Theory and Applications. He is a guest editor of Special Issue on “Petri Nets for System Control Automation” in Asian Journal of Control, Special Issue on “Petri Nets and Agile Manufacturing” in Transactions of the Institute of Measurement and Control, and Special Issue on Modeling and Verification of Discrete Event Systems in ACM Transactions on Embedded Computing Systems.

    He is a member of Discrete Event Systems Technical Committee of the IEEE Systems, Man, and Cybernetics Society. He serves as a frequent reviewer for more than 20 international journals including 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 Alexander von Humboldt Research Grant, Alexander von Humboldt Foundation, Germany. He is a senior member of IEEE and is the founding chair of Xi’an Chapter of IEEE Systems, Man, and Cybernetics Society.

    This work was supported in part by the Natural Science Foundation of China under Grants 60773001 and 61074035, the Fundamental Research Funds for the Central Universities under Grant No. JY10000904001, the National Research Foundation for the Doctoral Program of Higher Education, the Ministry of Education, P.R. China, under Grant No. 20090203110009, “863” High-tech Research and Development Program of China under Grant No 2008AA04Z109, and Alexander von Humboldt Foundation. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian R. Petersen.

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