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METHODS FOR THE SUPERVISORY CONTROL OF CONCURRENT SYSTEMS BASED ON PETRI NET ABSTRACTIONS A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Marian Valentin Iordache, B.S.E.E., M.S.E.E. Panos J. Antsaklis, Director Graduate Program in Electrical Engineering Notre Dame, Indiana December 2003 METHODS FOR THE SUPERVISORY CONTROL OF CONCURRENT SYSTEMS BASED ON PETRI NET ABSTRACTIONS Abstract by Marian Valentin Iordache This dissertation proposes new methodologies for the supervisory control of con- current systems. The focus of this work is on discrete-event concurrent systems. However, the extension of the discrete-event methods to concurrent systems with continuous dynamics is also approached. Petri nets are a convenient discrete-event representation of concurrent systems, and are used here for the modeling of concur- rent systems. Several discrete-event supervisory control problems are approached here. First, the enforcement of specifications described as linear inequalities in terms of three Petri net parameters is considered. Then, the decentralized supervisory control problem is considered for specifications described by linear marking inequalities. The decentralized supervision problem is approached in three settings: no com- munication, unrestricted communication, and restricted communication. Finally, procedures for deadlock prevention and liveness enforcement are presented. Addi- tionally, new results relating deadlock and liveness to the structure of a Petri net are also presented. The main feature of the supervision methods proposed in this disser- tation is that they rely on the structure of the Petri net. This structural approach has computational benefits and allows the supervisor design to be independent of the initial state of the system. The methods are given in a general supervision Marian Valentin Iordache setting, which makes no assumptions on the structure of the Petri nets and allows partial controllability and partial observability to be present. This dissertation addresses also the supervisory control problem in the more general framework of hybrid systems, that is, systems involving both discrete-event and continuous dynamics. A two-level approach is proposed. The lower level design is concerned with the development of controllers for the continuous part of the hybrid systems. The higher level design is concerned with the design of a supervisor coordinating the operation of the lower level controllers, according to given discrete- event specifications. At the higher level the controlled hybrid systems are abstracted as Petri nets. Petri net methodologies can then be applied to design the appropriate supervisor. The discrete-event setting here is extended to model some of the hard constraints arising in the supervision of hybrid systems. Extensions of discrete-event methods to this setting are also approached. Finally, the controller design at the lower level is considered in a discrete-time setting. The controller design produces both a controller and a Petri net abstraction for the higher level. This dissertation aims to contribute to the automated design of controllers for complex systems. This work is believed to be relevant for applications in a variety of areas, including automated manufacturing, robotics, computer networks, and traffic control. “To God belong wisdom and power; counsel and understanding are his.” Job 12:13 [NIV translation] CONTENTS TABLES......................................viii FIGURES...................................... ix SYMBOLS.....................................xiii ACKNOWLEDGMENTS............................. xv CHAPTER1:INTRODUCTION......................... 1 1.1ContributionandBackground...................... 1 1.2OutlineoftheDissertation........................ 5 CHAPTER2:ANINTRODUCTIONTOPETRINETS............ 9 CHAPTER 3: THE SUPERVISION OF PETRI NETS ............. 14 3.1Introduction................................ 14 3.2Preliminaries............................... 18 3.3NotationandDefinitions......................... 20 3.4RelatedWork............................... 21 3.5 Background: Supervision Based on Place Invariants . ........ 28 3.5.1 FullyControllableandObservablePetriNets.......... 29 3.5.2 Petri Nets with Uncontrollable and Unobservable Transitions . 30 3.6AdmissibleandFeasibleSetsofConstraints.............. 32 3.7EnforcingGeneralizedLinearConstraints................ 35 3.7.1 OntheSignificanceoftheConstraints............. 35 3.7.2 Supervisor Design in the Fully Controllable and Observable Case................................ 39 3.7.3 Admissibility ........................... 41 3.7.4 Supervisor Design in the Partially Controllable and Observ- ableCase............................. 45 3.7.5 Example.............................. 57 3.7.6 AnOptimalStructuralApproach................ 60 iv CHAPTER 4: DECENTRALIZED SUPERVISION OF PETRI NETS .... 74 4.1Introduction................................ 74 4.2RelatedWork............................... 78 4.3Preliminaries............................... 85 4.4TheModel................................. 87 4.5 Decentralized Admissibility ....................... 88 4.5.1 DefinitionandApplication.................... 88 4.5.2 Significance of D-Admissibility .................. 96 4.6DesignwithDistributionofCentralSupervisoryPolicies.......101 4.7DesignwithConstraintTransformations................107 4.7.1 SupervisionwithoutCommunication..............107 4.7.2 SupervisionwithCommunication................110 4.7.3 LivenessConstraints.......................111 4.8Example..................................112 CHAPTER 5: GENERALIZED CONDITIONS FOR DEADLOCK PREVEN- TION AND LIVENESS ENFORCEMENT IN PETRI NETS ........117 5.1Introduction................................117 5.2Preliminaries...............................120 5.3Results...................................123 5.3.1 Conditions for Deadlock Prevention and Liveness Enforcement 123 5.3.2 Deadlock and (T -)Liveness Characterization Based on Active Subnets ..............................133 5.4ImplicationsandDiscussion.......................144 5.4.1 DeadlockPrevention.......................144 5.4.2 LeastRestrictiveDeadlockPrevention.............147 5.4.3 T -livenessEnforcement......................149 5.5Algorithms.................................150 5.5.1 The Computation of Active Subnets ..............150 5.5.2 Transformation of Petri Nets to PT-ordinary Petri Nets . 152 5.5.3 Transformation of Petri Nets to EAC Nets...........153 CHAPTER 6: DEADLOCK PREVENTION AND T-LIVENESS ENFORCE- MENTINPETRINETS–PARTI.......................158 6.1Introduction................................158 6.2RelatedWork...............................162 6.3ProblemStatement............................166 6.3.1 DeadlockPrevention.......................166 6.3.2 T -livenessEnforcement......................167 6.4Motivation.................................168 6.4.1 TheRoleofLinearMarkingInequalities............169 6.4.2 TheRoleofIterations......................170 6.4.3 TheNeedforNetTransformations...............172 6.5ProcedureDefinition...........................173 6.5.1 Definition.............................173 6.5.2 SiphonsNotNeedingControl..................178 v 6.5.3 Generating the Sets of Inequalities (L, b)and(L0,b0) .....179 6.5.4 PetriNetTransformations....................180 6.5.5 The Effect of Net Transformation on Marking Constraints . 183 6.5.6 The Computation of a T -minimal Active Subnet ........185 6.6Examples.................................186 6.7Properties.................................191 6.7.1 Preliminaries...........................191 6.7.2 ProofofCorrectness.......................194 6.7.3 PermissivenessProperties....................197 6.8ExtendingthePermissivenessoftheProcedure.............202 CHAPTER 7: DEADLOCK PREVENTION AND T-LIVENESS ENFORCE- MENTINPETRINETS–PARTII......................205 7.1Introduction................................205 7.2ProblemStatement............................205 7.2.1 DeadlockPrevention.......................206 7.2.2 T -livenessEnforcement......................207 7.3Motivation.................................208 7.3.1 PartiallyControllableandObservablePetriNets........208 7.3.2 ConstraintTransformationsandDeadlock...........209 7.3.3 The Set T 0 ............................211 7.3.4 TheUseofInitialConstraints..................212 7.4ProcedureDefinition...........................213 7.4.1 Description............................213 7.4.2 Definition.............................215 7.4.3 TransformingConstraintstoAdmissibleConstraints......220 7.4.4 The Computation of The Active Subnet ............224 7.5Examples.................................225 7.6Properties.................................230 7.6.1 Preliminaries...........................231 7.6.2 ProofofCorrectness.......................231 7.6.3 Permissiveness...........................232 7.7ExtendingPermissiveness........................238 7.8ConvergenceIssues............................242 7.8.1 TerminationIssues........................242 7.8.2 ComputationalComplexity....................246 7.9Applications................................248 7.9.1 DeadlockPreventioninaManufacturingSystem........248 7.9.2 MinimizationoftheNumberofResources...........251 7.9.3 ResourcePreallocation......................254 CHAPTER 8: DES LEVEL CONTROL OF CONCURRENT HYBRID SYS- TEMS......................................256 8.1Introduction................................256 8.2RelatedWork...............................260

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