HighLevel Abstraction of Concurrent Finite Automata for the

High-Level Abstraction of Concurrent Finite Automata for the Purpose of Hierarchical Control Klaus Schmidt SVC - Seminar September 16 th, 2003 Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

Overview 1. Preliminaries 2. Concurrent Systems 3. Structural Decentralized Finite Automata 4. - Properties 5. - Advantages 4. High – Level Abstraction - Control Structure - Hierarchical Consistency - Example 5. Complexity Issues 6. Summary and Future Work Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

1. Preliminaries Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4 1. Preliminaries: Notations in the DES Framework automata and regular languages • deterministic finite automata G Automaton G 0 - states 1 - events - transition function - marked states 2 3 - initial state 4 • regular languages - Alphabet - strings - closed language - marked language Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

1. Preliminaries: Standard Control Structure Automaton G classification of events • controllable events 0 – can be disabled: • uncontrollable events – always enabled: 1 feedback structure Supervisor S disabled events Plant G 2 events occuring in G 3 4 supervisor • tracks strings occuring in G: • disables event after if Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

1. Preliminaries: Supervisor Design automaton G controlled language 0 1 specification 2 desired language: 3 Definition: Controllability A language 4 is controllable w. r. t. to a language If and only if generator for 0, 1 0, 3 0, 0 0, 2 4 supremal controllable sublanguage Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

1. Preliminaries: Supervisor Design nonblocking - condition there must exist a path to a marked state controlled language: Controllability and Non-blocking Theorem A non-blocking supervisor, which implements the Specification for an automaton (i) controllable w. r. t. (ii) is exists iff: -closed, i. e. Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

1. Preliminaries: Supervisor Design Advantages • systematic procedure for determining supervisors for finite automata • synthesize maximally permissive behavior: supremal controllable sublanguage • algorithms which are polynomial in time Problem: • state explosion for monolithic design • formulation of specifications for large systems modular design decentralized architectures hierarchical architectures Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

2. Concurrent Systems Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

2. Concurrent Systems: Decentralized Architecture subsystems: Finite Automata - Alphabet: - closed language: - marked language: - common events: overall system: Finite Automaton - synchronous composition: - closed language: common behavior subsystem - marked language: - alphabet: Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

2. Concurrent Systems: Blocking Issues Idea: local control • specifications for decentralized subsystems • supremal controllable sublanguage local supervisors • concurrent operation of • interaction may lead to blocking in the overall behavior nonconflicting-condition • concurrent operation is nonblocking and controllable iff are nonblocking local languages always check if condition is fulfilled Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

2. Concurrent Systems: Blocking Issues • if concurrent operation is conflicting try to adapt specifications construct overall specification and compute overall supremal controllable language • in case of conflict • in general for which systems equality holds? Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

3. Structural Decentralized Finite Automata Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

3. Structural Decentralized Finite Automata: Conditions 14 (i) shared event marking condition: marks predecessor state of each common event is marked (ii) mutual controllability condition: local languages is controllable w. r. t. external behavior of other as seen by Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

3. Structural Decentralized Finite Automata: Example two cooperating machine cells (Lee & Wong) 1 1 3 0 2 2 4 4 5 5 Controllability 0 1 Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

3. Structural Decentralized Finite Automata: Theorem (Lee and Wong 2002): Let the decentralized architecture be given as defined before. Suppose that for and (i) marks (ii) and are mutually controllable Then for any the following holds: and is nonblocking w. r. t. verify local specifications generate local supervisors concurrent operation of local supervisors is nonblocking concurrent operation of local supervisors is optimal (equals overall solution) Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Control Structure local system • Concurrent nonblocking Finite Automata • Structural Decentralized Architecture - shared event marking - mutual controllability local control • local specifications ( -closed) • local supervisor s. t. • equivalent overall supervisor • local closed-loop language local controls are non-conflicting (mutual controllability) closed loop language is nonblocking Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Control Structure global system • high-level events (common events) • natural projection • high-level language • high-level marked language • high-level supervisor • translation map with iff (i) (ii) concept of high-level abstraction: projection on common event set coordination of system interaction via high-level control translation of virtual high-level control action to low-level supervisor consists of low-level + translated high-level control Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Results problems to be solved: (1) complexity of projection is exponential in size of automaton (2) hierarchical consistency: translation of high-level control policy to low level (3) interaction of high- and low-level part of low-level supervisor (1): Theorem (Construction of High-Level Language) Let the proposed hierarchical control architecture be given. Then the high-level language can be constructed as follows: apply projection only to sublanguages compose abstracted sublanguages Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Results (2) and (3): Theorem (Hierarchical Consistency) Assume the proposed hierarchical control architecture, then (i) is hierarchical consistent w. r. t. for all high level specifications it follows that (ii) is nonblocking (iii) is maximally permissive high-level control action can be realized in low level the closed-loop language is nonblocking the translated low-level supervisor does not disable behavior if it is not necessary Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Control Loop High Level Low Level Supervisor Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Plant Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Example 1 1 3 0 2 2 4 4 5 5 low-level spezifications 1 1 3 0 2 5 b 4 5 a Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata 5 b 5 a 4 Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Example generator for 0 generator for Generator for 0 0, 0 1 1 1, 1 high-level specification 0 1 2 controllable in high level translate to low level Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

4. High-Level Abstraction: Example high-level specification 0 1 2 low-level realization 1 1 3 0 2 5 b 4 5 a Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata 5 b 5 a 4 Institute of Automation and Control University of Erlangen-Nuremberg

5. Complexity Issues Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

5. Complexity Issues: Structural Decentralized Architecture Mutual Controllability: • projection: exponential in number of states • controllability polynomial in number of states worst case: exponential complexity Shared-event marking: marks polynomial in number of states and transitions polynomial Low-Level Control: n times equvalent high level control nonconflicting condition need not be verified polynomial Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

5. Complexity Issues: High-Level Architecture High-Level Automaton: Theorem 1 projection of local automata: exponential synchronous composition of projected automata: global marked states: polynomial exponential in number of subautomata and states of subautomata High-Level Control: Theorem 2 supremal controllable sublanguage for high-level automaton: polynomial Translation High-Level to Low-Level Supervisor: Theorem 2 proposed architecture: more general architectures: ? polynomial ? Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

6. Summary and Future Work Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg

6. Summary and Future Work Summary • Structural Decentralized Architecture • Hierarchical Abstraction • Low-Level Supervision • High-Level Supervision • Hierarchical Consistency • Complexity Future Work • elaborate conditions for mutual controllability • formulate more general architecture • multi-level hierarchy Klaus Schmidt High-Level Abstraction of Concurrent Finite Automata Institute of Automation and Control University of Erlangen-Nuremberg