High-Level Abstraction of Concurrent Finite Automata for the Purpose of Hierarchical Control Klaus Schmidt SVC - Seminar September 16th, 2003

Overview Preliminaries Concurrent Systems Structural Decentralized Finite Automata - Properties - Advantages 4. High – Level Abstraction - Control Structure - Hierarchical Consistency - Example 5. Complexity Issues 6. Summary and Future Work

1. Preliminaries

1. Preliminaries: Notations in the DES Framework Automaton G automata and regular languages 1 2 3 4 deterministic finite automata G states events transition function marked states initial state 4 regular languages Alphabet strings closed language marked language

1. Preliminaries: Standard Control Structure Automaton G classification of events 1 2 3 4 controllable events – can be disabled: uncontrollable events – always enabled: feedback structure Plant G Supervisor S disabled events occuring in G 4 tracks strings occuring in G: disables event after if supervisor

1. Preliminaries: Supervisor Design automaton G controlled language 1 2 3 4 specification desired language: 0,0 0,1 0,2 0,3 4 generator for Definition: Controllability A language is controllable w.r.t. to a language If and only if 4 supremal controllable sublanguage

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 exists iff: (i) controllable w.r.t. (ii) is -closed, i.e.

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

2. Concurrent Systems

2. Concurrent Systems: Decentralized Architecture subsystems: Finite Automata Alphabet: closed language: marked language: common events: overall system: Finite Automaton synchronous composition: closed language: marked language: alphabet: common behavior subsystem

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 always check if condition is fulfilled are nonblocking local languages

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? Sword of damokles

3. Structural Decentralized Finite Automata

3. Structural Decentralized Finite Automata: Conditions (i) shared event marking condition: marks  predecessor state of each common event is marked (ii) mutual controllability condition: local language is controllable w.r.t. external behavior of other languages as seen by

3. Structural Decentralized Finite Automata: Example two cooperating machine cells (Lee & Wong) 1 1 3 3 2 2 4 4 5 5 Controllability 1 1

3. Structural Decentralized Finite Automata: Theorem 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)

4. High-Level Abstraction

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

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 low-level supervisor consists of low-level + translated high-level control

4. High-Level Abstraction: Results problems to be solved: (1) complexity of projection is exponential in size of automaton 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: States: left side: k^n states right side: k states * n apply projection only to sublanguages compose abstracted sublanguages

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

4. High-Level Abstraction: Control Loop Low Level Supervisor Plant

4. High-Level Abstraction: Example 1 1 3 3 2 2 4 4 5 5 1 2 5a 4 3 5b low-level spezifications

4. High-Level Abstraction: Example generator for Generator for 1,1 0,0 generator for 1 1 high-level specification 1 2 controllable in high level translate to low level

4. High-Level Abstraction: Example high-level specification 1 2 low-level realization 1 1 3 3 2 2 5b 4 5b 5a 4 5a

5. Complexity Issues

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:  polynomial in number of states and transitions marks  polynomial Low-Level Control:  n times  equvalent high level control  nonconflicting condition need not be verified  polynomial

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 ?

6. Summary and Future Work

6. Summary and Future Work 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