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SUPERVISORY CONTROL THEORY

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  1. SUPERVISORY CONTROL THEORY W.M. Wonham Systems Control Group ECE Department University of Toronto wonham@control.utoronto.ca MODELS AND METHODS Workshop on Discrete-Event Systems Control Eindhoven 2003.06.24

  2. WHAT’S BEEN ACCOMPLISHED? • Formal control theory • Basis – simple ideas about control and observation • Some esthetic appeal • Amenable to computation • Admits architectural composition • Handles real industrial applications

  3. WHAT MORE SHOULD BE ACCOMPLISHED? • Flexibility of model type • Flexibility of model architecture • Transparency of model structure (how to view and understand a complex DES?) • ... Accepting that most of the interesting problems are exponentially hard!

  4. MODEL FLEXIBILITY For instance Automata versus Petri nets or batrakhomuomakhia

  5. COMPUTATION OF SIMSUP 1.FMS = Sync (M1,M2,R) (20,34) 2. SPEC = Allevents (FMS) (1,8) 3. SUPER(.DES) = Supcon (FMS,SPEC) (15,24) 4. SUPER(.DAT) = Condat (FMS,SUPER) 5. SIMSUP = Supreduce (FMS,SUPER,SUPER) (computes control congruence on SUPER) (4,16)

  6. COMPUTATION OF MONITORS Based on “theory of regions” 1. Work out reachability graph of PN (20 reachable markings, 15 coreachable) 2. Find the 6 “dangerous markings” 3. Solve the 6 “event/state separation” problems (each a system of 15 linear integer inequalities) 4. Implement the 3 distinct solutions as monitors

  7. MODEL WITH THE BEST OF BOTH WORLDS ? (Algebraically) hybrid state set Q1 Q2 ··· Qm  k  l • Qi for (an unstructured) automaton component • for a naturallyadditive component (buffer...)  for a naturally boolean component (switch...)

  8. WHAT ABOUT LARGE SYSTEMS? For architecture, need algebraic “laws” for basic objects and operators E.g. languages, prefix-closure, synchronous product _____ DES Gnonblocking if Lm(G) = L(G). Suppose G = G1 G2. _____ ____________ Lm(G) Lm(G1)  Lm(G2) (computationally intensive!) _____ _____ =? Lm(G1)  Lm(G2) = L(G1)  L(G2) =L(G)

  9. TOP-DOWN MODELLING BY STATE TREES • Adaptation of state charts to supervisory • control • • Transparent hierarchical representation • of complex systems • • Amenable to efficient control computation • via BDDs

  10. AIP CONTROL SPECIFICATIONS • Normal production sequencing Type1 workpiece: I/O  AS1 AS2  I/O Type2 workpiece: I/O  AS2  AS1  I/O • AS3 backup operation if AS1 or AS2 down • Conveyor capacity bounds, ... • Nonblocking

  11. AIP COMPUTATION • Equivalent “flat” model ~ 1024 states, intractable by extensional methods • BDD controller ~ 7  104 nodes • Intermediate node count < 21  104 • PC with Athlon cpu, 1GHz, 256 MB RAM • Computation time ~ 45 min

  12. CONCLUSIONS • Base model flexibility, architecturalvariations among topics of current importance •Symbolic computation to play major role •Other topics: p.o. concurrency models, causality, lattice-theoretic ideas, ... •There is steady progress •There is lots to do