SUPERVISORY CONTROL THEORY MODELS AND METHODS W M

SUPERVISORY CONTROL THEORY MODELS AND METHODS W. M. Wonham Systems Control Group ECE Department University of Toronto wonham@control. utoronto. ca Workshop on Discrete-Event Systems Control Eindhoven 2003. 06. 24 1

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 2

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! 3

MODEL FLEXIBILITY For instance Automata versus Petri nets or batrakhomuomakhia 4

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COMPUTATION OF SIMSUP 1. FMS = Sync (M 1, M 2, 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) (computes control congruence on SUPER) (4, 16) 7

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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 10

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MODEL WITH THE BEST OF BOTH WORLDS ? (Algebraically) hybrid state set Q 1 Q 2 · · · Q m k l Qi for (an unstructured) automaton component Í for a naturally additive component (buffer. . . ) 15 for a naturally boolean component (switch. . . )

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WHAT ABOUT LARGE SYSTEMS? For architecture, need algebraic “laws” for basic objects and operators E. g. languages, prefix-closure, synchronous product _____ DES G nonblocking if Lm(G) = L(G). Suppose G = G 1 G 2. ____________ Lm(G) Lm(G 1) Lm(G 2) (computationally intensive!) 17 _____

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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 19

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AIP CONTROL SPECIFICATIONS • Normal production sequencing Type 1 workpiece: I/O AS 1 AS 2 I/O Type 2 workpiece: I/O AS 2 AS 1 I/O • AS 3 backup operation if AS 1 or AS 2 down • Conveyor capacity bounds, . . . • Nonblocking 26

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, 1 GHz, 256 MB RAM • Computation time ~ 45 min 27

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CONCLUSIONS • Base model flexibility, architectural variations 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 30