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SUPERVISORY CONTROL THEORY

MODELS AND METHODS

W.M. WonhamSystems Control Group

ECE DepartmentUniversity of Toronto

wonham@control.utoronto.ca

Workshop on Discrete-Event Systems Control

Eindhoven 2003.06.24

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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

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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!

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MODEL FLEXIBILITY

For instance

Automata versus Petri nets

batrakhomuomakhia

or

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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)

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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

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MODEL WITH THE BEST OF BOTH WORLDS ?

Q1 Q2 · · · Qm k l

(Algebraically) hybrid state set

Qi for (an unstructured) automaton component

for a naturally additive component (buffer...)

for a naturally boolean component (switch...)

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WHAT ABOUT LARGE SYSTEMS?

For architecture, need algebraic “laws” for basic objects and operators

_____ DES G nonblocking 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)

E.g. languages, prefix-closure, synchronous product

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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

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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

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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

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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