State Based Control of Timed Discrete Event Systems using Binary ...

Chapter 4. Synthesis Algorithm Based on Predicates 31Thus it can also be expressed asV ′ : Q × Σ → {0, 1}where⎧⎪⎨ 1, if σ ∈ V (s)V ′ (q, σ) =⎪⎩ 0, Otherwiseif q = δ(q 0 , s) and σ ∈ Σ a ∪ {tick}.We can express V ′ in the form of n + 1 scalar functions:V ′ ≡ {V ′i : Q → {0, 1}, i ∈ {0, 1, ..., n} }where n is the number of events except tick, V ′i is the predicate corresponds to event σ i ,σ 0 = tick and σ i ∈ Σ a for i = 1, ..., n.Thus V ′ is nothing but n + 1 predicates. Therefore supervisory control can be implementedby predicates in a straightforward way.The V ′ defined above can be obtained as follows:LetQ := Q p × Q sq = (q p , q s )∆(q, σ) = (δ p (q p , σ), δ s (q s , σ))We define⎧⎪⎨V i ′ (q) =⎪⎩1, if q P sup & ∆(q, σ i )! & ∆(q, σ i ) P sup0, otherwiseThus the state q satisfies the predicate for event σ i if q satisfies P sup and we can reachfrom q to another state that satisfies P sup by the event σ i .Obviously, for any σ i ∈ Σ u ,V ′i = 1, because P sup is controllable. If q F, then V ′0(q) = 1because P sup is controllable. Also, clearly L(V ′ /G) = L(G, P sup ) and L m (V ′ /G) =L m (G, P sup ), since exactly the same constraints are imposed upon G by V ′ and by P sup .