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

Chapter 4. Synthesis Algorithm Based on Predicates 273. [Q p × Q s ] Pbad2:= {(q p , q s ) | (∃w ∈ Σ ∗ u)( (δ p (q p , w), δ s (q s , w)) P bad1 )or((q p , q s ) F) ∧ (∃σ ∈ Σ u ∪ {tick})((δ p (q p , σ), δ s (q s , σ)) P bad1 )}4. [Q p × Q s ] Pre:= {(q p , q s ) | (∃w ∈ Σ ∗ )(δ p (q p,0 , w) = q p , δ s (q s,0 , w) = q s )∧(∀v ≤ w)(δ p (q p,0 , v), δ s (q s,0 , v)) P bad1 ∨ P bad2 }5. [Q p × Q s ] Pcr:= {(q p , q s ) | (∃w ∈ Σ ∗ )(δ p (q p , w) ∈ Q p,m , δ s (q s , w) ∈ Q s,m )∧(∀v ≤ w)(δ p (q p , v), δ s (q s , v)) P bad1 ∨ P bad2 }6. P good2 = P re ∧ P cr7. If P good2 ⊂ P good1 , repeat steps 2-7 with P good1 := P good2 . Otherwise let P sup =P newgood and the algorithm terminates here.• The algorithm will terminate in finite steps. Define |P | to be the number of stateswhere a predicate P holds.Obviously in each cycle, if the algorithm does notterminate, |P bad | must increase by at least 1 at step 7. Therefore, the algorithmcan iterate at most |Q p | × |Q s | times, which is finite.• L m (G meet , P sup ) ⊆ supC(L m (G p ), L m (G s ))- P sup is nonblocking, because we know that P good2 = P re ∩P cr ,i.e. the states whichsatisfy P good2 are both reachable and coreachable, so P sup is nonblocking.- P sup is controllable.Suppose(∃(q p, ′ q s) ′ P sup )(∃σ ∈ Σ u )(δ p (q p, ′ σ)! ∧ ¬δ s (q s, ′ σ)!)then according to step 2, (q p, ′ q s) ′ P bad1 which contradicts (q p, ′ q s) ′ P sup , sowe have(∀(q p , q s ) P sup )(∀σ ∈ Σ u )(δ p (q p , σ)! ⇒ δ s (q s , σ)!) (4.1)