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

Chapter 4. Synthesis Algorithm Based on Predicates 23we should at least have Σ p ⊇ Σ s , because it makes no sense to add “imaginary events” inthe specification. And if Σ p ⊃ Σ s , actually it will cause an ambiguity in the modelling,i.e., the events in Σ p − Σ s are not known to be freely enabled or totally disabled [30].For the common behavior of the plant and the specification, the meet[29] of G p and G scan be expressed asG meet = meet (G p , G s ) = Reachable states of (Q p × Q s , Σ, δ p × δ s , (q p,0 , q s,0 ), Q p,m × Q s,m )whereδ p × δ s ((q p , q s ), σ) = (δ p (q p , σ), δ s (q s , σ))Note: The closed behavior(marked behavior) of G meet is the intersection of the closedbehaviors(marked behaviors) of the plant and the specification.Remark: The predicates can also be defined on a “2-dimensional” set. For example:L(G meet , P ) = {w ∈ L(G meet ) | (∀v ≤ w)(δ p (q p,0 , v), δ s (q s,0 , v) ) P }where P ∈ P red(Q p × Q s ).Definition 4.1.3 ControllabilityFor the given G p and G s , P is controllable iff L(G meet , P ) is controllable with respectto L(G meet ).That is, at each state of G meet an uncontrollable transition will lead us to a state thatsatisfies P . The uncontrollable transition can be by an uncontrollable event or by tickwhen there is no forcible event defined at that state.Definition 4.1.4 NonblockingFor the given G p and G s , P is nonblocking iff L m (G meet , P ) = L(G meet , P )That is, from each state of G meet that satisfies P you can reach to a marker state througha path of the states that all satisfy P .In order to extend the theorems in untimed models to timed models, we define aforcing-free predicate.