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

Chapter 5. BDD based implementation 39the form:next = (act ′ = a ′ ) ∧ (∧τ≠σ,¬δ act(a ′ ,τ)!(t ′ τ = t τ,0 ) ) ∧ (∧τ≠σ, δ act(a ′ ,τ)!(t ′ τ = t τ ) ) ∧ (t ′ σ = u σ )The transition predicate of prospective event σ for the states of the TDES whoseactivity is “a” will be current∧(event = tick)∧next. In order to find the transitionpredicate for all the σ transitions of the TDES, we should find the disjunction ofthe transition predicates of σ transitions for the states of the TDES with differentactivities. According to these rules the transition predicate can be computed for σ:T σ =• Remote events.∨a∈A ∧δ act (a,σ)!δ(q, σ)! ⇔ δ act (a, σ)! ∧ t σ = 0{ {(act = a) ∧ (0 ≤ t σ ≤ u σ − l σ )}∧ {(event = σ)}∧ {(act ′ = a ′ ) ∧ ((∧τ≠σ, δ act (a ′ ,τ)!∧τ≠σ,¬δ act(a ′ ,τ)!(t ′ τ = t τ,0 ) )∧(t ′ τ = t τ ) ) ∧ (t ′ σ = u σ ) } }Thus the remote event σ will occur at state q if it is defined at activity “a” in G actand its delay has been passed. By occurring σ we will reach the state q ′ :δ(q, σ) = q ′ ⇔ a ′ = δ act (a, σ) and for any τ ∈ Σ act ,⎧⎪⎨ t τ,0 , if δ act (a ′ , τ) is not defined- if τ ≠ σ , t ′ τ :=⎪⎩ t τ , if δ act (a ′ , τ)!- if τ = σ , t ′ τ := l σ .The next state’s activity will be a ′ = δ act (a, σ) and the timer value of σ will resetto its default value. The timer value of other events will be remaind unchanged ifthey are defined at activity a ′ and will reset to their default value if they are notdefined at a ′ . So the transition predicate for remote event σ will be constructed in