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

Chapter 2. Supervisory Control of Timed Discrete Event Systems 7• σ = tick and ∀τ ∈ Σ spe ; δ act (a, τ)! ⇒ t τ > 0i.e. no deadline of a prospective event in q is 0.• σ ∈ Σ spe and δ act (a, σ)! and 0 ≤ t σ ≤ u σ − l σi.e. if σ is defined at activity a in G act and the delay in its occurrence has beenpassed, it can occur but it should occur before its deadline.• σ ∈ Σ rem and δ act (a, σ)! and t σ = 0i.e. if σ is defined at activity a in G act and the delay in occurring it has beenpassed, it can occur.When δ(q, σ)!,q ′ = δ(q, σ) = (a ′ , {t ′ τ|τ ∈ Σ act }) is defined as follows:1. If σ = tick, then a ′ := a and, for each τ ∈ Σ act ,• if τ ∈⎧Σ spe ,⎪⎨ u τ , if δ act (a, τ) is not definedt ′ τ :=⎪⎩ t τ − 1, if δ act (a, τ)! and t τ > 0• if τ ∈⎧Σ rem ,l τ , if δ act (a, τ) is not defined⎪⎨t ′ τ := t τ − 1, if δ act (a, τ)! and t τ > 0⎪⎩ 0, if δ act (a, τ)! and t τ = 02. if σ ∈ Σ act , then a ′ := δ act (a, σ) and for any τ ∈ Σ act ,• if τ ≠⎧σ and τ ∈ Σ spe ,⎪⎨ u τ , if δ act (a ′ , τ) is not definedt ′ τ :=⎪⎩ t τ , if δ act (a ′ , τ)!• if τ = σ and τ ∈ Σ spe , t ′ τ := u σ .• if τ ≠⎧σ and τ ∈ Σ rem ,⎪⎨ l τ , if δ act (a ′ , τ) is not definedt ′ τ :=⎪⎩ t τ , if δ act (a ′ , τ)!