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

Chapter 2. Supervisory Control of Timed Discrete Event Systems 9or uncontrollable. By forcing an enabled event in Σ for to occur, we can disable the eventtick. In this framework a supervisor decides to disable or enable each event in Σ c ∪{tick}.The simplest way to visualize the behavior of a TDES G under supervision is first toconsider the infinite reachability tree of G before any control is operative [29]. Each nodeof the tree corresponds to a unique string s of L(G). At each node of the tree we candefine the subset of eligible events byElig G (s) := {σ ∈ Σ| sσ ∈ L(G)}In order to define the notion of controllability we should consider a language K ⊆L(G)and writeElig K (s) := {σ ∈ Σ| sσ ∈ ¯K)}K is controllable with respect to G if, for all s ∈ ¯K⎧⎪⎨ Elig G (s) ∩ (Σ u ∪ {tick}),Elig K (s) ⊇⎪⎩ Elig G (s) ∩ Σ u ,if Elig K (s) ∩ Σ for = ∅if Elig K (s) ∩ Σ for ≠ ∅(2.1)Thus K controllable means that an event σ (in the full alphabet Σ including tick) mayoccur in K if σ is currently eligible in G and either (i) σ is uncontrollable, or (ii) σ = tickand no forcible event is currently eligible in K.The difference between controllability in TDES and in untimed DES is in the event tick.Event tick acts as an uncontrollable event in the states where no forcible event is present,but in the states with one or more forcible events, tick can be preempted (disabled) bya forcible event. So an event σ can occur in K if σ is currently eligible in G and eitherσ is uncontrollable or σ = tick and no forcible event is currently eligible in K.A supervisory control for G is any mapV : L(G) → 2 Σ