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

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Chapter 2Supervisory Control of TimedDiscrete Event SystemsTimed discrete event systems (TDES) are introduced in order to deal with not onlylogical specifications but also temporal specifications. A variety of frameworks for therepresentation of TDES have been presented in computer science and engineering disciplines.However, most of them are developed for the verification of the system, suchas Timed Petri Nets and Timed Transition Models.For synthesis purposes, a timedautomaton [25][4][5] is introduced.2.1 Timed Discrete Event SystemsIn this section, the Brandin and Wonham TDES model will be reviewed.First, weconsider a finite automaton G act= (A, Σ act , δ act , a 0 , A m ), called an activity transitiongraph (ATG) to describe the untimed behavior of the system. In G act , A is the finiteset of activities, Σ act is the finite set of events, a partial function δ act : A × Σ act −→ A isthe activity transition function, a 0 ∈ A is the initial activity, and A m ⊆ A is the set ofmarked activities.In order to construct a TDES model, timing information is introduced into G act . Let5
Chapter 2. Supervisory Control of Timed Discrete Event Systems 6N denote the set of all nonnegative integers.In Σ act , each event σ will be equippedwith a lower time bound l σ ∈ N and an upper time bound u σ ∈ N ∪ {∞} such thatl σ ≤ u σ . Then the set of events is decomposed into two subsets Σ spe = {σ ∈ Σ act |u σ ∈ N}and Σ rem = {σ ∈ Σ act |u σ = ∞}. Σ spe (respectively Σ rem ) is the set of prospective(respectively, remote) events whose upper time bounds are finite (respectively, infinite).The lower time bound would typically represent a delay, in communication or in controlenforcement; while an upper time bound is a hard deadline, imposed by legal specificationor physical necessity.For each σ ∈ Σ act , the timer interval T σ is defined as⎧⎪⎨ [0, u σ ] if σ ∈ Σ speT σ =⎪⎩ [0, l σ ] if σ ∈ Σ remThe TDES defined by Brandin and Wonham [5] is a finite automatonG = (Q, Σ, δ, q 0 , Q m )which can be displayed by timed transition graph (TTG). The state set Q is defined asQ = A × ∏ {T σ |σ ∈ Σ act }A state q ∈ Q is of the form q = (a, {t σ |σ ∈ Σ act }), where a ∈ A and t σ ∈ T σ . The initialstate q 0 ∈ Q is defined as q 0 = (a 0 , {t σ,0 |σ ∈ Σ act }) where⎧⎪⎨ u σ , if σ ∈ Σ spet σ,0 =⎪⎩ l σ , if σ ∈ Σ remThe set Q m ∈ Q is given by a subset of A m × ∏ {T σ |σ ∈ Σ act }. The event set Σ is definedas Σ = Σ act ∪ {tick},where the additional event tick represents the passage of one timeunit. The state transition function δ : Q × Σ −→ Q is defined as follows. For any σ ∈ Σand any q = (a, {t τ |τ ∈ Σ act }) ∈ Q, δ(q, σ) is defined, written δ(q, σ)!, if and only if oneof the following conditions holds:
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Chapter 7Conclusions and Future Res
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Appendix ABuDDy PackageBuDDy [17] i
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Bibliography[1] S. B. Akers. Functi
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Bibliography 77[18] C. Ma. Nonblock
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