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

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Chapter 4Synthesis Algorithm Based onPredicates4.1 PredicatesIn order to place conditions on the states of the system G, it will be convenient to use alogic formalism. A predicate P defined on Q is a function P : Q → {0, 1}. P can alwaysbe identified with the corresponding state subset (“Truth Set”)Q P := {q ∈ Q | P (q) = 1}.For clarity, if P (q) = 1, we write q P , otherwise q P .Note: P red(Q) is the set of all the predicates on Q.For s ∈ Σ ∗ we say t ∈ Σ ∗ is a prefix of s, and write t ≤ s, if s = tu for some u ∈ Σ ∗ . Wedefine the closed languageL(G, P ) induced by a predicate P ∈ P red(Q) to beL(G, P ) := {w ∈ L(G) | (∀v ≤ w)δ(q 0 , v) P }That is, the set of strings that are generated by G and leads us to a state which satisfiesP through a path of states that all satisfy P .21
Chapter 4. Synthesis Algorithm Based on Predicates 22and the marked language L m (G, P ) induced by a predicate P ∈ P red(Q) to beL m (G, P ) := {w ∈ L(G, P ) | δ(q 0 , w) ∈ Q m }That is, the set of strings in L(G, P ) that lead us to the marker states.The following definitions will help us to introduce the controllability and nonblockingproperty related to predicate P .Definition 4.1.1 ReachabilityFor a given predicate P , a state q is reachable iff(∃w ∈ L(G, P ))(δ(q 0 , w) = q)That is, it can be reached from the initial state through a path of states that all satisfyP .Definition 4.1.2 CoreachabilityFor a given predicate P , a state q is coreachable iff(∃w ∈ Σ ∗ )(δ(q, w) ∈ Q m ) and (∀v ≤ w)(δ(q, v) P )That is, from that state a marker state can be reached through a path of states that allsatisfy P .Suppose we have a simple TDES system, whose plant and specification are both givenin the form of a single automaton:• Plant: G p = (Q p , Σ, δ p , q p,0 , Q p,m )• Specification: G s = (Q s , Σ, δ s , q s,0 , Q s,m )Note that we consider that the event sets of the plant and specification are the same.Suppose that these event sets are different and denoted as Σ p and Σ s respectively. Then
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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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