,
, Monotone transition systems
146 7.2.2 Discrete-time control system as a transition system ,
148 7.3.2 Controller synthesis for monotone transition systems and directed safety specifications ,
,
,
, C
, To prove the second inclusion, it is sufficient to show that C is a safety controller for the transition system T and the safety specification Q S . We have that dom(C) =? dom(C * ) ?? Q S = Q S , where the first equality comes from (i), the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedness of Q S . Hence, the first condition of Definition B.1 is satisfied. Now let q ? dom(C) and v ? C(q). From construction of the controller C, we have the existence of q ? Q such that q ? Q q , q ? dom(C * ) and v ? C * (q ). Then, we have that ?(q, v) ?? ?(q , v) ?? dom(C * ) = dom(C), where the first inclusion comes from (ii), C, it follows immediately that ? dom(C * ) = dom(C). Let us prove that dom(C * ) = dom(C). Let q ? Q, since q ? (? q) we have that dom(C * ) ? dom(C)
,
, Let C * be the maximal safety controller enforcing the lower closed safety specification Q S ? Q. The following properties hold: (i) dom(C * ) is a lower closed set
, ii) for all q 1 , q 2 ? Q, if q 1 ? Q q 2 then C * (q 2 ) ? C *
, iii) for q ? Q, C * (q) is a lower closed set
, Then, ? dom(C * ) =? dom(C) = dom(C * ), where the last equality comes from (i) in Lemma 7.9. Hence, dom(C * ) is a lower closed set. (ii) Let q 1 , q 2 ? Q with q 1 ? Q q 2, i) We have from (ii) in Lemma 7.9 that dom(C * ) = dom(C)
, Hence, by maximality of C * , we have that v ? C * (q 1 ). Then, C * (q 2 ) ? C * (q 1 ). (iii) Let q ? Q, v ? C * (q) and v ?? v. We have that ?(q, v ) ?? ?(q, v) ?? dom(C * ) = dom(C * ), where the first inclusion comes from the monotonicity of the transition system T , the second inclusion comes from the fact that C * is a safety controller and the last equality comes from the lower closedeness of dom(C * ). Hence, we have ?(q, v ) ? dom(C * ). Then, by maximality of C * , v ? C * (x), Hence, we have that ?(q 1 , v) ?? ?(q 2 , v) ?? dom(C * ) = dom(C * ), where the first inclusion comes from the fact that T is a monotone transition system and the last equality comes from (i)
,
, = InvariantSet(T 1
, Cont(Q s ), vol.1
, 10 return C
, 5) we have that C(q) ? C * (q) for all q ? Cont(Q s ). Let Z i be defined as above, We have from (i) in Proposition 7
, N } such that q ? Z i?1 \ Z i = Z i?1 \ Z i , where the last equality comes from (ii) in Proposition 7.18. Hence, we have that v / ? V i . Then, v ? V i?1 . Moreover, C * is the maximal safety controller, then using the fact that ?(q, v) ? Cont(Q s )
, N } and for any (v i?1 , v i ) ? V i?1 × V i , we have v i ? v i?1 . The synthesis is more efficient. Indeed, for all q ? Z i?1 \ Z i , we have the existence of v ? V i?1 such that ?(q, v) ? Cont(Q s ). Hence, from (iii) in Proposition 7.10 and since V i?2 ?? v, we have that V i?2 ? C * (q). Then, at each step i ? {1, on the inputs ? V satisfies the following property: for all i ? {2
A subset A ? Q S is said to be a controlled invariant if for all q ? A there exists v ? V such that for any (q , y) ? ?(q, v), q ? A. It was shown in [Tab09] that there exists a maximal controlled invariant Cont(Q s ) which is the union of all controlled invariants. The maximal safety controller can be defined as follows: (i) for all q / ? Cont ,
, ii) for all q ? Cont(Q s ), C * (q) = {v ? enab(q) | for (q , y) ? ?(q, v), q ? Cont
, Let us remark that for any safety controller C we have that dom(C) ? Cont(Q s ), while for the maximal safety controller C * , we have dom(C * ) = Cont(Q s )
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