The sets of events that operators 1 and 2 can trigger are, respectively, 6 t 1 = f 1 ; 1 ; 1 g and 6 t 2 = f 2 ; 2 ; 2 g. Similarly, the sets of events operators 1 and 2 can disable are, respectively, 6 d 1 = f1;1; 1g and 6 d 2 = f2;2; 2; 1g.If user 1 wants to insure safety (i.e., prevent the system from reaching any of the illegal states) and user 2 is not cooperative, then 6s = 6 d 1 .The legal language in this case is not controllable and the supremal controllable sublanguage is empty. However, if user 2 is cooperative, then 6s = 6, and the legal language is controllable. In contrast, If user 2 wants to insure safety, then 6 s = 6 d 2 in case user 1 is uncooperative, and 6s = 6 in case user 1 is cooperative. In both cases the legal language is controllable.Let us now assume that the initial state IEI is the only marked state of the system, and that in addition to safety, the controller must satisfy the liveness condition specified by the marked language that consists of all the event strings that lead the system to this marked state.Let us now consider the two-user control problem with the requirement that both safety and liveness must be satisfied. In case user 1 wants to achieve safety and liveness, only the situation where user 2 is cooperative with respect to safety is relevant. Let us further assume that user 2 is cooperative also with respect to liveness, in which case, EA = ; and FA = f1; 2g.In case 1 (where 6 l = 6 t i 0([ j6 =i 6 d j )), we obtain 6 l = f 1 ; 1 g.By our synthesis algorithm, the resulting safe and live system consists of states IEI and REI.In case 2 (where 6 l = 6 t i 0 ([j2EA6 d j )), we obtain 6 l = f 1 ; 1 ; 1 g. By our synthesis algorithm, the resulting safe and live system consists of states IEI and REI.In case 3 (where 6 l = ([ i2FA 6 t i ) 0 ([ j2EA 6 d j )), we obtain 6 l = 6. By our synthesis algorithm, the resulting safe and live system consists of all the legal states.In case 4 (where 6 l = 6), we obtain 6 l = 6. By our synthesis algorithm, the resulting safe and live system consists of all the legal states.In a similar fashion, we can discuss how user 2 can achieve safety and liveness.
VII. CONCLUSIONWe have introduced an extended framework for discrete-event control where, in addition to the events that can be triggered by the environment, the user has at his/her disposal a set of events that he/she can trigger. Both the user and the environment can each disable certain events of the other. We examined the control problem where both safety and liveness requirements can be specified in a somewhat more general setting than in the traditional discrete-event control framework. A particularly interesting generalization is obtained when the environment consists of (or includes) one or more additional users. This leads to a variety of interesting scenarios where the users have each their own control objectives (specifications) and capabilities.
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