The formalism of probabilistic languages has been introduced for modeling the qualitative behavior of stochastic discrete event systems. A probabilistic language is a unit interval valued map over the set of traces of the system satisfying certain consistency constraints. Regular language operators such as choice, concatenation, and Kleene-closure have been defined in the setting of probabilistic languages to allow modeling of complex systems in terms of simpler ones. The set of probabilistic languages is closed under such operators thus forming an algebra. It also is a complete partial order under a natural ordering in which the operators are continuous. Hence recursive equations can be solved in this algebra. This is alternatively derived by using contraction mapping theorem on the set of probabilistic languages which is shown to be a complete metric space. The notion of regularity, i.e., finiteness of automata representation of probabilistic languages has been defined and shown that...

The behavior of timed DES can be described by sequences of event occurrence times. These sequences can be ordered to form a lattice. Since logical (untimed) DES behaviors described by regular languages also form a lattice, questions of controllability for timed DES may be treated in much the same manner as they are for untimed systems. In this paper we establish conditions for the controllability of timed DES performance specifications which are expressed as inequations on the lattice of sequences. These specifications may take the form of sets of acceptable event occurrence times, maximum or minimum occurrence times, or limits on the separation times between events. Optimal behaviors are found as extremal solutions to these inequations using fixed point results for lattices. Keywords: Discrete event systems, supervisory control, max-algebra, lattices. I. Introduction Discrete event systems (DES) are characterized by a collection of events, such as the completion of a job in a manufac...

. Constrained supremum and supremum operators are introduced to obtain a general procedure for computing supremal elements of upper semilattices. Examples of such elements include supremal (A; B)-invariant subspaces in linear system theory and supremal controllable sublanguages in discrete-event system theory. For some examples, we show that the algorithms available in the literature are special cases of our procedure. Our iterative algorithms may also provide more insight into applications; in the case of supremal controllable subpredicate, the algorithm enables us to derive a lookahead policy for supervisory control of discrete-event systems. Keywords. Discrete-event systems, linear systems, lattice theory, supervisory control, partition, supremal elements, supremum operators AMS subject classifications. 93B, 68Q20 1 Introduction In system theory, we sometimes encounter lattice structures [2], [5]. Examples are the lattice of equivalence relations in the theory of sequential machi...

The theory of supervisory control of discrete event systems is extended to the real-time setting. The real-time behavior of a system is represented by the set of all possible timed traces of the system. This is alternatively specified using timed automata where each transition is associated with an event occurrence time set during which time the transition can occur. Our model for time is more general in that the time advances continuously as compared to a model where time advances discretely. We extend the notion of prioritized synchronous composition to the real-time setting to use it as the control mechanism. It is shown that a suitable extension of the controllability condition to the real-time setting yields a condition for the existence of a supervisor achieving a desired timed behavior. Although the real-time controllability is similar in form to its untimed counterpart, they are different in the sense that one does not imply the other and vice-versa.

Algorithms for computing a minimally restrictive control in the context of supervisory control of discrete event systems have been well developed when both the plant and the desired behavior are given as regular languages. In this paper we extend such prior results by presenting an algorithm for computing a minimally restrictive control when the plant behavior is a deterministic Petri net language and the desired behavior is a regular language. As part of the development of the algorithm, we establish the following results that are of independent interest: (i) The problem of determining whether a given deterministic Petri net language is controllable with respect to another deterministic Petri net language is reducible to a reachability problem of Petri nets. (ii) The problem of synthesizing the minimally restrictive supervisor so that the controlled system generates the supremal controllable sublanguage is reducible to a forbidden marking problem. In particular, we can directly identi...

Abstract — This paper presents a modelling and an analysis of P-time Event Graphs in the field of (max, +) algebra. Under the hypothesis of the logical liveness of the event graph, temporal liveness is defined by the existence of a trajectory. Based on a particular serie of matrices, the extremal trajectories starting from an initial interval are deduced. The liveness of the static part and dynamic part are analysed.

specifications

Using (max, +) algebra, this paper presents a modeling and an analysis method of P-time Event Graphs whose behaviors are defined by lower and upper bound constraints. On the hypothesis of liveness of Event Graphs, consistency is defined by the existence of a temporal trajectory. The extremal trajectories starting from an initial interval are deduced by two dual polynomial algorithms based on a particular series of matrices. The analysis of the circuits introduces conditions of consistency.

Abstract — In this paper, we consider the (max, +) model of P-time Event Graphs whose behaviors are defined by lower and upper bound constraints. The extremal trajectories of the system starting from an initial interval are characterized with a particular series of matrices for a given finite horizon. Two dual polynomial algorithms are proposed to check the existence of feasible trajectories. The series of matrices are used in the determination of the maximal horizon of consistency and the calculation of the date of the first token deaths. Index Terms — P-time Petri Nets, (max,+) algebra, token death, Kleene star, fixed point. I.

The formalism of probabilistic languages has been introduced for modeling the qualitative behavior of stochastic discrete event systems. A probabilistic language is a unit interval valued map over the set of traces of the system satisfying certain consistency constraints. Regular language operators such as choice, concatenation, and Kleene-closure have been defined in the setting of probabilistic languages to allow modeling of complex systems in terms of simpler ones. The set of probabilistic languages is closed under such operators thus forming an algebra. It also is a complete partial order under a natural ordering in which the operators are continuous. Hence recursive equations can be solved in this algebra. This is alternatively derived by using contraction mapping theorem on the set of probabilistic languages which is shown to be a complete metric space. The notion of regularity, i.e., finiteness of automata representation of probabilistic languages has been defined and shown that...