On the existence of supervisory policies that enforce liveness indiscrete-event dynamic systems modeled by controlled Petri nets

We consider discrete-state plants represented by controlled Petri nets (CtlPNs), where a subset of transitions can be prevented from firing by a supervisor. A transition in a CtlPN can fire at a marking if there are sufficient tokens in its input places and it is permitted to fire by the supervisor. A CtlPN is live if it is possible to fire any transition from every marking that is reachable under supervision. In this paper we derive a necessary and sufficient condition for the existence of a supervisory policy that enforces liveness in CtlPNs. We show this condition cannot be tested for an arbitrary CtlPN. However, for bounded CtlPNs or CtlPNs, where each transition is individually controllable, we show the existence of a supervisory policy which enforces that liveness is decidable. We also show the existence of a supervisory policy that enforces liveness is necessary and sufficient for the existence of a minimally restrictive supervisory policy     

Observability and decentralized control of fuzzy discrete-event systems

Fuzzy discrete-event systems as a generalization of (crisp) discrete-event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete-event system has been modeled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete-event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development.     

On supervisory policies that enforce liveness in completelycontrolled Petri nets with directed cut-places and cut-transitions

The process of synthesizing a supervisory policy that enforces liveness in a Petri net (PN), where each transition can be prevented from firing by an external agent, can be computationally burdensome in general. We consider PNs that have a directed cut place or a cut-transition. A place (transition) in a connected PN is said to be a cut place (cut-transition) if its removal will result in two disconnected component PNs. A cut place is said to be a directed cut-place, if in the original PN, all arcs into this cut place emanate from transitions in only one of the two disconnected component PNs. The authors show there is a supervisory policy that enforces liveness in the original PN if and only if similar policies exist for two PNs derived from the disconnected components obtained after the removal of the directed cut-place (cut-transition). The utility of this observation in alleviating the computational burden of policy synthesis is illustrated via example     

On the existence of finite state supervisors for arbitrarysupervisory control problems

Given two prefix closed languages K, L ⊆ Σ*, where K ⊆ L represents the desired closed-loop behavior and L is the open-loop behavior, there exists a finite-state supervisor that enforces K in the closed loop if and only if there is a regular, prefix-closed language M ⊆ Σ*, such that: 1) MΣ_u∪L⊆M, and 2) M∪L=K. In this paper, we show that this is equivalent to: 1) the controllability of sup{P⊆K∪L¯|pr(P)=P} with respect to Σ*; and 2) the regularity of sup{P⊆K∪L¯|pr(P)=P}, where L¯=Σ*-L:and pr(·) is the set of prefixes of strings in the language argument. We use this property to investigate the issue of deciding the existence of a finite-state supervisor for different representations. We also present some properties of the language sup{P⊆K∪L¯|pr(P)=P}, along with implications to the synthesis of solutions to the supervisory control problem with the fewest states     

Robust nonblocking supervisory control of discrete-event systems under partial observation

In this paper, we examine the problem of robust nonblocking supervisory control. In the problem considered here, the exact model of the plant is not known but is assumed to be among a finite set of possible models. For each plant model a legal marked behavior is assumed given. We extend previous results for the case of control with full observation to the case of control under partial observation where only a subset of events are observable. Furthermore, we remove the limitations of previous results on ensuring the nonblocking property of the plant under supervision. We characterize the entire set of solutions of the robust control problem and obtain a set of necessary and sufficient conditions for the existence of a solution for the problem. As an illustrative example, we use our results on robust control to solve a fault recovery problem.     

Supervisory control of discrete-event systems with output: Application to hybrid systems

In this paper, the problem of supervisory control of discrete-event systems (DES) with output is presented and discussed at length. In such systems, causal output maps are employed to assign to each sequence of input events a corresponding sequence of output events. When the specification of desired behaviour is given by a formal language over the output alphabet, necessary and sufficient conditions are derived for the existence of non-blocking input as well as non-blocking output supervisory control. After making minor adjustments the theory is applied to non-deterministic discrete-event abstractions of hybrid systems, giving rise to the development of a theory for non-blocking supervisory control of hybrid systems. Our results enable one to apply classical supervisory control theory to design supervisors for DES approximations of hybrid systems, and to import many interesting concepts from classical theory such as modular and hierarchical control.     

Grid automata and supervisory control of dense real-time discrete event systems

We present a generalization of the classical supervisory control theory for discrete event systems to a setting of dense real-time systems modeled by Alur and Dill timed automata. The main problem involved is that in general the state space of a timed automaton is (uncountably) infinite. The solution is to reduce the dense time transition system to an appropriate finite discrete subautomaton, the grid automaton, which contains enough information to deal with the timed supervisory control problem (TSCP). The plant and the specifications region graphs are sampled for a granularity defined in a way that each state has an outgoing transition labeled with the same time amount. We redefine the controllability concept in the context of grid automata, and we provide necessary and sufficient solvability conditions under which the optimal solution to centralized supervisory control problems in timed discrete event systems under full observation can be obtained. The enhanced setting admits subsystem composition and the concept of forcible event. A simple example illustrates how the new method can be used to solve the TSCP.     

Supervisor synthesis for discrete event systems under partial observation and arbitrary forbidden state specifications

In this paper, we consider the forbidden state problem in discrete event systems modeled by partially observed and partially controlled Petri nets. Assuming that the reverse net of the uncontrollable subnet of the Petri net is structurally bounded, we compute a set of weakly forbidden markings from which forbidden markings can be reached by firing a sequence of uncontrollable/unobservable transitions. We then use reduced consistent markings to represent the set of consistent markings for Petri nets with structurally bounded unobservable subnets. We determine the control policy by checking if the firing of a certain controllable transition will lead to a subsequent reduced consistent marking that belongs to the set of weakly forbidden markings; if so, we disable the corresponding controllable transition. This approach is shown to be minimally restrictive in the sense that it only disables behavior that can potentially lead to a forbidden marking. The setting in this paper generalizes previous work by studying supervisory control for partially observed and partially controlled Petri nets with a general labeling function and a finite number of arbitrary forbidden states. In contrast, most previous work focuses on either labeling functions that assign a unique label to each observable transition or forbidden states that are represented using linear inequalities. More importantly, we demonstrate that, in general, the separation between observation and control (as considered in previous work) may not hold in our setting.     

On the maximum acyclic subgraph problem under disjunctive constraints

Disjunctively constrained versions of classic problems in graph theory such as shortest paths, minimum spanning trees and maximum matchings were recently studied. In this article we introduce disjunctive constrained versions of the Maximum Acyclic Subgraph problem. Negative disjunctive constraints state that a certain pair of edges cannot be contained simultaneously in a feasible solution. Positive disjunctive constraints enforces that at least one arc for the underlying pair is in a feasible solution. It is convenient to represent these disjunctive constraints in terms of an undirected graph, called constraint graph, whose vertices correspond to the arcs of the original graph, and whose edges encode the disjunctive constraints. For the Maximum Acyclic Subgraph problem under Negative Disjunctive Constraints we develop 1/2-approximative algorithms that are polynomial for certain classes of constraint graphs. We also show that determining if a feasible solution exists for an instance of the Maximum Acyclic Subgraph problem under Positive Disjunctive Constraints is an NP-Complete problem.     

Robust and adaptive supervisory control of discrete event systems

Both robust and adaptive supervisory control in discrete-event systems are discussed. It is assumed that the system G to be controlled is not known exactly. It is only known either that it belongs to a set or that it has certain lower and upper bounds. The task of robust supervision is to synthesize a supervisor that realizes a given desired behavior for all possible G. A necessary and sufficient condition for the existence of such a robust supervisor is derived. Based on this condition, a robust supervisory control and observation problem of synthesizing a robust supervisor whose behavior is both legal and acceptable is solved. Adaptive supervision is discussed. As the system progresses, the information on occurrences of events may help to resolve or reduce uncertainties     

On tolerable and desirable behaviors in supervisory control of discrete event systems

We formulate and solve a new supervisory control problem for discrete event systems. The objective is to design a logical controller—or supervisor—such that the discrete event system satisfies a given set of requirements that involve event ordering. The controller must deal with a limited amount of controllability in the form of uncontrollable events. Our problem formulation considers that the requirements for the behavior (i.e., set of traces) of the controlled system are specified in terms of a desired behavior and a larger tolerated behavior. Due to the uncontrollable events, one may wish to tolerate behavior that sometimes exceeds the ideal desired behavior if overall this results in achieving more of the desired behavior. The general solution of our problem is completely characterized. The nonblocking solution is also analyzed in detail. This solution requires the study of a new class of controllable languages. Several results are proved about this class of languages. Algorithms to compute certain languages of interest within this class are also presented.Research supported in part by the National Science Foundation under grants ECS-8707671, ECS-9057967, and ECS-9008947.     

On computation of supremal controllable, normal sublanguages

In this paper, we present an algorithm for the computation of the controllable, normal sublanguage of a given language, encountered in the solution of the supervisory control of discrete-event systems under partial observation. The algorithm produces the desired result under certain assumptions on the plant and the event projection map. In particular, the plant has to be nonblocking. The advantage of the algorithm over the solution available in the literature is that it does not involve iterations on the supremal controllable sublanguage and supremal normal sublanguage operators.     

Robust supervisory control of a class of timed discrete event systems under partial observation

This paper studies robust supervisory control of timed discrete event systems proposed by Brandin and Wonham. Given a set of possible models which includes the exact model of the plant, the objective is to synthesize a robust supervisor such that it achieves legal behavior for all possible models. We show that controllability for each possible model and observability for a suitably defined aggregate model are necessary and sufficient conditions for the existence of a solution to the robust supervisory control problem. Moreover, when there does not exist a solution, a maximally permissive robust supervisor is synthesized under the assumption that all controllable events are observable.     

Maximizing robustness of supervisors for partially observed discrete event systems

We consider a robust supervisory control problem to synthesize a supervisor for the nominal plant model which maximizes robustness. Cury and Krogh have first addressed this problem by unnecessarily restricting the upper bound of the legal behavior. In our previous work, we have solved the problem without restricting the upper bound of the legal behavior when the specification is described by prefix-closed languages. In this paper, we extend our previous work to the partial (event) observation case. First, we synthesize a supervisor that maximizes robustness. This result shows that robustness can be optimized under partial observation, while permissiveness cannot be optimized in general. We next consider a special case where all the controllable events are observable. In this special case, we synthesize a maximally permissive supervisor for the nominal plant model which maximizes not only the robustness but also permissiveness for the maximal set of admissible plant variations.     

A Large Margin Algorithm for Speech-to-Phoneme and Music-to-Score Alignment

We describe and analyze a discriminative algorithm for learning to align an audio signal with a given sequence of events that tag the signal. We demonstrate the applicability of our method for the tasks of speech-to-phoneme alignment (ldquoforced alignmentrdquo) and music-to-score alignment. In the first alignment task, the events that tag the speech signal are phonemes while in the music alignment task, the events are musical notes. Our goal is to learn an alignment function whose input is an audio signal along with its accompanying event sequence and its output is a timing sequence representing the actual start time of each event in the audio signal. Generalizing the notion of separation with a margin used in support vector machines for binary classification, we cast the learning task as the problem of finding a vector in an abstract inner-product space. To do so, we devise a mapping of the input signal and the event sequence along with any possible timing sequence into an abstract vector space. Each possible timing sequence therefore corresponds to an instance vector and the predicted timing sequence is the one whose projection onto the learned prediction vector is maximal. We set the prediction vector to be the solution of a minimization problem with a large set of constraints. Each constraint enforces a gap between the projection of the correct target timing sequence and the projection of an alternative, incorrect, timing sequence onto the vector. Though the number of constraints is very large, we describe a simple iterative algorithm for efficiently learning the vector and analyze the formal properties of the resulting learning algorithm. We report experimental results comparing the proposed algorithm to previous studies on speech-to-phoneme and music-to-score alignment, which use hidden Markov models. The results obtained in our experiments using the discriminative alignment algorithm are comparable to results of state-of-the-art systems.     

Supervisory control of nondeterministic discrete-event systems with driven events via masked prioritized synchronization

Studies the supervisory control problem of nondeterministic discrete event systems with driven events in the setting of masked prioritized synchronous composition (MPSC). MPSC was extended from prioritized synchronous composition (PSC) by Kumar and Heymann (2000) in order to permit systems interaction with their environment via interface masks. They also studied the supervisory control problem under the assumption that the set of driven events was empty. In this paper, the aforementioned assumption is relaxed. We first derive relations among behaviors at different levels of the system. Next, we solve the supervisor synthesis problem for controlling the plant behavior as observed at the interface level. Finally, we give a necessary and sufficient condition for the existence of the supervisor for controlling the plant behavior as observed at the plant level. We establish a link between MPSC and PSC by showing that a supervisory control problem in the setting of MPSC can be transferred to a supervisory control problem in the setting of PSC under certain conditions.     

An Optimization Approach to Petri Net Monitor Design

This note addresses the problem of enforcing generalized mutual exclusion constraints on a Petri net plant. First, we replace the classical partition of the event set into controllable and uncontrollable events from supervisory control theory, by associating a control and observation cost to each event. This leads naturally to formulate the supervisory control problem as an optimal control problem. Monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs. Second, we consider timed models for which the monitor choice may lead to performance optimization. If the plant net belongs to the class of mono-T-semiflow nets, we present an integer linear fractional programming approach to synthesize the optimal monitor so as to minimize the cycle time lower bound of the closed loop net. For strongly connected marked graphs the cycle time of the closed-loop net can be minimized     

Mind the gap: Expanding communication options in decentralized discrete-event control

Frameworks that incorporate communication into decentralized supervisory control theory address the following problem: find locations in the evolution of the plant behavior where supervisors send information so that a supervisor that was unable to make the correct control decision prior to receiving external information, is now capable of making the correct control decision. Existing solutions to this problem identify an earliest and a latest placement where the communication protocol leads to the synthesis of a correct control solution. In addition to the first and last communication opportunities, there may be a selection of intermediate possibilities where communication would also produce the correct control solution. We present a computable procedure to identify a broader range of suitable communication locations.     

An algorithm for computing the mask value of the supremal normalsublanguage of a legal language

We consider the problem of finding the mask value of the supremal normal sublanguage L_R of some given language L. We describe a straightforward algorithmic solution that can be applied to existing off-line procedures for determining the supremal controllable and normal sublanguage of L and that does not require an explicit calculation of L _R. This problem is fundamental because it is related to the supervisory control problem under partial observation. Our algorithm applies only to closed languages     

Supervisory control of switching control systems

In this paper, the problem of designing a switching policy for an adaptive switching control system is formulated as a problem of supervisory control of a discrete-event system (DES). Two important problems in switching control are then addressed using the DES formulation and the theory of supervisory control under partial observation. First, it is verified whether for a given set of controllers, a switching policy satisfying a given set of constraints on the transitions among controllers exists. If so, then a minimally restrictive switching policy is designed. Next, an iterative algorithm is introduced for finding a minimal set of controllers for which a switching policy satisfying the switching constraints exists. It is shown that in the supervisory control problem considered in this paper, limitations on event observation are the factors that essentially restrict supervisory control. In other words, once observation limitations are respected, limitations on control will be automatically satisfied. This result is used to simplify the proposed iterative algorithm for finding minimal controller sets.