Minimal Communication in a Distributed Discrete Event System by

Distributed discrete event systems in which agents or local sites are required to communicate in order to perform some speci ed monitoring and control tasks are considered Each agent is modeled as a nite state machine that must be able to distinguish between its states to perform some required task To help it disambiguate states an agent uses a combination of direct observation obtained from sensor readings available to that agent and communicated information obtained from sensor readings available to another agent Since communication may be costly a strategy to minimize communication between sites is developed The complexity of the solution re ects the interdependence of the agents communication protocols That is the decision to communicate the occurrence of an event relies on which event sequences are indistinguishable to an agent which in turn is a result of what has already been communicated to that agent Introduction Existing work on decentralized control of discrete event systems focuses on problems where decentralized agents each control and observe some events in a system and must together achieve some prescribed goal In this model it is assumed that agents make independent observations and control decisions with no communication between agents Here we examine models for decentralized discrete event This work was supported in part by the Natural Sciences and Engineering Research Council of Canada NSERC under Grant OGP by NSF under grant ECS by the Department of Defense Research Engineering DDR E Multidisciplinary University Research Initiative MURI on Low Energy Electronics Design for Mobile Platforms and man aged by the Army Research O ce ARO under grant ARO DAAH by NSF under grant INT and by NASA under grant NAG systems that incorporate explicitly in the model a degree of communication between agents In particular we are interested in solving problems where some degree of communication must take place for the problem to be solved and we would like to characterize the minimal degree of communication needed for the distributed agents to achieve the global prescribed goal Synthesis problems for decentralized control plus communication have only recently been investigated for discrete event systems Such problem formulations are important because of the pervasiveness of computer networks and capture real life problems in which a centralized controller is implemented by several independent components that communicate with each other via a network Some proposed models can be found in and Work on diagnostics i e monitoring with communication can be found in and We consider distributed monitoring and control systems in industrial automation where a set of agents or local sites are cooperating in order to perform a given system level function such as failure detection and identi cation or supervisory control Agents make local observations based on their own sensors there may be common observations in that some sensors may report to more than one site We are interested in situations where the agents are not able to perform the desired system level function without communicating with each other in other words their own local observations do not provide su cient information to accomplish the required task Thus the agents need to communicate during the operation of the automated system The problem that we solve is how to minimize the communication required between the agents in order to correctly implement the system level function We assume that if the agents were to exchange all of their local observations then they would be able to perform the system level function But this solution is not optimal in the sense that there may be unnecessary communications between the agents For various reasons communication may be costly For example in some applications it may be desirable to save bandwidth for absolutely necessary communication or in wireless networks it may be crucial to save battery power For simplicity we consider the case of two agents We formulate our problem in a discrete event system framework where each agent is modeled as a nite state automaton The state transitions of the automaton are due to local and remote observations remote observations require communication from the other agent With each state of the automaton is associated a function that must be performed by the local agent for monitoring or control purposes Therefore at any time each agent must know unambiguously what state it is in in order to correctly implement the desired system level function Moreover each agent must also know when it is supposed to communicate a local observation to the other agent if this observation is required by the other agent These two facets of the problem make it both interesting and di cult to solve We provide an algorithm that nds a minimal communication strategy between the two agents A communication strategy is a set of functions one for each agent that determines if an observation made by an agent should be communicated or not to the other agent The minimality property of our algorithm is to be interpreted as follows if any of the observations that is supposed to be communicated according to the communication strategy is not communicated then the agents will not be able to continue correctly implementing the desired system level function It should be noted that such solutions are not unique Some of the parameters of our algorithm can be tuned to synthesize di erent minimal incomparable communication strategies We expect our algorithm to be applicable to decentralized implementations with communication of the problems of i supervisory control of discrete event systems in the framework initiated by Ramadge and Wonham and ii failure diagnosis of discrete event systems in the frameworks proposed in The former problem is di cult because control a ects information and vice versa In particular producing a control solution means determining when controllers communicate and what information they exchange in each communication however the information that controllers exchange may depend on what control protocol they are using Since both the type of control algorithm and the communication protocol are free parameters of the algorithm it is not obvious at the outset how to separate control from communication to produce a solution Moreover the algorithm may need to account for nonnegligible time delays in message communication For example suppose a controller s decision to disable an event depends on it being able to distinguish the sequence from the sequence and suppose that the controller cannot observe directly A communication from another controller indicating that has occurred will not help in decision making if the message transmission could be delayed so that it is not received until after is observed by the rst controller even if the message was sent before actually occurred Finally in developing algorithms that solve distributed supervisory control problems the term minimal communication must be formalized Is it best to have a protocol where in the worst case the number of messages sent is less than the number of messages sent in any other protocol even if on average the number of messages sent is greater than the average of some other protocol For supervisory control problems where legal behavior is represented by a formal language which captures some set of event sequences the control versus communication problem reduces to How to achieve more in terms of language inclusion while communicating less in terms of message passing That is in having distributed agents inhibit the plant s behavior through event disablement the goal is to generate as many of the legal sequences as possible while disallowing any illegal sequences to be generated and subject to as little communication as possible A tradeo between control and communication seems likely In general one would expect that the more we require our controllers to communicate the more they can collectively achieve However if communication is costly a compromise must be made between control and communication A preliminary and partial version of this paper appeared as Background and Problem Description This work draws in part from the supervisory control framework for discrete event systems DESs de veloped by Ramadge and Wonham and in part on standard automata theory A brief review of the relevant concepts is given in this section Readers unfamiliar with the notation and de nitions may refer to or to Chapters and of The key tie between the work presented in this paper and that of the discrete event control approach is that as in standard discrete event control theory we assume that process behavior is typi ed by event sequences and that therefore the system or process under consideration can be modeled by an automaton often a nite state automaton In standard DES control a typical problem proceeds as follows one rst models the uncontrolled plant behavior by an automaton then one describes desirable behavior usually as a formal language or a pair of formal languages then one tries to nd conditions under which a supervisor could be found that would enable and disable plant events to yield the desired behavior Controllers are also typically modeled by automata where the interpretation is that they are devices hardware or software or human which make observations of plant sequences possibly only partial observations if some events are not accessible via sensors and then based on their observations disable various subsets of events throughout the event evolution of the plant In this paper we do not solve the more di cult problem of ensuring that appropriate supervisory control is exercised in the face of partial observation and using minimal communication between distributed supervisors Instead we abstract away the issue of means of control and assume that for whatever control objective must be met the agents responsible for the objective must be able to distinguish certain states for decision making In other words the control decision might be when to enable and disable which events but the control goal might also instead relate to diagnosing a system failure We do not specify how an agent would make whatever control decisions it needs to make once it can determine which state or subset of states it is in For supervisory control problems this would require determining a minimal communication scheme that relates to the property of co observability shown in to be necessary for decentralized control We use the term agent to mean a process of interest whose behavior is described by sequences of events or actions In practice an agent could be for example a wireless device a microcontroller a robot a piece of machinery a piece of hardware or software or even a human operator What we have in mind for supervisory control is that the agents of interest are supervisors or controllers that make control decisions or that do diagnostics We assume that an agent Ri i is represented by a nite state automaton Ri Xi i xi where is an alphabet of event labels Xi is a set of states xi Xi is the initial state and i Xi Xi the transition function is a partial function de ned at each state in Xi for a subset of For the case where Xi is nite Ri can be represented by a directed graph whose nodes are states and whose edges are transitions de ned by Sometimes it is more convenient to specify i by a set of transitions Transition Ri f x i x i x is de ned in Rig The behavior of an agent Ri is also characterized by a subset of called the closed behavior of Ri written L Ri de ned as L Ri fsjs and i xi s is de nedg and interpreted to mean the set of all possible event sequences which the agent may generate In this paper we will assume that i x is de ned for all x and Therefore L Ri In the course of describing the behavior of two agents we will often need a structure that captures their joint behavior and therefore we de ne here the product of two agents R and R