That Supervisor Is Best Which Supervises Least

Inspired by the well known motto of Henry David Thoreau that government is best which governs least we prove that in a general discrete event system whose behavior is described by strings of discrete events a supervisor or controller that supervises less that is gives more freedom to the system is indeed better in the sense that the system will evolve faster that is events will occur earlier if certain fairness condition is satis ed This interesting result is a consequence of recent development of a supervisory control theory for discrete event systems Consider the following scenario a manager sitting in his o ce supervises the operations of a plant He may conceive of these operations as a ow of discrete events For him events are generic abstractions of various occurrences activities decisions tasks etc Initiating a new program marketing a new product signing a contract negotiating a deal placing an order renovating certain equipments hiring new personal etc are typical example of events in his plant His aim is to ensure the orderly ow of events leading to the realization of certain objectives Formally we denote by the set of events A string of events describes a possible evolution of the plant One of such strings could be for example initiate a new program install new equipments hire more workers Each occurrence of events has a lifetime which for generality is assumed to be random and generated by some stochastic process of which deterministic process is a special case The event lifetime describes the time elapsed from the initiation of the event and the completion of the event Because of our general assumption di erent occurrences of the same event may have di erent lifetimes We denote the set of all strings of events of by Clearly in general not all strings in can physically occur in the plant We call the set of feasible or physically possible strings of in the absence of supervision the uncontrolled behavior UB The set UB may be thought of as the set of all free trajectories in the classical sense and thus serves as a model for the plant Naturally every pre x or initial substring of a string in UB is itself in UB For example if the aforementioned string is in UB then initiate a new program install new equipments is also a string in UB In other words UB is closed under pre x operation This will be assumed for all behaviors de ned in this paper The manager as a supervisor decides on the strings of events that are permissible legal or desirable and these that are not Thus he determines the target behavior TB of his plant In general TB is a proper subset of UB TB UB The manager pursues the objective of realizing this target behavior through a string of enablement disablement Each time he authorizes a transaction approves a plan rejects a request etc he is enabling or disabling event as the case may be We assume without lose of generality that his role is merely supervisory That is he refrains from issuing orders or enforcing actions which is implicit in the model itself Rather he responds to his environment that is his subordinates or the proposal put forward by them through approval or disapproval It is only in this permissive manner that he exercises control In short he leaves initiation of actions to the internal dynamics of his plant and those involved in the operational detail responds to the events of his environment by his approval or disapproval and leave enforcement entirely to his subordinates Clearly he does not have control over all events For instance unpredictable events that are often also unpreventable may occur Machine may break down market may be lost to competition personnel may resign workers may go on strike contractors may renege on their promises of timely delivery etc These events the manager cannot control may not disable Therefore we partition into controllable events c and uncontrollable events uc Furthermore it is reasonable to assume that the manager cannot observe the occurrence of all events some information is either not reported to him or is not available at the time he needs to make a relevant decision Thus we also partition into observable events o and unobservable events uo The manager s task is thus to enable or disable the controllable events based on the record of occurrences of the observable events in such a way that only the strings belonging to the target behavior survive To formalize the manager s function we rst de ne the projection P o as P where denotes the empty string P s P s if uo P s if o Thus P erases these events that are not observable to the manager Projecting UB on o results in P UB fs o t UB Pt sg The set P UB describes the behavior of the plant observable by the manager Based on the observed behavior the manager would like to implement a policy which is de ned as a feedback map