Limits of control in designing distributed C2 experiments under imperfect communications

The paper evaluates the controllability of hierarchically organized C² systems with distributed components under conditions of imperfect communications. Controllability in a C² system depends on the number of levels of reporting in the C² hierarchy and the amount of aggregation of events in the feedback to the command levels. Following the discrete event dynamic systems concepts developed by Ramadge and Wonham (1987), a formal control language was defined with its alphabet consisting of controllable and uncontrollable discrete events at the lowest level in the C² hierarchy. At the next level of command, specific patterns of these events are recognized as words. These words, in turn, form the alphabet for a higher level formal control language generated by this automaton and recognized by the automaton at the next higher level of command. In this context, we establish the theoretical limits of control in a C² system for performing a given mission using a selected course of action (COA). Furthermore, introduction of imperfect communications which result in a loss of observability at the command level may render the controllable formal language associated with a selected COA to become uncontrollable. Control by the command level may also be diminished due to the unobservability of events not reported to this level. Theoretical limits of control are computed for a specific C² system modeled as a two level hierarchy of interacting automata under conditions of perfect observability, i.e., perfect communications and total observation of all events. Introduction of observability issues and communication delays in this model is used to demonstrate how a previously controllable language for a selected COA may become uncontrollable resulting in a loss of the mission. We demonstrate that a more conservative COA is then required to ensure that its associated formal language is controllable