| Control and monitoring of discrete event dynamic systems (DEDS) have proven to be difficult since traditional analysis tools offer little value. For a certain class of DEDS, however, the dynamics of the event times can be modeled using a nonstandard algebra termed max-plus. This work examines methods to monitor and control manufacturing DEDS based on a max-plus algebraic model of the event-time dynamics. To fully understand the nature of these systems, properties such as reachability and observability that parallel classical system theory are studied. Working in the event domain, analogous to the time domain, a framework is established whereby definitions of reachability and observability can be constructed and subsequent linear system theoretic results can be obtained.By applying results obtained in the study of observability, a model-based methodology to condition monitoring is developed. A block-form state observer is introduced that although does not necessarily produce the actual event-time state, it reproduces the sequence of observed output measurements. The model state is updated periodically with a delayed version of the event-time state estimate. The approach makes use of on-line measurements and, after an initial delay, can be implemented in real-time.Finally, this work initiates a study of the extent to which well-known results in optimal control theory can be carried over from the context of traditional linear systems to the domain of max-plus systems. An objective is formed that measures the sum of the differences between parts' completion times and their respective due dates. First, an approach based on analysis of the output-input time relationship is shown to result in strong necessary optimality conditions for generalized flow-shop systems. The conditions, however, are somewhat awkward to use. Then, for transfer-line production systems, a costate-based approach is presented which leads to more straightforward optimality conditions. The optimality conditions developed in both approaches are believed to be sufficient as well as necessary. Finally, for single-machine systems, the costate-based conditions are shown to be sufficient as well as necessary and are shown to lead to an efficient algorithm that yields the optimal solution. |
