Modeling and control of complex systems in a dioid framework

Abstract

Many different kinds of manufacturing systems can be modeled by timed event graphs (TEG), a sub-class of Petri nets. The main advantage of timed event graphs is their linear representation in specific mathematical structures named dioids or idempotent semirings. For linear systems in dioids, in turn, exists an established control theory, which can be used to determine feedback and feedforward controllers. However, if the considered system shall be operated with a re-entrant workflow, resulting in a nested schedule, i.e., a resource may be occupied by the same part more than once and in between these resource allocations another part is processed on this very resource, it is not possible to determine a TEG modeling the system’s behavior. Furthermore, in standard Petri nets, and consequently in standard timed event graphs, timing information is always considered to be the minimal time of a (sub-) process. Nevertheless, often manufacturing systems are operated with time windows, i.e., a minimal time is necessary to complete a (sub-) process but, at the same time, a maximal time is given, at which the (sub-) process needs to be finished. Such time windows cannot easily be included in timed event graphs. Similarly, it is rather straightforward to include maximum capacities of (sub-) processes or resources, but not possible to include minimum capacities in timed event graphs. While the issue of time windows has been addressed in various publications, an extension of timed event graphs to model systems with nested schedules or minimum capacities has not been studied. In this work, we propose an approach to model manufacturing systems with nested schedules. This approach is based on an extension for timed event graphs which, in turn, results in non-causal dioid representations with respect to the standard definition of causality for linear systems in dioids. Consequently, the causality issue for systems with nested schedules is addressed. Eventually, the control theory developed for systems in a dioid framework can be applied to determine suitable controllers. In the second part of this thesis, timed event graphs with constraints are investigated. The constraints include time windows, i.e., minimal and maximal time bounds for some (sub-) processes as well as minimum and maximum capacities. Using results from residuation theory, an algorithm to determine linear systems for extended timed event graphs is developed. Finally, a method is introduced to compute suitable controllers for timed event graphs with the mentioned additional constraints. Using a real world example from high-throughput screening, the applicability of our approach is demonstrated.