Paul I. Barton

Department of Chemical Engineering

Massachusetts Institute of Technology

Cambridge, MA 02139

Many engineering applications call for the open loop optimization of a discontinuous (or hybrid) dynamic system. Examples include thedesign of operating procedures for process start-up, shut-down and changeovers, the design of emergency shut-down systems, or the optimal design of inherently dynamic processes such as those operated in a batch, semi-continuous and/or periodic manner. The most intriguing class of problems are those in which the optimal trajectories are characterized by a sequence of switches and/or jumps at events, some of which are dependent on the state of the system satisfying certain conditions (state or implicit events), and it is necessary to search over several alternative sequences of events to find the optimal one.

The potential of numerical optimization procedures to make optimal sequencing decisions in hybrid dynamic systems is explored. A general formulation of the hybrid optimal control problem is presented. However, the classical optimality conditions for such problems provide no guidance on how to find the optimal sequence. Instead, we develop novel existence and uniqueness results for the parametric sensitivity functions of hybrid dynamic system. These results show that parameter optimization of hybrid dynamic systems (including sequencing decisions) is in general nonsmooth, but also smooth in many important cases. Such parameter optimization problems may also be used to approximate optimal control problems numerically. For illustration, the design of a minimum time changeover operation for a pressure vessel avoiding the formation of explosive mixtures is considered. Results show that a direct stochastic search procedure can indeed make the sequencing decisions automatically given merely a statement of the system model and the path and point constraints. In closing, progress on more systematic approaches to the solution of the resulting nonsmooth optimization problems will be discussed.