Peter Caines

Department of Electrical and Computer Engineering

McGill University

**On the Optimal Control of Hybrid Systems:
Theory and Algorithms for Trajectory and Schedule Optimization**

Hybrid systems have state spaces with both continuous and discrete subspaces and correspondingly continuous and discrete dynamics; they appear in a vast range of contemporary engineering systems with some examples being given by chemical engineering and manufacturing plants, and aerospace and automotive control systems. In work with Shahid Shaikh, we formulate a general class of hybrid system optimal control problems (HOCPs), and provide a Hybrid Maximum Principle (HMP) and a Hybrid Dynamic Programming theory. Efficient HMP based HOCP algorithms are presented with proven convergence properties. The work to this point assumes that the sequence of discrete states (i.e. the schedule) is fixed while the switching times and corresponding continuous switching states, and the continuous controls are to be optimized. We next introduce the notion of Optimality Zones(OZ) which permits one to reach the global optimum with respect to the schedules as well as the continuous decision variables with computational complexity which is only linear in the number of switching times. It is hoped that the OZ notions will lead to a differential and algebro-geometric analysis of HOCPs. Computational examples will be presented.

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