In the design of controllers for governed physical processes, a central problem is to minimize a given quantity (time, amount of fuel, etc.) and to find a systematic way to characterize optimal trajectories. Due to technical and computational difficulties, it is impossible in most cases to find the optimal trajectories for a given criterion. In addition, often the situation is complicated by the existence of more than one quantity to be minimized.
This research project investigates a new approach to such minimization problems with two costs to be minimized. We do not try to find an optimal synthesis for a given cost. Instead, we start with the set of all extremals for one cost, say time, as guaranteed by the maximal principle. Then, using higher order necessary conditions coming from the maximum principle, coupled with differential geometric techniques, we reduce the set of candidate extremals to optimality for this cost to some significantly smaller set of extremals. We then minimize the other cost, say energy, over this smaller set of candidate time minimizers. We focus our method to apply to the very important class of controlled mechanical systems. The Lie algebra generated by the vector fields describing such systems possesses commutativity properties that make our computations possible.
The major application of our theory is the control of robotic underwater vehicles, which is an area of intense current interest among oceanographers, geophysicists, et al. The strategy described above is based on promising results previously obtained on controlling the motion of such a vehicle. The methods under development will be incorporated into a working vessel through collaboration with the University of Hawaii College of Engineering Autonomous Systems Laboratory. The results of the project will provide enhanced research tools to scientists in a wide variety of disciplines.
Robotic underwater vehicles are intended to be versatile and reliable tools designed, for instance, to intelligently and independently detect and monitor oceanographic phenomena, to collect samples, or to work on minesweeping where human intervention would be too risky. Unmanned underwater vehicles can be sent on long duration missions, can take measurements over large distances and variations in depth, and are not dependent on human skill for realization of mission goals. This research project develops mathematical control theory for direct application to improvement of robotic underwater vehicle capabilities.
