Time Optimal Control for a Class of Common Random Disturbances

This paper concerns the time optimal control of a system variable where the controlling input is bounded, as is usually the case, and the system is subject to arbitrary disturbances. An arbitrary disturbance is made up of uncontrollable portions followed by controllable sections. In industrial practice controllers are sized, as for example as to power, to fit the system so that the disturbances encountered are primarily made up of uncontrollable sections followed by controllable portions of sufficient duration for the controller to bring the system to equilibrium. The control designer wishes to have optimal control for any disturbance made up of such an uncontrollable portion followed by a sufficiently long controllable section. Here this problem is solved with the aid of the maximum principle for the class of second order systems which describe almost all governor-engine applications to first approximation accuracy. Previous attempts to solve this problem involved assuming statistical properties of the disturbance thus severely restricting the class of applications. Here only those statistical properties required to implement optimal control are determined. A single control function is derived which suffices to yield optimal trajectories.