Time Optimal Control Within a Dual Mode Constraint

A design is reported for a fast time positional control system of third order which has two modes of operation; a first bang-bang mode operates until the position is close to target, followed by a second linear feedback mode which gives asymptotic stability. The design is applicable to situations in which the positional state is to be moved repeatedly among a finite set of target positions. The switch times for the first mode are selected to minimize the time to reach the mode boundary plus a terminal penalty which is the settling time of the second mode. In the example given, the response time using this controller is compared with the optimal time to the target, and with the response time of equivalent saturating proportional control.