Optimal discrete-continuous control for the linear-quadratic regulator problem

In the conventional method of discrete-time control, the control-input is held constant across each sampling interval; i.e., "zero-order hold" type control. In this paper a previously introduced generalization of the conventional discrete-time control method, in which the control is allowed to vary with time (open-loop fashion) across each sampling interval is considered and applied to the optimal linear quadratic regulator (LQR) problem. This generalization of discrete-time control, called "discrete-continuous control", leads to significant performance improvements compared to conventional discrete-time control. An important special case of discrete-continuous control where control variations are constrained to be linear-in-time across each sampling-interval, is examined in detail and a new general LQR theory is developed for that special case. These latter results are illustrated by a worked numerical example with simulation plots that clearly demonstrate the LQR performance improvements obtained by (linear-in-time) discrete-continuous control.