When a second-order on-off or saturating control system is subject large step inputs, the best performance that can be achieved is to bring the error and its first derivative simultaneously to zero in the shortest possible time; there is then no overshoot. This response requires a single change of the output of the on-off component at the proper time before the error reaches zero. A proposal is made to achieve this response for all magnitudes of step input using a linear network and a relay in the feedback path of the control system. The proposal applies to linear and non-linear systems of second order with viscous damping, and to some higher-order systems.