Why can delayed feedback overshoot or oscillate?

A room can keep warming while a controller is still responding to an earlier, cooler reading. Feedback uses a measured deviation from a target to choose a correction. A correction intended to reduce that deviation can overshoot if it is too strong or based on old information. In the simple mathematical example below, the measured value can still be positive even after the current value has reached zero.

This unitless model has target zero and starts with x[−1] = x[0] = 1. Each update follows x[k+1] = x[k] − g × x[k−d]. The index k counts steps, g is the correction gain, and d is either zero for the current measurement or one for a measurement one step old.

At gain 0.5 without delay, the values begin 1, 0.5, 0.25 and 0.125, approaching zero. With a one-step delay they begin 1, 0.5, 0 and −0.25. Even when the current value reaches zero, the old positive measurement still commands a negative correction, carrying the state past its target.

Delay does not always cause instability. At gain 0.5, the sizes of successive oscillations shrink. At gain 1.2, the no-delay model alternates while shrinking, but the delayed model's peak sizes grow over time. Timing changes the effect of the same correction rule.

A controller can therefore keep correcting a condition that has already changed. The issue is not simply whether the response points toward the target when it was calculated; it is whether that response still fits when it takes effect. Correction strength and information age work together. The same rule can settle smoothly, cross the target and settle, or drive increasing oscillations as those settings change.