Where does overshoot collide?

Henry Wilbraham first described this behaviour in 1848: when a Fourier series (a sum of sine and cosine waves used to reproduce a repeating shape) approximates a function that jumps suddenly, the partial sums swing past the true value right beside the jump before settling. J. Willard Gibbs's 1899 letter Fourier's series in Nature analysed the effect, and Maxime Bôcher's 1906 Introduction to the theory of Fourier's series proved that the swing has a fixed size, close to nine percent of the jump, which does not shrink as more terms are added. The referent here is a fixed mathematical artefact in a series approximation near a discontinuity, and it has nothing to do with a machine settling on a target, a currency moving past its resting value, or a population exceeding its food supply.[1][2][3]

Unit

The peak excess of a Fourier partial sum near a jump, as a fraction of the jump height

Recovery means

Not applicable

Measured by

Computed as the limiting overshoot of the partial sums, about 8.95 percent of the jump, independent of the number of terms

James, Nichols, and Phillips set out the modern treatment in their 1947 Theory of servomechanisms, where overshoot names the amount by which a system's output rises above its final steady value while responding to a sudden change, before it settles back down. Katsuhiko Ogata's Modern control engineering gives the standard definition, usually reported as percent overshoot, the peak excess expressed as a fraction of the final value, which engineers trade off against how fast the system responds. The referent is a temporary excursion of a physical or electronic system past the level it is aiming for, distinct from a Fourier approximation's fixed swing near a jump, a currency's move past its long-run rate, or a population outrunning its resources.[4][5]

Unit

A response's peak excess above its final value, often reported as percent overshoot

Recovery means

The output settles back to the steady value over the settling time as the oscillations die out

Measured by

Read from the step-response curve as the peak value minus the final value, divided by the final value

Rudiger Dornbusch's 1976 Expectations and exchange rate dynamics introduced exchange-rate overshooting: because prices of goods adjust slowly while financial markets react at once, a change in the money supply drives the exchange rate past its eventual long-run level in the short term, after which it returns gradually. The paper became the standard account of why floating exchange rates can be so volatile even when the underlying economy moves smoothly. The referent is the temporary gap between a currency's immediate market price and its long-run value after a monetary shock, unrelated to a control system settling on a target, a Fourier sum's swing near a jump, or a population exceeding its habitat.[6]

Unit

The short-run gap between the spot exchange rate and its long-run level

Recovery means

The rate adjusts back toward its long-run value as goods prices slowly catch up

Measured by

Modelled as the deviation of the exchange rate from long-run equilibrium after a monetary shock

William Catton's 1980 book Overshoot: The ecological basis of revolutionary change gave the term its standard meaning in human ecology: overshoot is the condition in which a population grows beyond the number its environment can support, the carrying capacity, usually by drawing down resources faster than they renew. Because the excess is borrowed from a shrinking resource base, overshoot in this meaning tends to be followed by decline as that base gives out. The referent is a living population exceeding the limits of its habitat, which is a different thing from a machine's transient excursion, a currency moving past its resting rate, or the fixed swing of a Fourier series near a discontinuity.[7]

Unit

The amount by which a population exceeds its environment's carrying capacity

Recovery means

Often followed by decline as the depleted resource base gives out

Measured by

Estimated by comparing population size and resource demand against the carrying capacity of the habitat