Editing Proportional–integral–derivative controller (section)

===Relay (Åström–Hägglund) method===
Published in 1984 by [[Karl Johan Åström]] and Tore Hägglund,<ref>{{cite journal |last1=Åström |first1=K.J. |last2=Hägglund |first2=T. |title=Automatic Tuning of Simple Regulators |journal=IFAC Proceedings Volumes |date=July 1984 |volume=17 |issue=2 |pages=1867–1872 |doi=10.1016/S1474-6670(17)61248-5 |url=https://lup.lub.lu.se/record/8601786 |doi-access=free }}</ref> the relay method temporarily operates the process using [[bang-bang control]] and measures the resultant oscillations.  The output is switched (as if by a [[relay]], hence the name) between two values of the control variable.  The values must be chosen so the process will cross the setpoint, but they need not be 0% and 100%; by choosing suitable values, dangerous oscillations can be avoided.

As long as the process variable is below the setpoint, the control output is set to the higher value.  As soon as it rises above the setpoint, the control output is set to the lower value.  Ideally, the output waveform is nearly square, spending equal time above and below the setpoint.  The period and amplitude of the resultant oscillations are measured, and used to compute the ultimate gain and period, which are then fed into the Ziegler–Nichols method.

Specifically, the ultimate period <math>T_u</math> is assumed to be equal to the observed period, and the ultimate gain is computed as <math>K_u = 4b/\pi a,</math> where {{mvar|a}} is the amplitude of the process variable oscillation, and {{mvar|b}} is the amplitude of the control output change which caused it.

There are numerous variants on the relay method.<ref>{{cite journal |title=A Review of Relay Auto-tuning Methods for the Tuning of PID-type Controllers |first=Stephen |last=Hornsey |journal=Reinvention |volume=5 |issue=2 |date=29 October 2012 |url=http://www2.warwick.ac.uk/fac/cross_fac/iatl/reinvention/issues/volume5issue2/hornsey}}</ref>