Editing Proportional–integral–derivative controller (section)

===Origins===
The [[centrifugal governor]] was invented by [[Christiaan Huygens]] in the 17th century to regulate the gap between [[millstone]]s in [[windmill]]s depending on the speed of rotation, and thereby compensate for the variable speed of grain feed.<ref>{{citation|last=Hills|first=Richard L|authorlink=Richard L. Hills|title=Power From the Wind|publisher=Cambridge University Press|year=1996}}</ref><ref>{{cite book |title=Adaptive Control Processes: A Guided Tour |author=Richard E. Bellman |publisher=Princeton University Press |date=December 8, 2015 |url=https://books.google.com/books?id=iwbWCgAAQBAJ&q=%22Centrifugal+Governor%22+Huygens&pg=PA36 |isbn=9781400874668 }}</ref>

With the invention of the low-pressure stationary steam engine there was a need for automatic speed control, and [[James Watt]]'s self-designed "[[conical pendulum]]" governor, a set of revolving steel balls attached to a vertical spindle by link arms, came to be an industry standard. This was based on the millstone-gap control concept.<ref name="ben96" />

Rotating-governor speed control, however, was still variable under conditions of varying load, where the shortcoming of what is now known as proportional control alone was evident. The error between the desired speed and the actual speed would increase with increasing load. In the 19th century, the theoretical basis for the operation of governors was first described by [[James Clerk Maxwell]] in 1868 in his now-famous paper ''On Governors''. He explored the mathematical basis for control stability, and progressed a good way towards a solution, but made an appeal for mathematicians to examine the problem.<ref>{{cite journal |first=J. C. |last=Maxwell |author-link=James Clerk Maxwell |title=On Governors |date=1868 |journal=Proceedings of the Royal Society |volume=100 |url=https://upload.wikimedia.org/wikipedia/commons/b/b1/On_Governors.pdf}}</ref><ref name="ben96" />  The problem was examined further in 1874 by [[Edward Routh]], [[Charles Sturm]], and in 1895, [[Adolf Hurwitz]], all of whom contributed to the establishment of control stability criteria.<ref name="ben96" />
In subsequent applications, speed governors were further refined, notably by American scientist [[Willard Gibbs]], who in 1872 theoretically analyzed Watt's conical pendulum governor.

About this time, the invention of the [[Whitehead torpedo]] posed a control problem that required accurate control of the running depth. Use of a depth pressure sensor alone proved inadequate, and a pendulum that measured the fore and aft pitch of the torpedo was combined with depth measurement to become the [[pendulum-and-hydrostat control]]. Pressure control provided only a proportional control that, if the control gain was too high, would become unstable and go into overshoot with considerable [[instability]] of depth-holding. The pendulum added what is now known as derivative control, which damped the oscillations by detecting the torpedo dive/climb angle and thereby the rate-of-change of depth.<ref>{{cite book |last=Newpower |first=Anthony |title=Iron Men and Tin Fish: The Race to Build a Better Torpedo during World War II |publisher=Praeger Security International |year=2006 |isbn=978-0-275-99032-9}} p.&nbsp; citing {{citation |first=Edwyn |last=Gray |title=The Devil's Device: Robert Whitehead and the History of the Torpedo |location=Annapolis, MD |publisher=U.S. Naval Institute |date=1991 |page=33 |ref=none}}.</ref> This development (named by Whitehead as  "The Secret" to give no clue to its action) was around 1868.<ref>{{citation |last=Sleeman |first=C. W. |title=Torpedoes and Torpedo Warfare |date=1880 |location=Portsmouth |publisher=Griffin & Co. |pages=137&ndash;138 |url=https://archive.org/stream/torpedoestorpedo00sleerich#page/136/mode/2up/search/depth |quote=which constitutes what is termed as the secret of the fish torpedo. |ref=none}}</ref>

Another early example of a PID-type controller was developed by [[Elmer Sperry]] in 1911 for ship steering, though his work was intuitive rather than mathematically-based.<ref>{{cite web |url=http://www.building-automation-consultants.com/building-automation-history.html |title=A Brief Building Automation History |access-date=2011-04-04 |url-status=dead |archive-url=https://web.archive.org/web/20110708104028/http://www.building-automation-consultants.com/building-automation-history.html |archive-date=2011-07-08 }}</ref>

It was not until 1922, however, that a formal control law for what we now call PID or three-term control was first developed using theoretical analysis, by [[Russian American]] engineer [[Nicolas Minorsky]].<ref>{{cite journal |last=Minorsky |first=Nicolas |author-link=Nicolas Minorsky |title=Directional stability of automatically steered bodies |journal=Journal of the American Society for Naval Engineers |year=1922 |volume=34 |pages=280–309 |issue=2 |doi=10.1111/j.1559-3584.1922.tb04958.x}}</ref> Minorsky was researching and designing automatic ship steering for the US Navy and based his analysis on observations of a [[helmsman]]. He noted the helmsman steered the ship based not only on the current course error but also on past error, as well as the current rate of change;<ref>{{Harvnb|Bennett|1993|loc = [https://books.google.com/books?id=VD_b81J3yFoC&pg=PA67 p. 67]}}</ref> this was then given a mathematical treatment by Minorsky.<ref name="ben96">{{cite journal
| journal = IEEE Control Systems Magazine
| volume = 16
| issue = 3
| last = Bennett
| first = Stuart
| title = A brief history of automatic control
| year = 1996
| url = http://ieeecss.org/CSM/library/1996/june1996/02-HistoryofAutoCtrl.pdf
| pages = 17–25
| doi = 10.1109/37.506394
| access-date = 2014-08-21
| archive-url = https://web.archive.org/web/20160809050823/http://ieeecss.org/CSM/library/1996/june1996/02-HistoryofAutoCtrl.pdf
| archive-date = 2016-08-09
| url-status = dead
}}</ref>
His goal was stability, not general control, which simplified the problem significantly. While proportional control provided stability against small disturbances, it was insufficient for dealing with a steady disturbance, notably a stiff gale (due to [[#Steady-state error|steady-state error]]), which required adding the integral term. Finally, the derivative term was added to improve stability and control.

Trials were carried out on the [[USS New Mexico (BB-40)|USS ''New Mexico'']], with the controllers controlling the ''[[angular velocity]]'' (not the angle) of the rudder. PI control yielded sustained yaw (angular error) of ±2°. Adding the D element yielded a yaw error of ±1/6°, better than most helmsmen could achieve.<ref>{{cite book
| publisher = IET
| isbn = 978-0-86341-047-5
| last = Bennett
| first = Stuart
| title = A history of control engineering, 1800-1930
|date=June 1986
| pages = [https://books.google.com/books?id=1gfKkqB_fTcC&pg=PA142 142–148]
}}</ref>

The Navy ultimately did not adopt the system due to resistance by personnel. Similar work was carried out and published by several others{{who|date=December 2023}} in the 1930s.{{citation needed|date=December 2023}}