Editing Negative feedback (section)

===Error-controlled regulation===

[[File:Basic error-controlled regulator.svg|thumb|Basic error-controlled regulator loop]]

{{See also|Control engineering|Homeostasis|Allostasis}}

[[File:Regulator with feedback.png|thumb|200px|A regulator ''R'' adjusts the input to a system ''T'' so the monitored essential variables ''E'' are held to set-point values ''S'' that result in the desired system output despite disturbances ''D''.<ref name=Ashby/><ref name=Bhagade>{{cite book |title=Process Dynamics and Control |author1=Sudheer S Bhagade |author2=Govind Das Nageshwar |url=https://books.google.com/books?id=rD0xPl56hZEC&pg=PA9 |isbn=9788120344051 |year=2011 |publisher=PHI Learning Pvt. Ltd |pages=6, 9}}</ref>]]

One use of feedback is to make a system (say ''T'') [[Homeostasis|self-regulating]] to minimize the effect of a disturbance (say ''D''). Using a negative feedback loop, a measurement of some variable (for example, a  [[process variable]], say ''E'') is [[Subtraction|subtracted]] from a required value (the [[Setpoint (control system)|'set point']]) to estimate an operational error in system status, which is then used by a [[Regulator (automatic control)|regulator]] (say ''R'') to reduce the gap between the measurement and the required value.<ref name=Wilts>
{{cite book |author=Charles H. Wilts |title=Principles of Feedback Control |url=https://archive.org/details/principlesoffeed00wilt |url-access=registration |publisher=Addison-Wesley Pub. Co |year=1960 |page=[https://archive.org/details/principlesoffeed00wilt/page/1 1] |quote=In a simple feedback system a specific physical quantity is being controlled, and control is brought about by making an actual comparison of this quantity with its desired value and utilizing the difference to reduce the error observed. Such a system is self-correcting in the sense that any deviations from the desired performance are used to produce corrective action.}}</ref><ref name=Singh>{{cite book |title=Process Control: Concepts Dynamics And Applications |author=SK Singh |url=https://books.google.com/books?id=CRQr3HvzN40C&pg=PA222 |isbn=9788120336780 |year=2010 |publisher=PHI Learning Pvt. Ltd |page=222}}</ref> The regulator modifies the input to the system ''T'' according to its interpretation of the error in the status of the system. This error may be introduced by a variety of possible disturbances or 'upsets', some slow and some rapid.<ref name= Svrcek>For example, input and load disturbances. See {{cite book |title=A Real-Time Approach to Process Control |author1=William Y. Svrcek |author2=Donald P. Mahoney |author3=Brent R. Young |url=https://books.google.com/books?id=WnFPAgAAQBAJ&pg=PA57 |page=57 |isbn=9781118684733 |year=2013 |edition=3rd |publisher=John Wiley & Sons}}</ref> The [[Controller (control theory)|regulation]] in such systems can range from a simple 'on-off' control to a more complex processing of the error signal.<ref name=Exeter>{{cite web |title= Types of feedback control |url=http://newton.ex.ac.uk/teaching/cdhw/Feedback/ControlTypes.html |publisher=University of Exeter: Physics and astronomy |work=Feedback and temperature control |author=Charles D H Williams|access-date=2014-06-08}}</ref>

In this framework, the physical form of a signal may undergo multiple transformations. For example, a change in weather may cause a disturbance to the ''heat'' input to a house (as an example of the system ''T'') that is monitored by a thermometer as a change in ''temperature'' (as an example of an 'essential variable' ''E''). This quantity, then, is converted by the thermostat (a 'comparator') into an ''electrical'' error in status compared to the 'set point' ''S'', and subsequently used by the [[Regulator (automatic control)|regulator]] (containing a 'controller' that commands ''gas'' control valves and an ignitor) ultimately to change the ''heat'' provided by a furnace (an 'effector') to counter the initial weather-related disturbance in heat input to the house.<ref>{{Cite journal|last1=Giannini|first1=Alessandra|last2=Biasutti|first2=Michela|last3=Verstraete|first3=Michel M.|date=2008-12-01|title=A climate model-based review of drought in the Sahel: Desertification, the re-greening and climate change|journal=Global and Planetary Change|series=Climate Change and Desertification|volume=64|issue=3|pages=119–128|doi=10.1016/j.gloplacha.2008.05.004|issn=0921-8181|bibcode=2008GPC....64..119G}}</ref>

Error controlled regulation is typically carried out using a Proportional-Integral-Derivative Controller ([[PID controller]]). The regulator signal is derived from a weighted sum of the error signal, integral of the error signal, and derivative of the error signal. The weights of the respective components depend on the application.<ref name=Bechhoefer>{{cite journal | last = Bechhoefer | first = John | title = Feedback for Physicists: A Tutorial Essay On Control | journal = Reviews of Modern Physics | volume = 77 | issue = 3 | pages = 783–835 | doi=10.1103/revmodphys.77.783| citeseerx = 10.1.1.124.7043 | year = 2005 | bibcode = 2005RvMP...77..783B }}</ref>

Mathematically, the regulator signal is given by:
:<math>\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{d}{dt}e(t)\right)</math>
where
:<math>T_i</math>  is the ''integral time''
:<math>T_d</math>  is the ''derivative time''