Editing Closed-loop controller (section)

==PID feedback control==
{{main|PID controller}}
[[File:PID en.svg|right|thumb|400x400px|A [[block diagram]] of a PID controller in a feedback loop, {{math|''r''(''t'')}} is the desired process value or "set point", and {{math|''y''(''t'')}} is the measured process value.]]
A proportional–integral–derivative controller (PID controller) is a  [[control loop]] [[feedback mechanism]] control technique widely  used in control systems.

A PID controller continuously calculates an ''error value'' {{math|''e''(''t'')}} as the difference between a desired [[Setpoint (control system)|setpoint]] and a measured [[process variable]] and applies a correction based on [[Proportional control|proportional]], [[integral]], and [[derivative]] terms. ''PID'' is an initialism for ''Proportional-Integral-Derivative'', referring to the three terms operating on the error signal to produce a control signal.

The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers.
The PID controller is probably the most-used feedback control design.

If {{math|''u''(''t'')}} is the control signal sent to the system, {{math|''y''(''t'')}} is the measured output and {{math|''r''(''t'')}} is the desired output, and {{math|1=''e''(''t'') = ''r''(''t'') − ''y''(''t'')}} is the tracking error, a PID controller has the general form

:<math>u(t) =  K_P e(t) + K_I \int^t e(\tau)\text{d}\tau + K_D \frac{\text{d}e(t)}{\text{d}t}.</math>

The desired closed loop dynamics is obtained by adjusting the three parameters {{math|''K<sub>P</sub>''}}, {{math|''K<sub>I</sub>''}} and {{math|''K<sub>D</sub>''}}, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in [[process control]]). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if [[MIMO]] systems are considered.

Applying [[Laplace transform]]ation results in the transformed PID controller equation

:<math>u(s) =  K_P \, e(s) + K_I \, \frac{1}{s} \, e(s) + K_D \, s \, e(s)</math>
:<math>u(s) =  \left(K_P + K_I \, \frac{1}{s} + K_D \, s\right) e(s)</math>

with the PID controller transfer function
:<math>C(s) = \left(K_P + K_I \, \frac{1}{s} + K_D \, s\right).</math>

As an example of tuning a PID controller in the closed-loop system {{math|''H''(''s'')}}, consider a 1st order plant given by

:<math>P(s) = \frac{A}{1 + sT_P}</math>

where {{mvar|A}} and {{math|''T<sub>P</sub>''}} are some constants. The plant output is fed back through

:<math>F(s) = \frac{1}{1 + sT_F}</math>

where {{math|''T<sub>F</sub>''}} is also a constant. Now if we set <math>K_P=K\left(1+\frac{T_D}{T_I}\right)</math>, {{math|1=''K<sub>D</sub>'' = ''KT<sub>D</sub>''}}, and <math>K_I=\frac{K}{T_I}</math>, we can express the PID controller transfer function in series form as

:<math>C(s) =  K \left(1 + \frac{1}{sT_I}\right)(1 + sT_D)</math>

Plugging {{math|''P''(''s'')}}, {{math|''F''(''s'')}}, and {{math|''C''(''s'')}} into the closed-loop transfer function {{math|''H''(''s'')}}, we find that by setting

:<math>K = \frac{1}{A},   T_I = T_F,   T_D = T_P</math>

{{math|1=''H''(''s'') = 1}}. With this tuning in this example, the system output follows the reference input exactly.

However, in practice, a pure differentiator is neither physically realizable nor desirable<ref>{{cite journal |last1=Ang |first1=K.H. |last2=Chong |first2=G.C.Y. |last3=Li |first3=Y. |date=2005 |title=PID control system analysis, design, and technology |journal=IEEE Transactions on Control Systems Technology |volume=13 |issue=4 |pages=559–576|doi=10.1109/TCST.2005.847331 |s2cid=921620 |url=http://eprints.gla.ac.uk/3817/1/IEEE3.pdf |archive-url=https://web.archive.org/web/20131213200556/http://eprints.gla.ac.uk/3817/1/IEEE3.pdf |archive-date=2013-12-13 |url-status=live }}</ref> due to amplification of noise and resonant modes in the system. Therefore, a [[Lead–lag compensator|phase-lead compensator]] type approach or a differentiator with low-pass roll-off are used instead.