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{{Short description|Feedback controller}}
[[File:Industrial control loop.jpg|thumb|300px|Example of a single industrial control loop; showing continuously modulated control of process flow.]]

[[File:Closed Control Loop.svg|thumb|300px|Illustration of a Closed Loop Control consisting of [[Setpoint (control system)|Set Point]] <math>w(t)</math>, [[Feedback|Measured Output]] <math>y_m(t)</math>, Measured Error <math>e(t)</math>, Controller Output <math>u(t)</math>, System Input <math>u_s(t)</math>, Disturbance <math>d(t)</math>, and System Output <math>y(t)</math>]]

A '''closed-loop controller''' or '''feedback controller''' is a [[control loop]] which incorporates [[feedback]], in contrast to an ''[[open-loop controller]]'' or ''non-feedback controller''.
A closed-loop controller uses feedback to control [[state (controls)|states]] or [[Negative feedback#Overview|outputs]] of a [[dynamical system]]. Its name comes from the information path in the system: process inputs (e.g., [[voltage]] applied to an [[electric motor]]) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with [[sensor]]s and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.<ref>{{Cite journal |last=Bechhoefer |first=John |date=2005-08-31 |title=Feedback for physicists: A tutorial essay on control |url=https://link.aps.org/doi/10.1103/RevModPhys.77.783 |journal=Reviews of Modern Physics |volume=77 |issue=3 |pages=783–836 |doi=10.1103/RevModPhys.77.783|url-access=subscription }}</ref>

In the case of linear [[feedback]] systems, a [[control loop]] including [[sensor]]s, control algorithms, and actuators is arranged in an attempt to regulate a variable at a [[Setpoint (control system)|setpoint]] (SP). An everyday example is the [[cruise control]] on a road vehicle; where external influences such as hills would cause speed changes, and the driver has the ability to alter the desired set speed. The [[PID algorithm]] in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or [[Overshoot (signal)|overshoot]], by controlling the power output of the vehicle's engine.
Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent. [[Open-loop controller|Open-loop control systems]] do not make use of feedback, and run only in pre-arranged ways.

Closed-loop controllers have the following advantages over open-loop controllers:
* disturbance rejection (such as hills in the cruise control example above)
* guaranteed performance even with [[mathematical model|model]] uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
* [[instability|unstable]] processes can be stabilized
* reduced sensitivity to parameter variations
* improved reference tracking performance
* improved rectification of random fluctuations<ref>{{Cite journal |last=Cao |first=F. J. |last2=Feito |first2=M. |date=2009-04-10 |title=Thermodynamics of feedback controlled systems |url=https://link.aps.org/doi/10.1103/PhysRevE.79.041118 |journal=Physical Review E |volume=79 |issue=4 |article-number=041118 |doi=10.1103/PhysRevE.79.041118|arxiv=0805.4824 }}</ref>

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed ''[[feed forward (control)|feedforward]]'' and serves to further improve reference tracking performance.

A common closed-loop controller architecture is the [[PID controller]].

[[File:Ideal feedback model.svg|thumb|right | A basic feedback loop]]

== Open-loop and closed-loop ==
{{excerpt|Control loop#Open-loop and closed-loop}}

==Closed-loop transfer function==
{{Main|Closed-loop transfer function}}
The output of the system ''y''(''t'') is fed back through a sensor measurement ''F'' to a comparison with the reference value ''r''(''t''). The controller ''C'' then takes the error ''e'' (difference) between the reference and the output to change the inputs ''u'' to the system under control ''P''. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (''SISO'') control system; ''MIMO'' (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through [[coordinate vector|vectors]] instead of simple [[scalar (mathematics)|scalar]] values. For some [[distributed parameter systems]] the vectors may be infinite-[[Dimension (vector space)|dimensional]] (typically functions).

[[File:simple feedback control loop2.svg|center|A simple feedback control loop]]

If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are [[linear]] and [[time-invariant]] (i.e., elements of their [[transfer function]] ''C''(''s''), ''P''(''s''), and ''F''(''s'') do not depend on time), the systems above can be analysed using the [[Laplace transform]] on the variables. This gives the following relations:

: <math>Y(s) = P(s) U(s)</math>
: <math>U(s) = C(s) E(s)</math>
: <math>E(s) = R(s) - F(s)Y(s).</math>

Solving for ''Y''(''s'') in terms of ''R''(''s'') gives

: <math>Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).</math>

The expression <math>H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}</math> is referred to as the ''closed-loop transfer function'' of the system. The numerator is the forward (open-loop) gain from ''r'' to ''y'', and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If <math>|P(s)C(s)| \gg 1</math>, i.e., it has a large [[norm (mathematics)|norm]] with each value of ''s'', and if <math>|F(s)| \approx 1</math>, then ''Y''(''s'') is approximately equal to ''R''(''s'') and the output closely tracks the reference input.

==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|''KP''}}, {{math|''KI''}} and {{math|''KD''}}, 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|''TP''}} are some constants. The plant output is fed back through

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

where {{math|''TF''}} is also a constant. Now if we set <math>K_P=K\left(1+\frac{T_D}{T_I}\right)</math>, {{math|1=''KD'' = ''KTD''}}, 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.

==References==
{{reflist}}

[[Category:Control theory]]

Closed-loop controller - Revision history

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