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Closed-loop controller

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A closed-loop controller or feedback controller is a <a href="Control_loop" title="Control loop">control loop</a> which incorporates <a href="Feedback" title="Feedback">feedback</a>, in contrast to an <a href="Open-loop_controller" title="Open-loop controller">open-loop controller</a> or non-feedback controller.

A closed-loop controller uses feedback to control <a href="State_(controls)" class="mw-redirect" title="State (controls)">states</a> or <a href="Negative_feedback#Overview" title="Negative feedback">outputs</a> of a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. Its name comes from the information path in the system: process inputs (e.g., <a href="Voltage" title="Voltage">voltage</a> applied to an <a href="Electric_motor" title="Electric motor">electric motor</a>) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with <a href="Sensor" title="Sensor">sensors</a> and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.<a href="#cite_note-1">[1]</a>

In the case of linear <a href="Feedback" title="Feedback">feedback</a> systems, a <a href="Control_loop" title="Control loop">control loop</a> including <a href="Sensor" title="Sensor">sensors</a>, control algorithms, and actuators is arranged in an attempt to regulate a variable at a <a href="Setpoint_(control_system)" title="Setpoint (control system)">setpoint</a> (SP). An everyday example is the <a href="Cruise_control" title="Cruise control">cruise control</a> 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 <a href="PID_algorithm" class="mw-redirect" title="PID algorithm">PID algorithm</a> in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or <a href="Overshoot_(signal)" title="Overshoot (signal)">overshoot</a>, 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. <a href="Open-loop_controller" title="Open-loop controller">Open-loop control systems</a> do not make use of feedback, and run only in pre-arranged ways.

Closed-loop controllers have the following advantages over open-loop controllers:

<ul><li>disturbance rejection (such as hills in the cruise control example above)</li>

<li>guaranteed performance even with <a href="Mathematical_model" title="Mathematical model">model</a> uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact</li>

<li><a href="Instability" title="Instability">unstable</a> processes can be stabilized</li>

<li>reduced sensitivity to parameter variations</li>

<li>improved reference tracking performance</li>

<li>improved rectification of random fluctuations<a href="#cite_note-2">[2]</a></li></ul>

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed <a href="Feed_forward_(control)" title="Feed forward (control)">feedforward</a> and serves to further improve reference tracking performance.

A common closed-loop controller architecture is the <a href="PID_controller" class="mw-redirect" title="PID controller">PID controller</a>.

<div class="mw-heading mw-heading2"><h2 id="Open-loop_and_closed-loop">Open-loop and closed-loop</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Control_loop#Open-loop_and_closed-loop" title="Control loop">Control loop § Open-loop and closed-loop</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Control_loop&action=edit#Open-loop_and_closed-loop">edit</a>]</div><div class="excerpt">

Fundamentally, there are two types of control loop: <a href="Open-loop_control" class="mw-redirect" title="Open-loop control">open-loop control</a> (feedforward), and <a href="Closed-loop_control" class="mw-redirect" title="Closed-loop control">closed-loop control</a> (feedback).

<ul><li>In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable"). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature.</li>

<li>In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy, this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.<a href="#cite_note-Control_loop_auto-3">[3]</a></li></ul>

The definition of a closed loop control system according to the <a href="British_Standards_Institution" class="mw-redirect" title="British Standards Institution">British Standards Institution</a> is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."<a href="#cite_note-4">[4]</a>

Likewise; "A Feedback Control System is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."<a href="#cite_note-5">[5]</a></div></div>

<div class="mw-heading mw-heading2"><h2 id="Closed-loop_transfer_function">Closed-loop transfer function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Closed-loop_transfer_function" title="Closed-loop transfer function">Closed-loop transfer function</a></div>

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 <a href="Coordinate_vector" title="Coordinate vector">vectors</a> instead of simple <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> values. For some <a href="Distributed_parameter_systems" class="mw-redirect" title="Distributed parameter systems">distributed parameter systems</a> the vectors may be infinite-<a href="Dimension_(vector_space)" title="Dimension (vector space)">dimensional</a> (typically functions).

If we assume the controller C, the plant P, and the sensor F are <a href="Linear" class="mw-redirect" title="Linear">linear</a> and <a href="Time-invariant" class="mw-redirect" title="Time-invariant">time-invariant</a> (i.e., elements of their <a href="Transfer_function" title="Transfer function">transfer function</a> C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> on the variables. This gives the following relations:

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<annotation encoding="application/x-tex">{\displaystyle Y(s)=P(s)U(s)}</annotation>

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</math><img src="./a017597b49ed2396ca6b2dd6685c5b67091bee0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.099ex; height:2.843ex;" alt="{\displaystyle Y(s)=P(s)U(s)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle U(s)=C(s)E(s)}</annotation>

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</math><img src="./1b6f7688193f812a5fa3c3609c3d250529044993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.122ex; height:2.843ex;" alt="{\displaystyle U(s)=C(s)E(s)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle E(s)=R(s)-F(s)Y(s).}</annotation>

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</math><img src="./a787e2eecc1f81323f15d20384ddf2ad2c419032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.238ex; height:2.843ex;" alt="{\displaystyle E(s)=R(s)-F(s)Y(s).}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).}</annotation>

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</math><img src="./2ded87a715d1d1b77ddcf2cb4bd88997f540f9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.407ex; height:6.509ex;" alt="{\displaystyle Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).}" loading="lazy"></dd></dl>

The expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}">

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<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}</annotation>

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</math><img src="./f2490516044765db8e04dcd227c81a27716c5d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.853ex; height:6.509ex;" alt="{\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |P(s)C(s)|\gg 1}">

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<annotation encoding="application/x-tex">{\displaystyle |P(s)C(s)|\gg 1}</annotation>

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</math><img src="./2731f366bed2c595293672f798a869f770fbff2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.381ex; height:2.843ex;" alt="{\displaystyle |P(s)C(s)|\gg 1}" loading="lazy">, i.e., it has a large <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> with each value of s, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |F(s)|\approx 1}">

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<annotation encoding="application/x-tex">{\displaystyle |F(s)|\approx 1}</annotation>

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</math><img src="./37beeed5677b9485b73900c76cb56f10a68bc3af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.195ex; height:2.843ex;" alt="{\displaystyle |F(s)|\approx 1}" loading="lazy">, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input.

<div class="mw-heading mw-heading2"><h2 id="PID_feedback_control">PID feedback control</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="PID_controller" class="mw-redirect" title="PID controller">PID controller</a></div>

A proportional–integral–derivative controller (PID controller) is a <a href="Control_loop" title="Control loop">control loop</a> <a href="Feedback_mechanism" class="mw-redirect" title="Feedback mechanism">feedback mechanism</a> control technique widely used in control systems.

A PID controller continuously calculates an error value e(t) as the difference between a desired <a href="Setpoint_(control_system)" title="Setpoint (control system)">setpoint</a> and a measured <a href="Process_variable" title="Process variable">process variable</a> and applies a correction based on <a href="Proportional_control" title="Proportional control">proportional</a>, <a href="Integral" title="Integral">integral</a>, and <a href="Derivative" title="Derivative">derivative</a> 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 u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and e(t) = r(t) − y(t) is the tracking error, a PID controller has the general form

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<annotation encoding="application/x-tex">{\displaystyle 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}}.}</annotation>

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</math><img src="./a70350ae24e2b591fe820a70a79ea021d70c8800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.691ex; height:6.176ex;" alt="{\displaystyle 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}}.}" loading="lazy"></dd></dl>

The desired closed loop dynamics is obtained by adjusting the three parameters KP, KI and 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 <a href="Process_control" class="mw-redirect" title="Process control">process control</a>). 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 <a href="MIMO" title="MIMO">MIMO</a> systems are considered.

Applying <a href="Laplace_transform" title="Laplace transform">Laplace transformation</a> results in the transformed PID controller equation

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<annotation encoding="application/x-tex">{\displaystyle u(s)=K_{P}\,e(s)+K_{I}\,{\frac {1}{s}}\,e(s)+K_{D}\,s\,e(s)}</annotation>

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</math><img src="./d2e66877eb5e7242582e1481e46c729a02b08c8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.022ex; height:5.176ex;" alt="{\displaystyle u(s)=K_{P}\,e(s)+K_{I}\,{\frac {1}{s}}\,e(s)+K_{D}\,s\,e(s)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle u(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right)e(s)}</annotation>

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</math><img src="./a0299521032d9508677b8d0e855a214adfb08231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.703ex; height:6.176ex;" alt="{\displaystyle u(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right)e(s)}" loading="lazy"></dd></dl>

with the PID controller transfer function

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<annotation encoding="application/x-tex">{\displaystyle C(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right).}</annotation>

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</math><img src="./2d08c8a2d3710bbbe43f1504eed9a61519284e4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.803ex; height:6.176ex;" alt="{\displaystyle C(s)=\left(K_{P}+K_{I}\,{\frac {1}{s}}+K_{D}\,s\right).}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle P(s)={\frac {A}{1+sT_{P}}}}</annotation>

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</math><img src="./971eedbf43e9b4ae951eecf88a2d0363d427a80e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.497ex; height:5.676ex;" alt="{\displaystyle P(s)={\frac {A}{1+sT_{P}}}}" loading="lazy"></dd></dl>

where A and TP are some constants. The plant output is fed back through

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<annotation encoding="application/x-tex">{\displaystyle F(s)={\frac {1}{1+sT_{F}}}}</annotation>

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</math><img src="./4b7ee165b5becac24c5ba7e6b79cafe41b42d1a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.489ex; height:5.509ex;" alt="{\displaystyle F(s)={\frac {1}{1+sT_{F}}}}" loading="lazy"></dd></dl>

where TF is also a constant. Now if we set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}</annotation>

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</math><img src="./a9a4507c23017bc3b4a418604e0532deef298632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.202ex; height:6.176ex;" alt="{\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}" loading="lazy">, KD = KTD, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{I}={\frac {K}{T_{I}}}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{I}={\frac {K}{T_{I}}}}</annotation>

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</math><img src="./ae35c621895197d1b4326d9a78cc15614a349317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.387ex; height:5.509ex;" alt="{\displaystyle K_{I}={\frac {K}{T_{I}}}}" loading="lazy">, we can express the PID controller transfer function in series form as

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<annotation encoding="application/x-tex">{\displaystyle C(s)=K\left(1+{\frac {1}{sT_{I}}}\right)(1+sT_{D})}</annotation>

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</math><img src="./2fa1f945339571ff1e8fbb4ae781e00906b361b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.227ex; height:6.176ex;" alt="{\displaystyle C(s)=K\left(1+{\frac {1}{sT_{I}}}\right)(1+sT_{D})}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}</annotation>

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</math><img src="./601cadb75337fbd250b242c078d4d9a106751e4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.022ex; height:5.343ex;" alt="{\displaystyle K={\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}" loading="lazy"></dd></dl>

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<a href="#cite_note-6">[6]</a> due to amplification of noise and resonant modes in the system. Therefore, a <a href="Lead%E2%80%93lag_compensator" title="Lead–lag compensator">phase-lead compensator</a> type approach or a differentiator with low-pass roll-off are used instead.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBechhoefer2005" class="citation journal cs1">Bechhoefer, John (2005-08-31). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.77.783">"Feedback for physicists: A tutorial essay on control"</a>. Reviews of Modern Physics. 77 (3): 783–836. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.77.783">10.1103/RevModPhys.77.783</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFCaoFeito2009" class="citation journal cs1">Cao, F. J.; Feito, M. (2009-04-10). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevE.79.041118">"Thermodynamics of feedback controlled systems"</a>. Physical Review E. 79 (4): 041118. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0805.4824">0805.4824</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.79.041118">10.1103/PhysRevE.79.041118</a>.</cite>

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<li id="cite_note-Control_loop_auto-3"><a href="#cite_ref-Control_loop_auto_3-0">^</a> "Feedback and control systems" - JJ Di Steffano, AR Stubberud, IJ Williams. Schaums outline series, McGraw-Hill 1967

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFMayr1970" class="citation book cs1"><a href="Otto_Mayr" title="Otto Mayr">Mayr, Otto</a> (1970). The Origins of Feedback Control. Clinton, MA US: The Colonial Press, Inc.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMayr1969" class="citation book cs1"><a href="Otto_Mayr" title="Otto Mayr">Mayr, Otto</a> (1969). The Origins of Feedback Control. Clinton, MA US: The Colonial Press, Inc.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFAngChongLi2005" class="citation journal cs1">Ang, K.H.; Chong, G.C.Y.; Li, Y. (2005). <a rel="nofollow" class="external text" href="http://eprints.gla.ac.uk/3817/1/IEEE3.pdf">"PID control system analysis, design, and technology"</a> (PDF). IEEE Transactions on Control Systems Technology. 13 (4): 559–576. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCST.2005.847331">10.1109/TCST.2005.847331</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:921620">921620</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131213200556/http://eprints.gla.ac.uk/3817/1/IEEE3.pdf">Archived</a> (PDF) from the original on 2013-12-13.</cite>

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