Aclosed-loop controller orfeedback controller is acontrol loop which incorporatesfeedback, in contrast to anopen-loop controller ornon-feedback controller.A closed-loop controller uses feedback to controlstates oroutputs of adynamical system. Its name comes from the information path in the system: process inputs (e.g.,voltage applied to anelectric motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured withsensors and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.^[1]

In the case of linearfeedback systems, acontrol loop includingsensors, control algorithms, and actuators is arranged in an attempt to regulate a variable at asetpoint (SP). An everyday example is thecruise 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. ThePID algorithm in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay orovershoot, 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 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 withmodel uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstable processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance
• improved rectification of random fluctuations^[2]

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termedfeedforward and serves to further improve reference tracking performance.

A common closed-loop controller architecture is thePID controller.

The output of the systemy(t) is fed back through a sensor measurementF to a comparison with the reference valuer(t). The controllerC then takes the errore (difference) between the reference and the output to change the inputsu to the system under controlP. 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 throughvectors instead of simplescalar values. For somedistributed parameter systems the vectors may be infinite-dimensional (typically functions).

If we assume the controllerC, the plantP, and the sensorF arelinear andtime-invariant (i.e., elements of theirtransfer functionC(s),P(s), andF(s) do not depend on time), the systems above can be analysed using theLaplace transform on the variables. This gives the following relations:

${\displaystyle Y(s)=P(s)U(s)}$

${\displaystyle U(s)=C(s)E(s)}$

${\displaystyle E(s)=R(s)-F(s)Y(s).}$

Solving forY(s) in terms ofR(s) gives

${\displaystyle Y(s)=\left(\frac{P(s)C(s)}{1+P(s)C(s)F(s)}\right)R(s)=H(s)R(s).}$

The expression${\displaystyle H(s)=\frac{P(s)C(s)}{1+F(s)P(s)C(s)}}$ is referred to as theclosed-loop transfer function of the system. The numerator is the forward (open-loop) gain fromr toy, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If${\displaystyle |P(s)C(s)|\gg 1}$, i.e., it has a largenorm with each value ofs, and if${\displaystyle |F(s)|\approx 1}$, thenY(s) is approximately equal toR(s) and the output closely tracks the reference input.

A proportional–integral–derivative controller (PID controller) is acontrol loopfeedback mechanism control technique widely used in control systems.

A PID controller continuously calculates anerror valuee(t) as the difference between a desiredsetpoint and a measuredprocess variable and applies a correction based onproportional,integral, andderivative terms.PID is an initialism forProportional-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.

Ifu(t) is the control signal sent to the system,y(t) is the measured output andr(t) is the desired output, ande(t) =r(t) −y(t) is the tracking error, a PID controller has the general form

${\displaystyle u(t)=K_{P}e(t)+K_I{\int }^{t}e(\tau )\text{d}\tau +K_D\frac{\text{d}e(t)}{\text{d}t}.}$

The desired closed loop dynamics is obtained by adjusting the three parametersK_P,K_I andK_D, 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 inprocess 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 ifMIMO systems are considered.

ApplyingLaplace transformation results in the transformed PID controller equation

${\displaystyle u(s)=K_{P}e(s)+K_I\frac{1}{s}e(s)+K_{D}se(s)}$

${\displaystyle u(s)=\left(K_P+K_I\frac{1}{s}+K_{D}s\right)e(s)}$

with the PID controller transfer function

${\displaystyle C(s)=\left(K_P+K_I\frac{1}{s}+K_{D}s\right).}$

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

${\displaystyle P(s)=\frac{A}{1+s{T}_P}}$

whereA andT_P are some constants. The plant output is fed back through

${\displaystyle F(s)=\frac{1}{1+s{T}_F}}$

whereT_F is also a constant. Now if we set${\displaystyle K_P=K\left(1+\frac{T_D}{T_I}\right)}$,K_D =KT_D, and${\displaystyle K_I=\frac{K}{T_I}}$, we can express the PID controller transfer function in series form as

${\displaystyle C(s)=K\left(1+\frac{1}{s{T}_I}\right)(1+s{T}_D)}$

PluggingP(s),F(s), andC(s) into the closed-loop transfer functionH(s), we find that by setting

${\displaystyle K=\frac{1}{A},T_I=T_F,T_D=T_P}$

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^[6] due to amplification of noise and resonant modes in the system. Therefore, aphase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.

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