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A closed-loop controller or feedback controller is a
control loop A control loop is the fundamental building block of control systems in general and industrial control systems in particular. It consists of the process sensor, the controller function, and the final control element (FCE) which controls the proce ...
which incorporates
feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handle ...
, in contrast to an '' open-loop controller'' or ''non-feedback controller''. A closed-loop controller uses feedback to control
states State most commonly refers to: * State (polity), a centralized political organization that regulates law and society within a territory **Sovereign state, a sovereign polity in international law, commonly referred to as a country **Nation state, a ...
or outputs of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
. Its name comes from the information path in the system: process inputs (e.g.,
voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
applied to an
electric motor An electric motor is a machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a electromagnetic coil, wire winding to gene ...
) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with
sensor A sensor is often defined as a device that receives and responds to a signal or stimulus. The stimulus is the quantity, property, or condition that is sensed and converted into electrical signal. In the broadest definition, a sensor is a devi ...
s and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop. In the case of linear
feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handle ...
systems, a
control loop A control loop is the fundamental building block of control systems in general and industrial control systems in particular. It consists of the process sensor, the controller function, and the final control element (FCE) which controls the proce ...
including
sensor A sensor is often defined as a device that receives and responds to a signal or stimulus. The stimulus is the quantity, property, or condition that is sensed and converted into electrical signal. In the broadest definition, a sensor is a devi ...
s, control algorithms, and actuators is arranged in an attempt to regulate a variable at a 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, 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 with
model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...
uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact *
unstable In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
processes can be stabilized * reduced sensitivity to parameter variations * improved reference tracking performance * improved rectification of random fluctuations In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed '' feedforward'' and serves to further improve reference tracking performance. A common closed-loop controller architecture is the PID controller.


Open-loop and closed-loop


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 vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite- dimensional (typically functions). If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
and time-invariant (i.e., elements of their
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
''C''(''s''), ''P''(''s''), and ''F''(''s'') do not depend on time), the systems above can be analysed using the
Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
on the variables. This gives the following relations: : Y(s) = P(s) U(s) : U(s) = C(s) E(s) : E(s) = R(s) - F(s)Y(s). Solving for ''Y''(''s'') in terms of ''R''(''s'') gives : Y(s) = \left( \frac \right) R(s) = H(s)R(s). The expression H(s) = \frac 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 , P(s)C(s), \gg 1, i.e., it has a large norm with each value of ''s'', and if , F(s), \approx 1, then ''Y''(''s'') is approximately equal to ''R''(''s'') and the output closely tracks the reference input.


PID feedback control

A proportional–integral–derivative controller (PID controller) is a
control loop A control loop is the fundamental building block of control systems in general and industrial control systems in particular. It consists of the process sensor, the controller function, and the final control element (FCE) which controls the proce ...
feedback mechanism control technique widely used in control systems. A PID controller continuously calculates an ''error value'' as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional,
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
, and
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 is the control signal sent to the system, is the measured output and is the desired output, and is the tracking error, a PID controller has the general form :u(t) = K_P e(t) + K_I \int^t e(\tau)\text\tau + K_D \frac. The desired closed loop dynamics is obtained by adjusting the three parameters , and , 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 Industrial process control (IPC) or simply process control is a system used in modern manufacturing which uses the principles of control theory and physical industrial control systems to monitor, control and optimize continuous Industrial processe ...
). 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 In radio, multiple-input and multiple-output (MIMO) () is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wirel ...
systems are considered. Applying
Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
ation results in the transformed PID controller equation :u(s) = K_P \, e(s) + K_I \, \frac \, e(s) + K_D \, s \, e(s) :u(s) = \left(K_P + K_I \, \frac + K_D \, s\right) e(s) with the PID controller transfer function :C(s) = \left(K_P + K_I \, \frac + K_D \, s\right). As an example of tuning a PID controller in the closed-loop system , consider a 1st order plant given by :P(s) = \frac where and are some constants. The plant output is fed back through :F(s) = \frac where is also a constant. Now if we set K_P=K\left(1+\frac\right), , and K_I=\frac, we can express the PID controller transfer function in series form as :C(s) = K \left(1 + \frac\right)(1 + sT_D) Plugging , , and into the closed-loop transfer function , we find that by setting :K = \frac, T_I = T_F, T_D = T_P . 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 due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.


References

{{reflist Control theory