Linear control: Difference between revisions

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'''Linear control''' are [[control system]]s and [[control theory]] based on ''[[negative feedback]]'' for producing a [[control signal]] to maintain the controlled [[process variable]] (PV) at the desired [[Setpoint (control system)|setpoint]] (SP). There are several types of linear control systems with different capabilities.

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Proportional control is a type of linear feedback control system in which a correction is applied to the controlled variable which is proportional to the difference between the desired value (SP) and the measured value (PV). Two classic mechanical examples are the toilet bowl [[Ballcock|float proportioning valve]] and the [[Centrifugal governor|fly-ball governor]].

The proportional control system is more complex than an [[On–off control|on–off control system,]] but simpler than a [[PID controller|proportional-integral-derivative]] (PID) control system used, for instance, in an automobile [[cruise control]]. On–off control will work for systems that do not require high accuracy or responsiveness, but isare not effective for rapid and timely corrections and responses. Proportional control overcomes this by modulating the manipulated variable (MV), such as a [[control valve]], at a gain level that avoids instability, but applies correction as fast as practicable by applying the optimum quantity of proportional correction.

A drawback of proportional control is that it cannot eliminate the residual SP–PV error, as it requires an error to generate a proportional output. A [[PI controller]] can be used to overcome this. The PI controller uses a proportional term (P) to remove the gross error, and an integral term (I) to eliminate the residual offset error by integrating the error over time.

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When controlling the temperature of an [[industrial furnace]], it is usually better to control the opening of the fuel valve ''in proportion to'' the current needs of the furnace. This helps avoid thermal shocks and applies heat more effectively.

At low gains, only a small corrective action is applied when errors are detected. The system may be safe and stable, but may be sluggish in response to changing conditions. Errors will remain uncorrected for relatively long periods of time and the system is [[overdamped]]. If the proportional gain is increased, such systems become more responsive and errors are dealt with more quickly. There is an optimal value for the gain setting when the overall system is said to be [[critically damped]]. Increases in loop gain beyond this point lead to oscillations in the PV and such a system is [[underdamped]]. Adjusting gain to achieve critically damped behavior is known as ''tuning'' the control system.

In the underdamped case, the furnace heats quickly. Once the setpoint is reached, stored heat within the heater sub-system and in the walls of the furnace will keep the measured temperature rising beyond what is required. After rising above the setpoint, the temperature falls back and eventually heat is applied again. Any delay in reheating the heater sub-system allows the furnace temperature to fall further below the setpoint and the cycle repeats. The temperature oscillations that an underdamped furnace control system produces are undesirable.

In a critically damped system, as the temperature approaches the setpoint, the heat input begins to be reduced, the rate of heating of the furnace has time to slow and the system avoids overshoot. Overshoot is also avoided in an overdamped system but an overdamped system is unnecessarily slow to initially reach a setpoint respondresponse to external changes to the system, e.g. opening the furnace door.

==PID control==

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Some controllers include the option to limit the "ramp up % per minute". This option can be very helpful in stabilizing small boilers (3 MBTUH), especially during the summer, during light loads. A utility boiler "unit may be required to change load at a rate of as much as 5% per minute (IEA Coal Online - 2, 2007)".<ref>{{cite web |url=http://www.seeei.org.il/prdFiles/2702_desc3.pdf |access-date=2014-04-07 |url-status=live |archive-url=https://web.archive.org/web/20140805131600/http://www.seeei.org.il/prdFiles/2702_desc3.pdf |archive-date=2014-08-05 |publisher=ABB |title=Energy Efficient Design of Auxiliary Systems in Fossil-Fuel Power Plants |page=262 }}</ref>{{failed verification|reason=Source does not talk about controllers on this page.|date=May 2020}}

===Other techniques===

It is possible to [[filter (signal processing)|filter]] the PV or error signal. Doing so can help reduce instability or oscillations by reducing the response of the system to undesirable frequencies. Many systems have a [[resonant frequency]]. By filtering out that frequency, stronger overall feedback can be applied before oscillation occurs, making the system more responsive without shaking itself apart.

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==See also==

*[[Linear system]]

**[[Linear time-invariant system]]

*[[Nonlinear control]]