PID controller: Encyclopedia II - PID controller - Theory

PID controller - Theory

Development of PID control originated from the observation that a proportional-only control can only eliminate the error between setpoint and process variable at one particular setpoint. At any other setting, there would be an offset between the setpoint and the true process value. Metaphorically, an operator could reset the controller setpoint by hand, until the actual process eventually stabilized at the desired value. In older control literature this is referred to "reset" action as a result. The derivative term reflects the ability to observe the rate of change of the process variable and again adjust the setpoint in anticipation of the final value. Again, an older term for this action is "rate".

A PID loop can be mathematically characterized as a filter applied to a frequency-domain system.

The basic idea is that the controller reads a sensor. Then it subtracts the measurement from a desired "reference" to determine an "error". It then manages the error in three ways, to handle the present, recover from the past and anticipate the future:

1. Proportional - To handle the present, the error is multiplied by a proportional constant P, and sent to the output. P is only valid in the band over which a controller's output is proportional to the error of the system. E.g. for a heater, a controller with a proportional band of 10 °C and a setpoint of 20 °C would have an output of 100% at 10 °C, 50% at 15 °C and 10% at 19 °C. Note that when the error is zero, a proportional controller's output is zero. There may be a predefined bias value that is set to equal the output value required to keep the desired setpoint.
2. Integral - To handle the past, the error is integrated (or summed) over a period of time, and then multiplied by a constant I, and added to the proportional output. I represents the steady state error of the system. Using the proportional term alone often leaves a steady state error to the controlled variable. Real world processes are not perfect and are subject to changing operating conditions. Think of a tank that has a controllable inflow and a constant outflow. Keeping a setpoint of the level can be achieved with a P-controller and a bias (see proportional above). If the outflow is changed, for example the pipe becomes clogged, the level can not be regulated with such a controller as there will be some steady-state error. This can be corrected by replacing the pre-set bias value with an integral term, which will adjust its value to remove errors that last for some time. The Integral part can be thought of as a self-adjusting bias value for a P controller.
3. Derivative - To handle the future, the first derivative of the error (its rate of change) is calculated with respect to time, and multiplied by another constant D, and summed with the proportional and integral terms, before adding the result to the output. The derivative term is used to govern a controller's response to a change in the system. The larger the derivative term the more rapidly the controller will respond to changes in the process value. This is a good thing to reduce when trying to dampen a controller's response to short term changes.

The generic transfer function for a PID controller of the interacting form is

with C being a constant (typically 0.01 or 0.001).

Adapted from the Wikipedia article "Theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki