# PID--The Basic Technique for Feedback Control

01/01/1997

A feedback controller is designed to generate an output that causes some corrective effort to be applied to a process so as to drive a measurable process variable towards a desired value known as the setpoint. Shown is a typical feedback control loop with blocks representing the dynamic elements of the system and arrows representing the flow of information, generally in the form of electrical signals. Virtually all feedback controllers determine their output by observing the error between the setpoint and the actual process variable measurement.

PID control
A proportional-integral-derivative or 'PID' controller looks at the current value of the error, the integral of the error over a recent time interval, and the current derivative of the error signal to determine not only how much of a correction to apply, but for how long. Those three quantities are each multiplied by a tuning constant and added together to produce the current controller output CO(t) thusly:

(eq. 1)

In equation (1), P is the proportional tuning constant, I is the integral tuning constant, D is the derivative tuning constant, and e(t) is the error between the setpoint SP(t) and the process variable PV(t) at time t.

e(t) = SP(t) - PV (t)

(eq. 2)

If the current error is large, has been sustained for some time, or is changing rapidly, the controller will attempt to make a large correction by generating a large output. Conversely, if the process variable has matched the setpoint for some time, the controller will leave well enough alone.

Tuning a PID Controller

Conceptually, that's all there is to a PID controller. The tricky part is 'tuning' it; i.e., setting the P, I, and D tuning constants so that the weighted sum of the proportional, integral, and derivative terms produces a controller output that steadily drives the process variable in the direction required to eliminate the error.

marily by the derivative term in equation (1). This will cause the controller to initiate a burst of corrective efforts the instant the error changes from zero. The proportional term will then come in to play to keep the controller's output going until the error is eliminated.

iminated, the controller will continue to generate an output based on the history of errors that have been accumulating in the controller's integrator. The process variable may then overshoot the setpoint, causing an error in the opposite direction.

cumulated error are eliminated. Meanwhile, the derivative term will continue to add its share to the controller output based on the derivative of the oscillating error signal. The proportional term too will come and go as the error waxes and wanes.

apidly.

l Engineering magazine in 1991. Related subjects such as feedforward control, frequency domain analysis techniques, and self-tuning control will be addressed in future installments of this series.

Vance VanDoren
Consulting Editor

Vance J. VanDoren has a BS and MS in Control Engineering from Case Western Reserve University. He holds a Ph.D. in Control Engineering from Purdue University's School of Mechanical Engineering.

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