A Proportional-Integral-Derivative controller is a standard feedback loop component in industrial control applications. It measures an "output" of a process and controls an "input", with a goal of maintaining the output at a target value, which is called the "setpoint". A common application is to control a process temperature, in which case the PID controller acts as a sophisticated thermostat. It can also be used to control pressure, flow rate, chemical composition, force, speed or a number of other variables. Automobile cruise control is a consumer application of a PID controller.

The basic idea is that the controller reads a sensor. Then it subtracts the measurement from a desired "setpoint" to determine an "error".

The error is then treated in three different ways simultaneously:

• To handle the present, the error is multiplied by a proportional constant P. P is always negative, to drive the output toward the setpoint.
• To handle the past, the error is integrated (or averaged, or summed) over a period of time, and then multiplied by a constant I.
• 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.
• If the sum of the above is nonzero, but too small to make a difference, produce the smallest value with the same sign (usually -1 or 1).

The sum of the above is added to the last output of the PID loop. This eliminates any constant offset in the control behavior.

The output variable is used to calculate the rate-of-change, rather than the error, because a change of setpoint induces a non-linear step-function in the error. This can make the loop oscillate. The output variable is always continuous, and yet still closely reflects the error's actual behavior.

The generic transform function for a PID controller is

H(s)=,

with C being a constant (typically .01 or .001).

Tuning a PID loop

There are several methods for tuning a PID loop. The choice of method will depend largely on whether or not the loop can be taken "offline" for tuning, and the response speed of the system. If the system can be taken offline, the best tuning method often involves subjecting the system to a step change in input, measuring the output as a function of time, and using this response to determine the control parameters.

If the system must remain online, one tuning method is to first set the I and D values to zero. Increase the P until the output of the loop oscillates. Then increase I until oscillation stops. Finally, increase D until the loop is acceptably quick to reach its setpoint. The best PID loop tuning usually overshoots slightly to reach the set-point more quickly, however some systems cannot accept overshoot.

Another tuning method is formally known as the "Ziegler-Nichols method". It starts in the same way as the method described before: first set the I and D gains to zero and then increase the P gain until the output of the loop starts to oscillate. Write down the critical gain (Kc) and the oscillation period of the output (Pc). Then adjust the P, I and D controls as the table shows:

Problems

One common problem is "integral windup." It might take too long for the output value to ramp up to the necessary value when the loop first starts up. Sometimes this can be fixed with a more aggressive differential term. Sometimes the loop has to be "preloaded" with a starting output. Another option is to disable the integral function until the measured variable has entered the proportional band.

Some PID loops control a valve or similar mechanical device. Wear of the valve or device can be a major maintenance cost. In these cases, the PID loop may have a "deadband." The calculated output must leave the deadband before the actual output will change. Then, a new deadband will be established around the new output value.

Another problem with the differential term is that small amounts of noise can cause large amounts of change in the output. Sometimes it's helpful to filter the measurements, with a running average, or a low-pass filter. Alternatively, the differental band can be turned off in some systems with little loss of control. The differential term can also produce undesirable results in systems subjected to instantaneous "step" inputs.

Theory

A PID loop can be mathematically characterized as a filter applied to a frequency-domain system. Mathematical PID loop tuning induces an impulse in the system, and then uses the controlled system's frequency response to design the PID loop values. In loops with response times of several minutes, mathematical loop tuning is recommended, because trial and error can literally take days just to find a stable set of loop values. Optimal values are harder, and yet can save a company huge amounts of money. Commercial software is available from several sources, and can easily pay for itself if a PID loop runs a large, or expensive process.

Nomenclature

• Proportional Band is sometimes referred to as Gain
• Integral Band is sometimes referred to as Reset
• Derivative Band is sometimes referred to as Rate

How to get one

PID controller functionality is a common feature of programmable logic controllers (PLC). They can also be implemented with any physical system that can produce ratiometric behavior and integration. Mechanical systems (usually the cheapest) can use a lever, spring and a mass. Pneumatic controllers were once common, but have been largely replaced by digital electronic controllers. Electronic systems are very cheap, and can be made by using an amplifier, a capacitor and a resistance. Software PID loops are the most stable, because they do not wear out, and their high expense has been decreasing.

External Link

• sci.engr.* FAQ on PID controller tuning (http://www.tcnj.edu/~rgraham/PID-tuning.html)
• Information, including tutorial, on PID control algorithm (http://www.jashaw.com/pid)
• good, basic PID simulation in Excel (http://www.htservices.com/Applications/Process/PID2.htm)