Proportionalintegralderivative (PID) control is the most common strategy used for industrial processes. As the name implies, the controller consists of three interacting control functions that affect the power applied to change a process variable, for example, power supplied to the heating elements in a furnace to maintain a set temperature. The proportional function is used to adjust the speed of the system, the integral function provides the required accuracy, and the derivative function dampens the system to prevent excessive overshoot and minimize oscillatory behavior in the controlled variable. The PID controller is adjusted using three parameters, usually one parameter for each control function. Unfortunately, the mathematical algorithm and the parameter names used for the PID controllers vary with each manufacturer.
There are three basic algorithms used for PID controllers. The first form is called "series" or "interacting" or "analog" or "classical." The second form is called "parallel" or "noninteracting" or "ideal" or "standard ISA." It should be noted that ISA, or the Instrumentation, Systems and Automation Society, has no association with this algorithm. At one time this second algorithm was thought to become the standard, but as yet has not. The third form is called "parallel" or "ideal parallel" or "noninteracting" or "independent" or "gain independent." The purpose of mentioning these names is to show that the name does not necessarily depict the correct algorithm being used, and the only sure method of identifying the correct algorithm is to examine the equation, which may be obtained from either the instruction manual or the manufacturer.
For the purpose of this article, the second algorithm will be discussed in the equation where CO(t) is the controller output at time t, PV(t) is the process variable at time t, and e(t) is the difference or error between the set point, SP(t), and PV(t). Virtually all feedback controllers determine their output by observing e(t). The proportional action of the controller is weighted by the factor P, whereas the integral and derivative actions are weighted by P/TI and PTD respectively. The names associated with the control variables also may change between manufactures, but typically are defined by the following:

P = controller gain = 100/proportional band
 TI = integral time = 1/reset (units of time)
 TD = derivative time = rate = preact (units of time)
In some instances the term "reset" is defined as time per repeat, so consultation with the instruction manual or manufacturer is strongly recommended before attempting to tune the controller.
Now consider how a PID controller responds to an increase in the temperature set point for a furnace. Initially, the controller output is governed by the proportional and derivative functions, which are additive because the derivative of PV(t) (temperature) would be negative, and, thus, provides an initial power boost. The integral term represents the sum of all previous errors, starting at the beginning of the set point change and produces a corrective action, which grows in time the longer the error exists. As the temperature begins to recover, the derivative turns positive and begins to offset the growing contribution of the integral term. Figure 1 shows the temperature response that might be expected from a PID controlled furnace. Ideally, the three parameters are set to minimize the temperature overshoot and stabilize the temperature in a timely manner. In some instances, a larger derivative contribution may be required to eliminate overshoot, such as solution treatment of aluminum alloys susceptible to incipient melting.
If the integral action is too aggressive, the controller may produce an overshoot of larger magnitude than the original disturbance and create a closedloop instability or "hunting" scenario where the controller cycles between full on and off. A longer integral time would reduce the integral action and prevent this instability. Tuning the controller is nontrivial, and each furnace load may require a different set of parameters.
In 1942, John Ziegler and Nathaniel Nicols described a simple "closed loop" tuning technique for PID controllers. First, a steady state condition at the set point is obtained. The integral and derivative functions are now turned off, and the controller gain is increased until a disturbance causes a sustained oscillation in the process variable. The smallest controller gain that produces a sustained oscillation is defined as the ultimate gain (PU). The period of the oscillations is called the ultimate period (TU) and the parameters for the algorithm discussed in this article would be: