Find proportional, integral and derivative

Process control refers to measuring a process variable, comparing it to some setpoint and manipulating the process in a way that keeps the process variable at its setpoint when either the setpoint changes or a disturbance changes the process. An example is shown in Figure 1. The temperature of the water leaving the heat exchanger is to be held at its setpoint by manipulating the flow of steam to the exchanger using the steam flow valve. In this example, the temperature is the measured or controlled process variable and the steam flow (or the position of the steam valve) is the manipulated variable.

Figure 1. A temperature-indicating controller maintains water temperature by converting a thermocouple reading into a steam valve position.

Many algorithms can be used to control the process. The most common is the simplest: an on/off switch. Most appliances use a thermostat to turn on the heat when the temperature falls below the setpoint and then turn it off when the temperature reaches the setpoint. This results in the temperature cycling above and below the desired temperature, but it’s sufficient for most common home appliances and some industrial equipment.

Using mathematical algorithms that compute a change in the output based on the controlled variable achieves better control. Of these, by far the most common is known as the proportional, integral and derivative (PID) algorithm. Control loops contain:

• A sensor to measure temperature, liquid level, pressure, flow rate or other variable, and convert the reading into a signal (typically 4 ma to 20 ma).
• A control algorithm that calculates the output signal to be transmitted to the final control element.
• A final control element that could be a valve, damper, motor speed controller or other device. It receives a signal from the controller and manipulates the process, for example by changing the flow rate of some material (Figure 2).

Figure 2. Interconnection of the control loop elements.

 Most control systems allow the operator to place individual loops into either manual or automatic mode. In manual mode, the operator adjusts the output manually to bring the measured variable to the desired value. In automatic mode, the control loop manipulates the output to hold the process measurements at the desired value.

Most plants start a process with all the loops in manual mode and then transfer them to automatic mode individually. Sometimes during normal process operation individual loops might be transferred to manual temporarily.

The PID algorithm

The most common algorithm (almost the only algorithm used in industrial process control) is the time-proven PID algorithm.

The PID control algorithm doesn’t “know” the correct output that brings the process variable to the setpoint value. The PID algorithm changes the output in the direction that moves the process variable toward the setpoint. The algorithm needs feedback (process measurement) to perform. If the path between the output and input is broken or limited, the algorithm has no way to “know” what the output should be. Under these (open loop) conditions, the output is meaningless.

“The PID control algorithm doesn’t “know” the correct output that brings the process variable to the setpoint value.”

- John A. Shaw

The PID algorithm must be tuned for the particular process loop if it’s to function in a stable manner. To tune a PID loop, one needs to understand each of the terms of the PID equation. The tuning is based on the process dynamics.

A controller’s action determines the direction of the output change that results from a change in the input. If a controller is direct acting, an increase in its input results in an increase in its output. With reverse action, an input increase results in decreased output. The controller action is always the opposite of the process action.

The first control mechanism, is called “proportional” or gain. In its pure form, controller output is equal to the error (difference between the measurement and the setpoint) times the gain added to a constant known as “manual reset.” This variable, subject to operator adjustment, is the output value when the measurement equals the setpoint.

O = e x g + k

Where:
O = output signal to the control element
e = error
g = gain
k = manual reset