This article was featured in in the April 2000 issue of Tech Times, a newsletter by the ISA, and originally in the January 1999 InTech. Written by Nicholas Scheble. Tables and figures have been re-done by Lighthouse PLCs. Click on a term for its definition, or to see the corresponding figures.
Modern process control is a relatively new field. Controllers are widely used in all of the process industries. Continuous feedback process controllers ... using the ubiquitous proportional-integral-derivative (PID) algorithm have been around only since the 1940s.
Fourier, Laplace, Kirchoff, Kelvin, and others had laid out the mathematical basics for this control theory by the end of the 1800s. All of the terms, proportional, integral, and derivative, refer to certain arithmetic manipulations. The latter two are the heart of calculus. The mathematical complexities need not be explored to understand PID control theory. Graphical insight and low-tech interpretation is more than adequate.
Take a system, any system
To begin, consider a parameter that needs to be controlled. Take the temperature of water in a tank that is continuously fed with cold water and is continuously pumping 100ºC water. If the temperature of the water coming out of the tank is less than 100ºC, a variable gas flame is applied until the temperature goes back up. This is the process.
The controlled parameter in this process is temperature. The set point (SP) is 100ºC. The set point minus the actual temperature (PV) of the water is the error, or offset. When there's an error, the controller sends a message (CO) to crank up the heat.
Proportional leads off
The proportional mode (P) is the least complicated of the three. The mathematical expression is:
The KP is called the constant of proportionality. That's a fancy term for the number on a dial of the controller apparatus that is adjustable and that can be used by the operator to tweak the process.
If the KP is negative .5 (- .5), the PV is 98ºC, and the SP is 100ºC, then the CO would be 1. This 1, then, has a calibrated meaning to the gas burner underneath the tank that will adjust the flame to a certain intensity.
As well, if the water coming out of the tank is 100ºC, (PV-SP) would equal zero, the CO would be zero, and this number would be associated with a setting for the flame under the tank, possibly zero or no flame.
Because the P mode has only one adjustment, KP, it is the easiest to operate. Used by itself (see Figure 1), the mode never is able to completely eliminate the difference between the set point and the actual value (PV). The P mode does, however, provide rapid response and it is stable.
The integral term is next
The expression for the integral term of a PID algorithm stems from calculus, a kind of mathematics invented by Isaac Newton.
This expression says we're integrating (S) the error (PV-SP) with respect to time (dt), multiplying it by a tweaking factor (Ki), and getting a number that is the controller output signal (CO). Never mind that.
In calculus, when we integrate, we are finding the area of a space underneath a line. Plotting the error versus time, in this case, forms that line (see Figure 2). Thus, if at time one (t1) the error is 2 and at time two (t2) the error is 3, the number arrived at by integrating is the shaded area in Figure 2.
Likewise, if the error is zero at time one and zero at time two, then there is no area under the line and the above expression for CO equals zero.
As with the proportional mode, adjustments are made using a constant. In addition, the time period over which the line is plotted can be manipulated.
Unlike the proportional mode, integral action can eliminate error on its own (see Figure 3). Because of its dependence on long-time interval to average out the error, the correction is slow and suitable only to smaller systems.
The expressions for P mode and I mode are often added together to take advantage of both types of action. The integral component eliminates error, and the proportional component provides response speed and stability.
Using slopes for control
The D term of the PID control expression stands for the word derivative and also comes from calculus.
The derivative control mode provides a controller output that is proportional to the rate of change of the difference between the actual value of the parameter (temperature) and the set point (PV-SP).
The visual interpretation of this is clearer. Using the same data used to construct the line in Figure 2, an identical line is shown in Figure 4. The [d(PV-SP)/dt] component of the derivative expression merely says divide the change in the error by the change in time. In this example, that means divide one (Ñy) by the elapsed time (Ñt). If the elapsed time is 10 seconds, our result would be 1/10. This number is then multiplied by a tweaking factor (Kd), resulting in a CO.
The derivative mode cannot, by itself, control a process. One reason for this is that a constant deviation from the set point makes the above expression equal to zero. As well, if a sudden change in the process variable occurs, an infinite signal is sent to the controller, which causes the relevant mechanical apparatus to fully open or close. This leads to an unending instability.
Derivative action adds lead time in the controller, compensating for the time delays present in nearly all process control loops. When correctly applied, it stabilizes the process.
It's no PIDdling system
The crux of each of these three modes is they each produce a number (CO) from the same input data (Kx, SP, and PV). That number is different for each mode. The interplay of these numbers as data is input is the magic of the PID relationship.
When the three modes' individual numbers are combined, a PID controller controls the system. The proportional mode is the basic control. The integral mode deals with the long-term errors that the proportional is unsuited to handle. The derivative mode takes care of the more pronounced disturbances occurring in the ongoing process.
The PID algorithm is used when the system is large, when there are rapid changes in some process variables, and when these changes are big. It's a complex system and necessitates tinkering at start-up to justify the various proportionality constants (Kp, Ki, Kd) of each individual mode to efficiently reach and maintain the set point.
| C | Celsius | Offset | error |
| CO | Controller output | P | proportional mode |
| D | derivative mode | PID | proportional-integral-derivative mode |
| I | integral mode | PV | process variable actual measurement |
| Kd | derivative gain constant | PV-SP | error |
| Ki | constant of integration | SP | set point, or the goal measurement in a process |
| Kp | constant of proportionality |
Figure 1: Proportional control
Figure 2: Integrating error over time
Figure 4: The derivative of the error
