# P, I, and D together, separately control the process

The PID control algorithm is the combination of three methods of control.

The most basic continuous control mode is proportional (P) control where the controller output is algebraically proportional to the error signal to the controller.

Therefore, if the set point for a process is 70°F and the temperature of the process is 60°F, there is an error signal of 10 degrees. The controller tells the process to heat up by a proportion of those 10 degrees, which sends a new error signal that may now be only eight degrees because of the previous "heat up" direction.

Proportional control is the easiest of the continuous controllers to tune, that is, there is only one parameter to adjust. It also provides good stability, very rapid response, and is relatively stable dynamically.

Proportional control has one major disadvantage, however. At steady state, it exhibits offset; that is, there is a difference at steady state between the desired value or set point and the actual value of the controlled variable.

### Integral control

Integral (I) control is really an integration of the input error signal. In effect, this means the value of the manipulated variable changes at a rate proportional to the error.

Thus, if the deviation doubles over a previous value, the final control element moves twice as fast. When the controlled variable is at the set point (no deviation), the final control element remains stationary. In effect, this means at steady state, when reset control is present there can be no offset. The steady-state error must be zero.

Recalling your calculus, one knows an integral is the area underneath the plot of the curve of the error signal or the controlled variable vs. time. As that area between the curve and the set point gets smaller, the contribution of integral term in the PID becomes smaller and eventually zero when the controlled variable equals the set point.

Integral control action usually combines with proportional control. The combination is proportional-integral PI control.

The combination is favorable in that some of the advantages of both types of control are available. The advantage of including the integral mode with the proportional mode is the integral mode eliminates offset.

Typically, there is some decreased stability due to the presence of the integral mode; that is, the addition of the integral mode makes the total loop slightly less stable.

 Response for a step change in disturbance with tuned P, PI, and PID controllers.

### Derivative control

Recall a derivative (D) of curve is the rate of change or the slope of that curve at a point on the curve. So look at the error signal as we did before, and one can see this term of the PID algorithm is rate of change (slope) of the error curve.

We would never use control based solely on the rate of change of the error signal because, if the error were huge but unchanging, the correction would be zero. Thus, derivative control usually works in combination with proportional control.

By adding derivative mode to the controller, "lead" is added in the controller to compensate for "lag" around the loop. Almost any process has lag around the loop, and, therefore, the theoretical advantages of lead in the controller are attractive.

The addition of rate control to the controller makes the loop more stable if it is appropriately tuned. Since the loop is more stable, the proportional gain may be higher, and thus it can decrease offset more than can proportional action alone. It does not eliminate offset.

In three-mode control-PID control-we have the most complex controller algorithm that is available routinely. It gives rapid response and exhibits no offset, but it is very difficult to tune.

As a result, it is useful for only a small number of applications, and it often requires extensive and continuing adjustment to stay properly tuned and when properly tuned it offers very good process control.

Nicholas Sheble (nsheble@isa.org) writes and edits Automation Basics. A source for this piece was Paul Murrill's Fundamentals of process control theory, 3rd edition, ISA Press, 2000.