If you have work with some kind of industrial or marine automation, then most probably you've heard before the term PID. PID controllers are very common in closed-loop systems today. Here is how this system can calculate and minimize the error with great precision.

The whole story began as a marine application, when people were trying to find ways to make reliable and accurate ship steering systems. But the problem was that, if the automation turns the rudder let's say left, the ship will not turn instantaneously, instead it needs a long course, for ships do not steer like like cars, instead they have a big hysteresis. Another problem is also that when the ship finally turns to the right direction and the automation turns the rudder straight, the ship keeps turning left due to inertia and many other parameters like waves, wind, speed etc.

At first, proportional systems were developed to do this. A proportional systems reads the feedback (electronic compass) and turns the rudder according to the angle that the ship needs to turn. If for example the ship had to turn 45 degrees left, the rudder would turn 20 degrees, and as the ship slowly turns to this direction, the rudder decreases its angle proportionally. But this system has a great disadvantage: Either the rudder will oscillate left and right because the ship will never stay on course precisely due to external disturbances, or the system will stabilize with a small constant error in angle.

Elmer Sperry was the first to introduce an early PID-type controller back in 1911, and it took another 11 years until engineer Nicolas Minorsky published the first theoretical analysis of a PID controller (Minorsky 1922). In order to design an automatic steering system for the US navy, Minorsky spent several hours observing helmsmen while on duty. He noticed that the helmsman turned the rudder not only according to the current error, but also according to the past errors and the rate of change. During first trials with a PI controller on the USS New Mexico, the system achieved an error of 2 degrees. When the parameter D was added (PID) the system managed to achieve error of 1/6 of a degree, better than most helmsmen could achieve.

The word PID is composed by the first letters from the terms Proportional, Integral and Derivative. The Proportional, Integral, and Derivative terms are summed to calculate the output of the PID. Here is the diagram to get a first (just a first) idea how this system works:

As you see, the setpoint is compared with the feedback. With the error (setpoint-feedback) the controller calculates the three terms (P-I-D), and finally these terms are added to calculate the new output.

Before i go into details, i need to define some terms in advance:

The proportional term changes the output according to the error. A simple proportional controller has only one control parameter, the Kp. By changing this parameter, the controller becomes more or less reactive in changes. The proportional term is given by this form:

And the output of the controller:

The higher the Kp parameter, the faster the system will try to reach the set-point, but also high proportional gain turns the system unstable and it oscillates around the set-point. On the other hand, if the Kp is low, then the the system will stabilize with a constant error bellow (or above) the set-point, because the output will not provide sufficient power for the system to reach the final position. Here is a graph to better understand this parameter:

The red line shows the response of the system with a high Kp. The system will try to reach the set-point faster, but the overshoot is high as well as the error. This drives the system into an unstable state in which it oscillates around the set-point. The blue line shows the response with a low Kp. The system will reach the set-point slower and it will cause a much smaller overshoot. But after a few oscillations, the controller output does not provide enough power to change the system, thus it will remain stable a little bellow the set-point.

A proportional system is usually not sufficient to eliminate the error. The system must not only change its output according to the current error, but it must also be able to watch and change the output according to the past errors. The integral is the sum of the errors over time. What this means is that, if the error is big, then the integral builds up as time passes and the output changes rapidly to eliminate the error.

Now suppose that we have the same example as before, a proportional system that has a low Kp. First of all, the system will respond much faster at the beginning, because the error will build up a big integral. Then, when the system is stabilized little bellow the set-point, the integral will take over. This small error is added over time to increase the integral, and finally it will change the system's output. Finally, the system will stabilize onto the set-point.

Like before, the integral term is multiplied by the integral gain parameter to increase or decrease the effect and it is given by this form:

Now, suppose that we have a proportional system with low Kp and an integral term. Such a system is named PI, and here is how the system reacts as Ki changes:

A high integral gain (Ki) (red line) causes a faster system response, but also causes overshoot. Unlike the pure proportional system, despite the overshoot, the system is not driven into an unstable state. After a few oscillations the system will stabilize on the set-point. As the integral gain gets smaller (blue line), the system responses slower, but it stabilizes much faster with less oscillations. The optimal integral gain (purple line) causes the system to respond very slow but it stabilizes on the set-point with almost no oscillations.

A PI system output is mathematically represented as:

In other words, the output is the sum of the Proportional term plus the Integral term.

The third and last term is the Derivative. This term calculates the rate of change of the error and adds to the output accordingly. If the error changes slow, then the derivative is increased in order to make the PID system respond faster. On the other hand, if the error is changes rapidly, the derivative is decreased in order to make the system more stable and avoid oscillations. The formula to describe the derivative is:

The output of a full PID controller is given my this mathematical form:

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