You have probably heard of a PID controller.
Many digital controllers are based on PID error correctors.
The term stands for Proportional, Integral, Derivative.
The idea is to monitor the difference between the controlled value and the set point and correct the system response in 3 different ways.
The proportional response simply amplifies the error so the output is a direct representation of the measured error. This type of response is likely to find a stable point. The main problem is that the stable point does not correspond to the set point. Moreover, the difference between the set point and the output is not constant and thus cannot be compensated easily. This leads to the addition of an integral element.
In addition to the proportional effect, let’s imagine a module that integrates the error. Simply put, as long as there is a residual error, the output slightly changes proportionally every cycle. During the first cycles, the effect is almost non-existent. When the stable point is reached (proportional effect), the residual error will be slightly compensated by small changes in the output over time. This integral module now provides us with the cancellation of the error and now our set point and our output will always be exactly the same (after the transient effect). This is one major difference between PI type controller and Lead/Lag type for example and the reason why most digital controllers are based on proportional-integral correction.
Now there is one last element in our controller, the derivative effect. There are various reasons but let’s have a look at an important one. When controlling second order systems, the maximum phase lag is 180 Degs. By introducing an integral effect, we add another potential 90 Deg in the loop due to our corrector lag (iterative response). This leads to a closed loop max phase lag of 270 Deg and with a bit of amplification can lead to unstable systems. The derivative effect introduces a phase lead of 90 Deg and thus allows stable settings. An additional benefit of the derivative effect is its ability to react to sudden changes of the error which improves transient response for slow loops