PILearner

What is the difference between PI Control and PID Control ?

Proof through Mathematical Formulas would highly appreciated

PID control and associated controller types

PI controllers don't have the D-Part in calculation for initial acceleration on command.PID has all 3 main parts and they are enabled by switches.

In brief,PID controllers implemented in function blocks available in SIMATIC are blocks based on numerical calculations.Perhaps, the core of calculation of the functions are mathematical transfer functions, but several options are added to make them applicable in practice.Reading provided materials is sufficient and may not need to any "Proof through Mathematical Formulas ".

I hope these comments help you!
Hamid Hosseini

P- simplest controller, it only amplifies the errror (SP-PV)*Kp. This controller never reaches steady state zero error, if there aren't any integrating parts int the system. It makes sense that if SP=PV the output of controller is zero, so no error no output.

PI- the error is amplified and also integrated (P-proportional gain Kp; I-integral time Ti). The main benefit is that steady state error is zero, when using it. The main drawback is that integrator part can saturate, many algorithms under name anti-windup were made to overcome this problem. If well tuned it settles faster than P-regulator and hase zero steady state error, so it is most used in automation.

PID-like PI, but with extra derivation of error. It's the fastest, it looks forward in the future. Like riding the bycycle, you will correct position before you are falling, because you will detect the begining of a fall. But drawbacks are plenty: if signal is not 100% noise free it will got mad. When derivating the result is the rate of change of the signal, the amplitude of signal doesn't pay a big role. Therefore a small signal spikes with high slopes (noise) makes this controller unusable. This problemcan be moderated by using aproximative D part, it is the derivator filtered with low pass filter, also called like gamma parameter. Applying large filter on D part makes it useless, you will get PI regulator. With PID controler is also important to choose the right sampling time, for P,PI it would be the faster sampling the better, with PID is not so. The best is 1/20th to 1/10th of systems dominant time.