7/12/2014 8:26 AM  
Posts: 80 Rating: (1) 
What is the difference between PI Control and PID Control ? Proof through Mathematical Formulas would highly appreciated

7/12/2014 8:39 AM  
Joined: 1/28/2009 Last visit: 2/21/2023 Posts: 6794 Rating: (1341) 
Hello, I suggest reading the following topic plus all related external links: PID control and associated controller typesPI controllers don't have the DPart 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 
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7/18/2014 12:48 AM  
Joined: 10/10/2010 Last visit: 11/23/2022 Posts: 296 Rating: (14) 
the PI is the most controller used practically 
7/18/2014 9:35 AM  
Joined: 9/8/2009 Last visit: 3/15/2023 Posts: 1410 Rating: (146) 
P simplest controller, it only amplifies the errror (SPPV)*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 (Pproportional gain Kp; Iintegral 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 antiwindup were made to overcome this problem. If well tuned it settles faster than Pregulator and hase zero steady state error, so it is most used in automation. PIDlike 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. 
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7/18/2014 10:29 PM  
Joined: 10/12/2010 Last visit: 1/30/2021 Posts: 52 Rating: (4) 
You can find tons of articles and other kind of readings about PID on the internet. Marko has made a good explanation already. As you asked about math behind PID controller, I want to add something. Theoretically, PI controllers are used for 1st degree systems. In a perfect world heating systems are first degree systems and PI is the best solution. Using derivative in such system has no positive impact (response may get faster but probably you get an overshoot). However, in our lonely flawed world you may consider using derivative for heating systems. Still, I would suggest low derivative coeff. for that kind of systems. For second degree systems (almost all of the realworld systems) you should use all 3 coeff. of PID to get optimal response. But as Marko has stated derivative is something you should use carefully. If your signal has noise then your system response will be unstable with high derivative values. There is also a system behaviour called "derivative kick" which can give you hard times. If you are using Siemens's PID blocks then you can forget about it. If you are planning to create your own PID block, you can find an easy solution for this problem in some references. I would recommend Antonio Visioli's "Practical PID Control" book from Springer. It has given me a good insight about practical use of PID with theoretical background while I was a student at the university. And if you are going to use Siemens's PID blocks then you should read manuals of the blocks.

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