I just would like to make some minor correction and clarification of Mr. Bhase's statement.

It is true that a fractional-order controller may provide more adjustable time and frequency responses of the control system, but this only happens if the controller has more degrees-of-freedom (DOF) than its integer-order counterpart. For example, a classical integer-order PID controller only have 3 parameters that could be tuned, therefore, you by using this, you will only be able to satisfy at most 3 criteria of your choice (in assumption that there is a solution). A fractional-order PID controller can have as much as 5 DOF which means that you can satisfy more than 3. This what makes a FOPID better than PID in this particular case.

But it doesn't mean that it will always provide a better response. The only way to know if it is better is to compare the optimal FO controller to its optimal IO controller counterpart.

I agree with Mr. Bhase's comment that if the FOC is tuned properly, then it will definitely be better.

Here is a link, that will help you:

http://people.tuke.sk/igor.podlubny/pspdf/uef0394.pdf

It is not always the case in which FOPID used for integer-order plants will be better than integer-order PID. However, it has been shown in many papers from YangQuan Chen, Concha Monje, Ivo Petras, etc., that the use of FOPID could make the entire control system perform better. This is because using an FOPID could satisfy at most 5 robustness criteria as compare to the usual classical PID controller which only have 3 parameters to be tuned for 3 robustness criteria. If you plan to satisfy 4 or 5 robustness criteria, then you should use FOPID because you can tune 5 parameters: Kp, Ki, Kd, mu, and delta. Classical PID only has Kp, Ki and Kd.

Hope this helps.

Dear Jessica,

It has many advantages, if you have background on FO systems you will get an idea. Look in layman's language FOPID s have more degrees of freedom as here you get more parameters to tune since u have more kind of poles such as hyperdamped poles. Since ur Statibility region is now larger u have more flexibility to design ur controller.

To have a brief overview I would Suggest u read these two papers-

Especially the Journal paper will give u better idea on the use of FOPID over PID.

I tried to keep the language simple in the paper so might be helpful for beginners.

Also I suggest u go through the Book by - Shantanu Das on Fractional Calculus. He is one of my research collaborators and he is a vdeteran in the field of FO systems.

Link given below-

1. http://www.sciencedirect.com/science/article/pii/S0165168413002910

2. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6396000

Feel Free to shoot me with further queries. I would love to discuss and enrich my understanding.

Best,

Anish Acharya

PhD Student

Computer Science

University of California Irvine

Webpage: https://sites.google.com/a/uci.edu/anish-acharya/home

e-mail: [email protected]

The main advantage of Fractional-order controller is that it provide more adjustable time and frequency responses of the control system allowing fulfillment of better as well as robust performance. So a fractional-order controller will always provide better response than integer-order controller if it is properly tuned whatever may be the type of plant (integer or fractional). 

I just would like to make some minor correction and clarification of Mr. Bhase's statement.

It is true that a fractional-order controller may provide more adjustable time and frequency responses of the control system, but this only happens if the controller has more degrees-of-freedom (DOF) than its integer-order counterpart. For example, a classical integer-order PID controller only have 3 parameters that could be tuned, therefore, you by using this, you will only be able to satisfy at most 3 criteria of your choice (in assumption that there is a solution). A fractional-order PID controller can have as much as 5 DOF which means that you can satisfy more than 3. This what makes a FOPID better than PID in this particular case.

But it doesn't mean that it will always provide a better response. The only way to know if it is better is to compare the optimal FO controller to its optimal IO controller counterpart.

I agree with Mr. Bhase's comment that if the FOC is tuned properly, then it will definitely be better.

Hello everyone,

 have a question regarding FOPID is that

How to explain FOPID is better than PID controller in Mathematics

PID Controller is a speical case of FOPID. You can explain mittag lefflerfunction for one variable. And, explain that alpha=1, it is exp. function. 

http://mathworld.wolfram.com/Mittag-LefflerFunction.html

Thanks a lot sir  @Pritesh Shah. Can you please tell if there is a way to show the better performance of FOPID in term of stability? As stability region of FOPID controller is higher than PID

Yes, you can compute even generalized robustness regions for fractional PID controller, see e.g.

Conference Paper Generalized robust stability regions for fractional PID controllers

It can be shown really that if you have more robustness requirements in frequency domain, the set of classical PID controllers satisfying all of them is often empty (there is empty intersection of regions). However you may luckily get a non-empty intersection for fractional PID controller thanks to higher flexibility of the controller structure.

Dear Rishika, fractional operators (integrator, derivator) are linear, so there is no principal reason to deal better with non-linearities.

Emmanuel Gonzalez Could you please share a reference paper for this explanation.

Fractional order controller can be applied to integer order plants and they have better response than classiscal PID. Thiats the main advantage of fractional calculus.