For magnetic levitation control I had great results using high gain observer. It worked better than any differetiatior I had tried. (if you have problem accesing the attached paper please let me know and I'll send You full text)

Article Observer-based feedback for the magnetic levitation system

Hi Matheus

As you said, sliding mode differentiators provide some theoretical properties for the case of noisy signals. one of them is  the Kolmogorov bounds of error for the derivative of a noisy signal. Note that the Levant proposal is based on high gain design. Modifications for the noisy cases also have been proposed (See the attachement)

The first approaches for the design of algebraic are not easy to implement, however they present high performance. New algebraic differentiators have been proposed based on Jacobi approximators (See the attachement). Authors claim a high performance in presence of noise and made a comparison with other techniques.

I guess it is your task to start with a comparison scheme in which you use a criteria that allow you to make a good choice. In my opinion, all are good tools, depending on the case, including the Khalil high gain differentiator.

Regards.

https://hal.inria.fr/file/index/docid/986345/filename/ccc14-Final.pdf

https://hal.archives-ouvertes.fr/file/index/docid/1063052/filename/On-line_robot_dynamic_identification_based_on_power_model_modulating_functions_and_causal_Jacobi_estimator.pdf

https://hal.archives-ouvertes.fr/file/index/docid/1003992/filename/ILF_dif.pdf

http://cas.ensmp.fr/~praly/Telechargement/Journaux/2013-IJRNC-Khalil-Praly.pdf

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6426147

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6426662

It is best to obtain a model of the entire system and then design H-infinity filters (observer) for the system.  In filtering, a nonsingularity condition is that all measurement channels are noisy.  Since you are talking about numerical differentiation, you probably have a clean measurement of the output.  Just assume a small amount of noise in measurement, \epsilon, and design the filter.  It will lead to good estimates for the entire state vector, not only for the derivative of the output.  

Sounds like everybody is giving very good advice. 

I would actually choose the filter depending on the application. Would you like to rely on your model? Then go ahead with the model based observer approach. 

But, if you prefer to have a high bandwidth low delay differentiator than the Levant's nonlinear exact differentiators are quite good. They are basically similar to the Sliding mode observers. They perform quite well once you can determine a good parameter set for convergence. 

One last thought would be to utilise kaman approach. If you have a good idea bout the feedback devices resolution then you can tune your kalman filter quickly....

To be honest, in my experience, I do not feel the H-infinity has been a good tool for every application. They perform well case by case. They do not answer every problem just like they have been advertised....

Anyways, have fun! 

Thank you for all the answers!

Regards.

Please refer to this article:

https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design

Article On the Improved Nonlinear Tracking Differentiator based Nonl...

For a RG discussion on the meaning of the 'online' (or 'on-line') terminology at this forum:

https://www.researchgate.net/post/What_the_meaning_of_online_experimentation