Given a Fractional Order PID controller designed for trajectory control of mobile robot. The question is How to implement it using one of the arduino platforms knowing that when approximating the fractional integrator and derivative using discrete transform it gives infinite orders of terms in the Z-domain.
This link may answer on your question,
http://brettbeauregard.com/blog/2011/04/improving-the-beginners-pid-introduction/
Unfortunately, the link to Brett's blog doesn't address the design of Fractional-order PID controller (PIλDμ).
Since the fractional-order system is the direct extension of classical integer-order system, the fractional-order transfer function of a single variable system can be obtained. But the fractional-order Laplace transformations and their inverses are very complicated, making it unsuitable for computation.
An alternative solution is to implement the Oustaloup recursive filter as an approximate realization for the fractional-order derivative in the frequency range of interest.
Oustaloup, A. et al., 2000. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 47(1), pp.25–39. Available at: http://dx.doi.org/10.1109/81.817385.
Podlubny, I. et al., 2002. Analogue Realizations of Fractional-Order Controllers. Nonlinear Dynamics, 29(1/4), pp.281–296. Available at: http://dx.doi.org/10.1023/a:1016556604320.
Xue, D., Zhao, C. & Chen, Y., 2006. A Modified Approximation Method of Fractional Order System. 2006 International Conference on Mechatronics and Automation. Available at: http://dx.doi.org/10.1109/icma.2006.257769.
Dear Ibraheem,
As an alternative to Fractional PID controller, for nonlinear systems, I would recommend to implement the Neural Network-Based Self-Tuning PID Control (please find attached a link for the paper describing the Underwater Vehicles application). Using one of the Arduino platforms, I would suggest to use 3 neurons for the hidden layer.
http://www.mdpi.com/1424-8220/16/9/1429/pdf
Regards,
Alfonso Gómez
If you want to keep on fractional PID controller, low-order filters are essential for approximation of fractional elements. You may look at the method presented here: Article Optimal continuous approximation of basic fractional element...
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