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Fractional order [proportional derivative] controller for a class of fractional order systems. (English) Zbl 1183.93053
Summary: Recently, Fractional Order Systems (FOS) have attracted more and more attention in various fields. But the control design techniques available for the FOS suffer from the lack of direct systematic approaches. In this paper, we focus on a given type of simple model of FOS. A Fractional Order [Proportional Derivative] (FO-[PD]) controller is proposed for this class of FOS, and a practical and systematic tuning procedure has been developed for the proposed FO-[PD] controller synthesis. The fairness issue in comparing with other controllers such as the traditional integer order PID (IO-PID) controller and the fractional order proportional derivative (FO-PD) controller has been addressed under the same number of design parameters and the same specifications. Fair comparisons of the three controllers (i.e., IO-PID, FO-PD and FO-[PD]) via the simulation tests illustrate that, the IO-PID controller designed may not always be stabilizing to achieve flat-phase specification while both FO-PD and FO-[PD] controllers designed are always stabilizing. Furthermore, the proposed FO-[PD] controller outperforms FO-PD controller for the class of fractional order systems.

93B50 Synthesis problems
93B51 Design techniques (robust design, computer-aided design, etc.)
93D15 Stabilization of systems by feedback
34A08 Fractional ordinary differential equations
Matlab; irid_fod
Full Text: DOI
[1] Bagley, R.L.; Calico, R.A., Fractional-order state equations for the control of viscoelastic damped structures, Journal of guidance, control and dynamics, 14, 2, 304-311, (1991)
[2] Bagley, R.L.; Torvik, P., On the appearance of the fractional derivative in the behavior of real materials, Journal of applied mechanics, 51, 294-298, (1984) · Zbl 1203.74022
[3] Caputo, M., Elasticita e dissipacione, (1969), Zanichelli Bologna
[4] Chen, Y. (2008a). Impulse response invariant discretization of fractional order integrators/differentiators compute a discrete-time finite dimensional \(z\) transfer function to approximate \(s^r\) with \(r\) a real number. Category: Filter Design and Analysis, MATLAB Central. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do objectId = 21342 objectType = FILE
[5] Chen, Y. (2008b). Impulse response invariant discretization of fractional order low-pass filters discretize \([1 /(\tau s + 1)]^r\) with \(r\) a real number. Category: Filter Design and Analysis, MATLAB Central. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do objectId = 21365 objectType = FILE
[6] Chen, Y., Xue, D., & Dou, H. (2004). Fractional calculus and biomimetic control. In IEEE int. conf. on robotics and biomimetics (pp. (PDF-robio2004-347))
[7] Li, H., Luo, Y., & Chen, Y. (2008). A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments. IEEE Transactions on Control System Technology (in press)
[8] Luo, Y., & Chen, Y. (2008). Fractional-order [proportional derivative] controller for robust motion control: Tuning procedure and validation. IEEE Transactions on Automatic Control (under revision)
[9] Luo, Y., Li, H., & Chen, Y. (2008). Fractional order proportional and derivative controller synthesis for a class of fractional order systems: Tuning rule and hardware-in-the-loop experiment. IET Control Theory and Applications (revised)
[10] Magin, R.L., Fractional calculus in bioengineering, (2006), Begell House Publishers Inc
[11] Mehaute, A.L.; Crepy, G., Introduction to transfer and motion in fractal media: the geometry of kinetics, Solid state ionics, 9-10, 17-30, (1983)
[12] Monje, C.A.; Calderon, A.J.; Vinagre, B.M.; Chen, Y.; Feliu, V., On fractional \(\operatorname{PI}^\lambda\) controllers: some tuning rules for robustness to plant uncertainties, Nonlinear dynamics, 38, 1-4, 369-381, (2004) · Zbl 1134.93338
[13] Nakagawa, M.; Sorimachi, K., Basic characteristics of a fractance device, IEICE transactions on fundamentals E75-A, 12, 1814-1819, (1992)
[14] Nonnenmacher, T.F.; Glockle, W.G., A fractional model for mechanical stress relaxation, Philosophical magazine letters, 64, 2, 89-93, (1991)
[15] Oldham, K.B.; Zoski, C.G., Analogue instrumentation for processing polarographic data, Journal of electroanalytical chemistry, 157, 27-51, (1983)
[16] Oustaloup, A.; Sabatier, J.; Lanusse, P., From fractal robustness to the CRONE control, Fractional calculus and applied analysis, 2, 1, 1-30, (1999) · Zbl 1111.93310
[17] Podlubny, I., Fractional-order systems and \(\operatorname{PI}^\lambda \operatorname{D}^\mu\) controller, IEEE transactions on automatic control, 44, 1, 208-214, (1999)
[18] Westerlund, S., Capacitor theory, IEEE transactions on dielectrics and electrical insulation, 1, 5, 826-839, (1994)
[19] Zhao, C., Xue, D., & Chen, Y. Q. (2005). A fractional order PID tuning algorithm for a class of fractional order plants. In: Proc. of the IEEE ICMA (pp. 216-221)
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