Abstract
We consider the fractional-integrator as a feedback design element. It is shown, in a simple setting, that the fractional integrator ensures zero steady-state tracking. This observation should be contrasted with the typical formulation of the internal model principle which requires a full integrator in the loop for such a purpose. The use of a fractional integrator allows increased stability margin, trading-off phase margin against the rate of convergence to steady-state. A similar rationale can be applied to tracking sinusoidal signals. Likewise, in this case, fractional poles on the imaginary axis suffice to achieve zero steady-state following and disturbance rejection. We establish the above observations for cases with simple dynamics and conjecture that they hold in general. We also explain and discuss basic implementations of a fractional integrating element.
Mathematics Subject Classification. 93C80.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Duffy, Transform methods for solving partial differential equations. Boca Raton, FL: Crc Press, 2004.
N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism,” Antennas and Propagation, IEEE Transactions on, vol. 44, no. 4, pp. 554–566, 1996.
W. Chen, “Fractional and fractal derivatives modeling of turbulence,” eprint arXiv:nlin/0511066, 2005.
M. Ortigueira and J. Machado, “Special section: fractional calculus applications in signals and systems,” Signal Processing, vol. 86, no. 10, 2006.
K. Oldham and J. Spanier, The fractional calculus, ser. Mathematics in Science and Engineering, R. Bellman, Ed. New York: Academic Press, 1974, vol. 111.
I. Podlubny, Fractional differential equations, ser. Mathematics in Science and Engineering. San Diego: Academic Press, 1999, vol. 198.
J. Machado, “Discrete-time fractional-order controllers,” Fractional Calculus and Applied Analysis, vol. 4, no. 1, pp. 47–66, 2001.
Y. Chen, “Ubiquitous fractional order controls,” in Proc. of the second IFAC symposium on fractional derivatives and applications (IFAC FDA06, Plenary Paper). Citeseer, 2006, pp. 19–21.
A. Oustaloup, P. Melchior, P. Lanusse, O. Cois, and F. Dancla, “The CRONE toolbox for Matlab,” in Computer-Aided Control System Design, 2000. CACSD 2000. IEEE International Symposium on. IEEE, 2000, pp. 190–195.
K.J. Aström, “Model uncertainty and feedback,” in Iterative identification and control: advances in theory and applications, P. Albertos and A.S. Piqueras, Eds. New York: Springer Verlag, Jan. 2002, pp. 63–97.
B.M. Vinagre and Y. Chen, “Fractional calculus applications in automatic control and robotics,” in IEEE Proc. on Conference on Decision and Control, Las Vegas, Dec. 2002, Tutorial Workshop – 316 pages.
P.A. McCollum and B.F. Brown, Laplace Transform Tables and Theorems. New York: Holt, Rinehart and Winston, 1965.
A.D. Poularikas, The handbook of formulas and table for signal processing, ser. The Electrical Engineering Handbook Series. New York: CRC Press LLC and IEEE Press, 1999, vol. 198.
G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, 6th ed. San Diego: Harcourt, 2005.
B.A. Francis and W.M. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, Sept. 1976.
E. Sontag, “Adaptation and regulation with signal detection implies internal model,” Systems & control letters, vol. 50, no. 2, pp. 119–126, 2003.
H.S. Wall, Analytic theory of continued fractions. New York: D. Van Nostrand Company, Inc., 1948.
C. Brezinski, History of continued fractions and Padé approximations, ser. Computational Mathematics. New York: Springer-Verlag, 1991, vol. 12.
G.A.J. Baker, Essentials of Padé approximants. New York: Academic Press, 1975.
M. Ichise, Y. Nagayanagi, and T. Kojima, “An analog simulation of non-integer order transfer functions for analysis of electrode processes,” Journal of Electroanalytical Chemistry and Interfacial Electrochemistry, vol. 33, no. 2, pp. 253–265, 1971.
K. Oldham, “Semiintegral electroanalysis. Analog implementation,” Analytical Chemistry, vol. 45, no. 1, pp. 39–47, 1973.
G. Bohannan, “Analog realization of a fractional control element – revisited,” 2002.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to Bill Helton on his 65th birthday
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Takyar, M.S., Georgiou, T.T. (2012). Fractional-order Systems and the Internal Model Principle. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_21
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0411-0_21
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0410-3
Online ISBN: 978-3-0348-0411-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)