Fractional calculus in control theory

Key words: fractional calculus, control theory, fractional-order controlled system, fractional-order controller.

1. Introduction

The use of fractional calculus for modelling physical systems has been considered (Oldham et al., 1974; Torvik et al., 1984; Westerlund et al., 1994). We can find also works dealing with the application of this mathematical tool in control theory (Axtell et al., 1990; Dorcak, 1994; Outstaloup, 1995; Podlubny, 1994). We can use this mathematical tool for description of the controlled object and also for a new type of controller:  fractional-order controller. Motivation on using the fractional-order controllers was that PID controllers belong to the dominating industrial controllers and therefore there is a continuous effort to improve their quality and robustness. One of the possibilities to improve PID controllers is to use fractional-order PID controllers with non-integer differentiation and integration parts.

2. Fractional-order controlled system

The fractional-order controlled system can be described by the fractional-order model with the continuous transfer function in the form:

Fractional transfer function of controlled system

3. Fractional-order controller

The fractional-order controller can be described by the fractional-order continuous transfer function

Fractional transfer function of controller

where lambda is an integral order, delta is a derivation order, K is a proportional constant, Ti is an integration constant and Td is a derivation constant.

4. Main research topics in control theory

Fractional calculus in control theory is a new area of research. The most important topics are:

  1. Axtell, M. and Bise, M. E. Fractional Calculus Applications in Control Systems. IEEE 1990 National Aerospace and Electronics Conference, New York, 1990, pp. 563 - 566.
  2. Dorcak, L. Numerical Models for Simulation of the Fractional-Order Control Systems. UEF - 04 - 94, Slovak Academy of Science, Institute of Experimental Physics, Kosice, 1994.
  3. Gorenflo, R.Fractional Calculus: Some Numerical Methods: Fractals and Fractional Calculus in Continuum Mechanics. CISM Lecture Notes, Springer Verlag, Wien, 1997, pp. 277 - 290.
  4. Lubich, Ch. Discretized fractional calculus. SIAM J. Math. Anal., vol. 17, no. 3, 1986, pp. 704 - 719.
  5. Oldham, K. B. and Spanier, J. The Fractional Calculus. Academic Press, New York, 1974.
  6. Oustaloup, A. La Derivation non Entiere. Hermes, Paris,1995.
  7. Petras, I. List of publications.
  8. Podlubny, I.. Fractional - Order Systems and Fractional - Order Controllers. UEF - 03 - 94, Slovak Academy of Sciences, Institute of Experimental Physics, Kosice, 1994.
  9. Torvik, P. J. and Bagley, R. L.. On the appearance of the fractional derivative in behaviour of real materials, Trans. of the ASME, vol. 51, 1984, pp. 294 - 298.
  10. Westerlund, S. and Ekstam, L. Capacitor theory. IEEE Trans. on Dielectrics and Electrical Insulation, vol. 1, no. 5, 1994, pp. 826 - 839.

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