Fractional calculus in control theory
Key words: fractional
calculus, control
theory, fractionalorder controlled system, fractionalorder
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: fractionalorder
controller. Motivation on using the fractionalorder 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 fractionalorder
PID controllers with noninteger differentiation and integration parts.
2. Fractionalorder controlled system
The fractionalorder controlled system can be described by the fractionalorder
model with the continuous transfer function in the form:
3. Fractionalorder controller
The fractionalorder controller can be described by the fractionalorder
continuous transfer function
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:

controller parameters design methods,

control algorithm,

digital and analogue realization of fractionalorder controllers,

identification methods,

modelling and simulation of control systems,

stability.
References

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.

Dorcak, L. Numerical Models for Simulation of the FractionalOrder Control
Systems. UEF  04  94, Slovak Academy of Science, Institute of
Experimental Physics, Kosice, 1994.

Gorenflo, R.Fractional Calculus: Some Numerical Methods: Fractals and
Fractional Calculus in Continuum Mechanics. CISM Lecture Notes,
Springer Verlag, Wien, 1997, pp. 277  290.

Lubich, Ch. Discretized fractional calculus. SIAM J. Math. Anal.,
vol. 17, no. 3, 1986, pp. 704  719.

Oldham, K. B. and Spanier, J. The Fractional Calculus. Academic
Press, New York, 1974.

Oustaloup, A. La Derivation non Entiere. Hermes, Paris,1995.

Petras, I. List of publications.

Podlubny, I.. Fractional  Order Systems and Fractional  Order Controllers.
UEF  03  94, Slovak Academy of Sciences, Institute of Experimental
Physics, Kosice, 1994.

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.

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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