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:

3. Fractional-order controller

The fractional-order controller can be described by the fractional-order 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 fractional-order controllers,
• identification methods,
• modelling and simulation of control systems,
• stability.

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