Fractional calculus in control theory
Key words: fractional
theory, fractional-order controlled system, fractional-order
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,
digital and analogue realization of fractional-order controllers,
modelling and simulation of control systems,
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