Abstract
The object of research in the work is electromechanical systems, a characteristic feature of which is the presence of significant power dependence in the mathematical description. Because of this, problems arise when choosing the structure and parameters of controllers. In particular, in a DC motor with series excitation, a switched reluctance motor and electromagnetic retarders, saturation of the magnetic system in static and dynamic modes can occur. The apparatus of fractional-integral calculus used in the work allows us to describe such nonlinear objects with high accuracy by linear transfer functions of fractional order. So, when approximating the anchor circuit of a DC motor with series excitation by a fractional transfer function, the smallest standard error was obtained. The combination of a conventional PID controller with fractional integral components of the order of 0.35 and 1.35 ensured the best quality of the transient process - the current reaches the set value as quickly as possible without overshoot. Secondly, the switched reluctance motor, in the model of which it is necessary to take into account the power dependences, is described by the aperiodic function of the order of 0.7 when describing the transient processes of the speed during a voltage jump. From the family of controllers studied, the traditional PI controller with additional fractional-integral components of the order of 0.7 and 1.7 ensured the astaticism of the speed loop of the order of 1.7 and the smallest overshoot. Thirdly, the electromagnetic retarders of the driving wheels of a car, used to tune the internal combustion engine, are also most accurately described after testing by the fractional transfer function. Using the PIDIγIµ controller, which ensured closed loop astaticism of the order of 1.63, stabilization of the rotation speed of two wheels without out-of-phase oscillations and the accurate development of a triangular tachogram were achieved. Thus, thanks to the apparatus of fractional-integral calculus, a more accurate identification of object parameters is provided, the mathematical description is reduced to linear transfer functions of fractional order. And in closed systems it is possible to ensure astaticism of fractional order 1.3–1.7 and to achieve a better quality of transient processes than using classical methods.
Highlights
The beginning of the development of fractional calculus is considered to be 1695, when Leibniz in a letter to Francois L’Hospital discussed the differentiation of the order 1/2 [1]
A new surge of interest in it is noticeable in recent decades
This is primarily due to the fact that differential equations with fractional order made it possible to describe some physical processes with greater accuracy than integer ones [2, 3]
Summary
The beginning of the development of fractional calculus is considered to be 1695, when Leibniz in a letter to Francois L’Hospital discussed the differentiation of the order 1/2 [1]. A new surge of interest in it is noticeable in recent decades. This is primarily due to the fact that differential equations with fractional order made it possible to describe some physical processes with greater accuracy than integer ones [2, 3]. PIγDμ controllers are used that can improve the quality of transients in comparison with classical integer PID controllers, especially in nonlinear systems [6, 7]
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