Abstract
Control systems with a fractional order which provide better dynamic and static indicators for many technical objects in comparison with systems with integer order of astaticism were studied. Based on the analysis of frequency characteristics, transient processes and a modified criterion for quality assessment, optimal relationships between parameters of the desired transfer function were obtained. Normalized transition functions of closed systems with the order of astaticism from 1 to 2 were presented with overregulation less than 2...5 % on the basis of which parameters can be chosen and the controller structure determined. The process of stabilizing cutting power was analyzed for a milling machine as an example of the systems with nonlinear parametric and structural dependences in control and perturbation channels. It was shown that fractional integral-differentiating controllers make it possible to provide the order of astaticism from 1.3 to 1.7 and permissible level of overregulation in a wide range of external perturbing influences. A method for approximate calculation of fractional integrals based on the approximation of the highest coefficients of expansion in a series of geometric progressions was developed. It provides reduction of the memory capacity required to store the coefficient arrays and the history of the input signal and requires significantly less CPU time to calculate the controller signal. For example, for controllers based on the Intel® Quark ™ SoC X1000 or FPGA Altera Cyclone V, the quantization period is 6...15 μs and several milliseconds for Atmega328. This makes it possible to implement fractional integral-differentiating controllers based on widely used modern processors and apply fractional-integral calculus methods for synthesis of high-speed automatic control systems. The proposed methods can be used in the control of the objects both with fractional and integer orders of differential equations.
Highlights
The fractional-integral calculus extends the theory of differential equations to the domain of non-integral order of derivatives eliminating discontinuities of this parameter
From 1993 to 2007, when developing the theory of Mittag-Leffler functions, solutions to some fractional differential equations were obtained in general form
These functions are named after their developers: Robotnov-Hartley, Erdei, Miller-Ross
Summary
The fractional-integral calculus extends the theory of differential equations to the domain of non-integral order of derivatives eliminating discontinuities of this parameter. From 1993 to 2007, when developing the theory of Mittag-Leffler functions, solutions to some fractional differential equations were obtained in general form. These functions are named after their developers: Robotnov-Hartley, Erdei, Miller-Ross. Methods for identifying such objects by means of differential equations with an arbitrary fractional order are already well tested To control this kind of processes, it is advisable to use PIαDβ-controllers with fractional-integral and fractional-differentiating components, which ensure setting of the closed loop to obtain the required dynamic and static parameters. The problems of synthesis of optimized systems with a fractional order of astaticism for objects with variable parameters and development of methods for technical implementation of fast-acting fractional integral-differentiating controllers remain topical
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