DOI: https://doi.org/10.15587/1729-4061.2018.139892

Synthesis and technical realization of control systems with discrete fractional integral-differentiating controllers

Victor Busher, Ali Aldairi

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.


Keywords


astatic system; fractional integration; fractional differentiation; algorithm of fast fractional integral calculation

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References


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GOST Style Citations


Mandelbrot B. B. Fraktalyi i haos. Mnozhestvo Mandelbrota i drugie chudesa. [Fractals and chaos. Mandelbrot set and other wonders]. Izhevsk: NIC "Regulyarnaya i haoticheskaya dinamika", 2009. 392 p.

Freeborn T. J., Maundy B., Elwakil A. S. Fractional-order models of supercapacitors, batteries and fuel cells: a survey // Materials for Renewable and Sustainable Energy. 2015. Vol. 4, Issue 3. doi: https://doi.org/10.1007/s40243-015-0052-y 

Uchaikin V. Metod drobnyh proizvodnyh [Method of fractional derivative]. Ul'yanovsk: «Artishok», 2008. 512 p.

Das S. Functional Fractional Calculus for System Identification and Controls. Springer-Verlag Berlin Heidelberg, 2008. 240 p. doi: https://doi.org/10.1007/978-3-540-72703-3 

Tarasov V. E. Fractional Liouville and BBGKI equations // Journal of Physics: Conference Series. 2005. Vol. 7. P. 17–33. doi: https://doi.org/10.1088/1742-6596/7/1/002 

Svobodnokonvektivnyie techeniya, teplo- i massoobmen [Freeconventional flow, heat and mass transfer] / Gebhart B., Dzhaluriya Y., Mahadzhan R., Sammakiya B. Moscow: Mir, 1991. 678 p.

Aoki Y., Sen M., Paolucci S. Approximation of transient temperatures in complex geometries using fractional derivatives. Technical Note of department of aerospace of Notre Dam, 2005. 21 p.

Petras I. The fractional – order controllers: Methods for their synthesis and application. URL: https://arxiv.org/abs/math/0004064

Jumarie G. Path probability of random fractional systems defined by white noises in coarse-grained time. Application of fractional entropy // Fractional Differential Calculus. 2011. Issue 1. P. 45–87. doi: https://doi.org/10.7153/fdc-01-03 

Fractional dynamics and control / D. Baleanu, J. A. T. Machado, A. C. J. Luo (Eds.). New York: Springer, 2012. 309 p. doi: https://doi.org/10.1007/978-1-4614-0457-6 

Xue D., Chen Y. Q., Atherton D. P. Linear Feedback Control: Analysis and Design with MATLAB. SIAM, 2007. 354 p. doi: https://doi.org/10.1137/1.9780898718621 

Petras I., Podlubny I., O'Leary P. Analogue realization of Fractional Order Controllers. FBERG, Tech. University of Kosice, 2002. 84 p.

Marushchak Y., Kopchak B. Synthesis of automatic control systems by using binomial and Butterworth standard fractional order forms // Computational problems of electrical engineering. 2015. Vol. 5, Issue 2. P. 89–94.

Krivolapova L. I., Kravcova O. A., Sokolov S. V. Fraktal'naya razmernost' – ocenochnaya mera kachestva poverhnosti metalloprokata // Doklady TUSURa. 2015. Issue 1. P. 142–147.

Busher V., Yarmolovich V. Modeling and Identification of Systems with Fractional Order Integral and Differential // Electrotechnical and Computer Systems. 2014. Issue 15 (91). P. 52–56.

Spravochnik po obrabotke metallov rezaniem / Abramov F. N., Kovalenko V. V., Lyubimov V. E. et. al. Kyiv: Tekhnіka, 1983. 239 p.

Vodichev V. A., Gulyu M. V., Muhammed M. A. Primenenie fazzi-regulyatora v elektromekhanicheskoy sisteme avtomatizacii metalloobrabotki // Visnyk NTU «Kharkivskyi politekhnichnyi instytut ». 2005. Issue 45. P. 504–505.

Development of the method for determining optimal parameters of the process of displacement of technological objects / Lutsenko I., Tytiuk V., Oksanych I., Rozhnenko Z. // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 6, Issue 2 (90). P. 41–48. doi: https://doi.org/10.15587/1729-4061.2017.116788 







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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061