Please use this identifier to cite or link to this item: http://ena.lp.edu.ua:8080/handle/ntb/55979
Title: Application of Frequency Stability Criterion for Analysis of Dynamic Systems with Characteristic Polynomials Formed in j1/3 Basis
Other Titles: Застосування частотного критерію стійкості для аналізу динамічних систем з характеристичними поліномами, сформованими в базисі j1/3
Authors: Лозинський, Орест
Марущак, Ярослав
Лозинський, Андрій
Копчак, Богдан
Каша, Лідія
Lozynskyi, Orest
Marushchak, Yaroslav
Lozynskyi, Andriy
Kopchak, Bohdan
Kasha, Lidiya
Affiliation: Lviv Polytechnic National University
Rzeszow University of Technology
Bibliographic description (Ukraine): Application of Frequency Stability Criterion for Analysis of Dynamic Systems with Characteristic Polynomials Formed in j1/3 Basis / Orest Lozynskyi, Yaroslav Marushchak, Andriy Lozynskyi, Bohdan Kopchak, Lidiya Kasha // Computational Problems of Electrical Engineering. — Lviv : Lviv Politechnic Publishing House, 2020. — Vol 10. — No 1. — P. 11–18.
Bibliographic description (International): Application of Frequency Stability Criterion for Analysis of Dynamic Systems with Characteristic Polynomials Formed in j1/3 Basis / Orest Lozynskyi, Yaroslav Marushchak, Andriy Lozynskyi, Bohdan Kopchak, Lidiya Kasha // Computational Problems of Electrical Engineering. — Lviv : Lviv Politechnic Publishing House, 2020. — Vol 10. — No 1. — P. 11–18.
Is part of: Computational Problems of Electrical Engineering, 1 (10), 2020
Issue: 1
Issue Date: 24-Feb-2020
Publisher: Видавництво Львівської політехніки
Lviv Politechnic Publishing House
Place of the edition/event: Львів
Lviv
Keywords: dynamic system
fractional derivative
stability
characteristic polynomial
frequency stability criterion
Number of pages: 8
Page range: 11-18
Start page: 11
End page: 18
Abstract: В даній статті розглянуто питання стійкості динамічних систем, які описуються диференціальними рівняннями з дробовими похідними. На відміну від ряду робіт, де диференціальне рівняння, яке описує систему, може мати набір різних значень показників дробових похідних, а характеристичний поліном формується на основі найменшого спільного кратного для знаменників цих показників, в даній статті пропонується сформувати такий поліном в конкретному базисі j13 і далі проводити дослідження стійкості систем з таким дробовим описом на основі результуючих кутів повороту вектора lm (H j n w) при зміні частоти від нуля до нескінченності Така методика є аналогічною до дослідження стійкості систем за частотними критеріями, які використовуються для подібної задачі при описі системи диференціальними рівняннями в цілочисельних похідних. Саме застосування для опису процесів в динамічних системах характеристичних поліномів сформованих в базисі j13 і аналіз стійкості таких систем на основі частотного критерію становлять суть наукової новизни даного матеріалу. Стаття містить наступні розділи: постановка проблеми, мета роботи, виклад основного матеріалу, висновки, список літератури.
This paper considers the stability of dynamical systems described by differential equations with fractional derivatives. In contrast to a number of works, where the differential equation describing the system may have a set of different values of fractional derivatives, and the characteristic polynomial is formed on the basis of the least common multiple for the denominators of these indicators, this article proposes forming such a polynomial in a specific j13 basis and studying the stability of systems with such fractional description based on the resulting rotation angles of lm (H j n w) vector at a frequency change from zero to infinity. This technique is similar to the investigation of system stability by frequency criteria used for a similar problem in describing the system by differential equations in integer derivatives. The application of characteristic polynomials formed in the j13 basis for the description of the processes in dynamic systems and the analysis of the stability of such systems on the basis of the frequency criterion are the essence of the scientific novelty of this paper. The article contains the following sections: problem statement, work purpose, presentation of the research material, conclusions, list of references.
URI: http://ena.lp.edu.ua:8080/handle/ntb/55979
Copyright owner: © Національний університет “Львівська політехніка”, 2020
References (Ukraine): [1] D. Matignon, “Stability result on fractional differential equations with applications to control processing”, In Proc. International Meeting on Automated Compliance Systems and the International Conference on Systems, Man, and Cybernetics (IMACS-SMC '96), pp. 963–968, Lille, France, 1996.
[2] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “On the stability of linear system with fractional order elements”, Chaos Solutions & Fractals, no. 40, pp. 2317–2328, 2009.
[3] M. S. Tavazoei, and M. Haeri, “A note on the stability of fractional order systems”, Mathematics and Computers in Simulation, no. 79, pp. 1566–1576, 2009.
[4] M. Rivero, S. Rogosin, J. A. T. Machado, and J. Trujillo, “Stability of fractional order systems. Mathematical problems in engineering”. New Challenges in Fractional Systems, vol. 2013, 2013.
[5] S. Liang, C. Peng, and Y. Wang, “Improved linear matrix inequalities stability criteria for fractional order systems and robust stabilization synthesis: the 0<α<1 case”, Control Application, vol. 30, no 4, 2013.
[6] A. Banzaonia, A. Hmamed, F. Mesquine, B. Enhayoun, and F. Tadeo, “Stabilization of continuous-time fractional positive systems by using Lyapunov function”, IEEE Trans. Aut. Control, vol. 59, no. 8, pp. 2203–2208, 2014.
[7] M. Buslowicz, “Stability analysis of linear continuoustime fractional systems of commensurate order”, Journal Automation, Mobile Robotics and Intelligent Systems, vol. 3, no. 1, pp. 12–17, 2009.
[8] O. Yu. Lozynskyy, P. I. Kalenyuk, and A. O. Lozynskyy, “Frequency criterion for stability analysis of the systems with derivatives of fractional order”, Mathematical Modelling and Computing, vol. 7, no. 2, 2020. (Unpublished).
[9] Y. Marushchak, B. Kopchak, and L. Kasha, “Robust stability of fractional electromechanical systems”, Electrical Power and Electromechanical Systems, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, no. 900, pp. 47–51, 2018. (Ukrainian)
[10] O. Lozynskyy, A. Lozynskyy, B. Kopchak, Y. Paranchuk, P. Kalenyuk, and Y. Marushchak, “Synthesis and research of electromechanical systems described by fractional order transfer functions”, in Proc. International Conference on Modern Electrical and Energy Systems (MEES-2017), pp. 16–19, Kremenchuk, Ukraine, 2017.
References (International): [1] D. Matignon, "Stability result on fractional differential equations with applications to control processing", In Proc. International Meeting on Automated Compliance Systems and the International Conference on Systems, Man, and Cybernetics (IMACS-SMC '96), pp. 963–968, Lille, France, 1996.
[2] A. G. Radwan, A. M. Soliman, and A. S. Elwakil, "On the stability of linear system with fractional order elements", Chaos Solutions & Fractals, no. 40, pp. 2317–2328, 2009.
[3] M. S. Tavazoei, and M. Haeri, "A note on the stability of fractional order systems", Mathematics and Computers in Simulation, no. 79, pp. 1566–1576, 2009.
[4] M. Rivero, S. Rogosin, J. A. T. Machado, and J. Trujillo, "Stability of fractional order systems. Mathematical problems in engineering". New Challenges in Fractional Systems, vol. 2013, 2013.
[5] S. Liang, C. Peng, and Y. Wang, "Improved linear matrix inequalities stability criteria for fractional order systems and robust stabilization synthesis: the 0<α<1 case", Control Application, vol. 30, no 4, 2013.
[6] A. Banzaonia, A. Hmamed, F. Mesquine, B. Enhayoun, and F. Tadeo, "Stabilization of continuous-time fractional positive systems by using Lyapunov function", IEEE Trans. Aut. Control, vol. 59, no. 8, pp. 2203–2208, 2014.
[7] M. Buslowicz, "Stability analysis of linear continuoustime fractional systems of commensurate order", Journal Automation, Mobile Robotics and Intelligent Systems, vol. 3, no. 1, pp. 12–17, 2009.
[8] O. Yu. Lozynskyy, P. I. Kalenyuk, and A. O. Lozynskyy, "Frequency criterion for stability analysis of the systems with derivatives of fractional order", Mathematical Modelling and Computing, vol. 7, no. 2, 2020. (Unpublished).
[9] Y. Marushchak, B. Kopchak, and L. Kasha, "Robust stability of fractional electromechanical systems", Electrical Power and Electromechanical Systems, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, no. 900, pp. 47–51, 2018. (Ukrainian)
[10] O. Lozynskyy, A. Lozynskyy, B. Kopchak, Y. Paranchuk, P. Kalenyuk, and Y. Marushchak, "Synthesis and research of electromechanical systems described by fractional order transfer functions", in Proc. International Conference on Modern Electrical and Energy Systems (MEES-2017), pp. 16–19, Kremenchuk, Ukraine, 2017.
Content type: Article
Appears in Collections:Computational Problems Of Electrical Engineering. – 2020 – Vol. 10, No. 1



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