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  4. A Frequency criterion for analysis of stability of systems with fractional-order derivatives

A Frequency criterion for analysis of stability of systems with fractional-order derivatives

MMC.
2020;
Volume 7, Number 2
: pp. 389–399
https://doi.org/10.23939/mmc2020.02.389
Received: May 11, 2020
Accepted: September 04, 2020

Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 389–399 (2020)

Authors:
  1. Orest Lozynskyy
  2. P. I. Kalenyuk
  3. Andriy Lozynskyy
  4. Lidiya Kasha
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University
4
Lviv Polytechnic National University

Based on an analysis of the influence of the roots of a characteristic polynomial on the increment of the argument of the frequency characteristic of the system, the frequency criterion of stability of a system with fractional-order derivatives has been suggested.  The boundaries of the zone of location of the roots of the characteristic polynomial of a stable system have been determined in a complex plane when the index $\alpha$ of the basis of the characteristic polynomial changes.

fractional-order derivative
stability criterion
stability boundaries
  1. Monje C. A., Chen Y., Vinagre B. M., Xue D., Feliu V.  Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, New York (2010).
  2. Sheng H., Chen Y., Qiu T.  Fractional Processes and Fractional-Order Signal Processing. Springer, London (2012).
  3. Kumar D. M., Mudaliar H. K., Cirrincione M., Mehta U., Pucci M.  Design of a Fractional Order PI (FOPI) for the Speed Control of a High-Performance Electrical Drive with an Induction Motor.  2018 21st International Conference on Electrical Machines and Systems (ICEMS), Jeju. 1198–1202 (2018).
  4. Lozynskyy O., Lozynskyy A., Marushchak Y., Kopchak B., Kalenyuk P., Paranchuk Y.  Synthesis and research of electromechanical systems described by fractional order transfer functions.  Modern Electrical and Energy Systems (MEES 2017). Kremenchuk, Ukraine, 15–17 November 2017. 16–19 (2017).
  5. Leuzzi R., Lino P., Maione G., Stasi S., Padula F., Visioli A.  Combined fractional feedback-feedforward controller design for electrical drives.  ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014, Catania. 1–6 (2014).
  6. Tytiuk V., Ilchenko O., Chornyi O., Zachepa I., Serhiienko S., Berdai A.  SRM Identification with Fractional Order Transfer Functions.  2019 IEEE 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON), Lviv, Ukraine. 271–274 (2019).
  7. Das S.  Functional Fractional Calculus for System Identification and Controls.  Springer, Berlin (2008).
  8. Kaczorek T.  Stability Analysis of Fractional Linear Systems in Frequency Domain.  In: Selected Problems of Fractional Systems Theory. Lecture Notes in Control and Information Sciences, vol. 411.  Springer, Berlin, Heidelberg (2011).
  9. Rivero M., Rogosin S. V., Tenreiro Machado J. A., Trujillo J. J.  Stability of Fractional Order Systems.  Mathematical problems in engineering.  New Challenges in Fractional Systems. Vol. 201, Article ID 356215, 14 pages (2013).
  10. Petras I.  Stability of fractional-order systems with rational orders: a survey.  Fractional Calculus & Applied Analysis. 12 (3), 269–298 (2009).
  11. Matignon D.  Stability result for fractional differential equations with applications to control processing.  Computational Engineering in Systems and Application Multiconference, IMACS, IEEE-SMC, Lille, France. Vol. 2, 963–968 (1996).
  12. Ahmed E., El-Sayed A. M. A., El-Saka H. A. A.  On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems.  Physics Letters A. 358 (1), 1–4 (2006).
  13. Busłowicz M.  Stability analysis of linear continuous-time fractional systems of commensurate order.  J. Automation, Mobile Robotics and Intelligent Systems. 3 (1), 12–17 (2009).
  14. Mikhailov A. V.  Methods for harmonic analysis in automatic control systems.  Avtomat. i Telemekh. 3, 27–81 (1938), (in Russian).
  15. Gao Z., Liao X., Shan B., Huang H.  A stability criterion for fractional-order systems with $\alpha $-order in frequency domain: The $1<\alpha < 2$ case.  2013 9th Asian Control Conference (ASCC), Istanbul. 1–6 (2013).
  16. Li Y., Chen Y. Q., Podlubny I.  Stability  of  fractional-order  nonlinear  dynamic  systems:  Lyapunov  direct  method  and generalized Mittag–Leffler stability.  Comput. Math. Appl. 59 (5), 1810–1821 (2010).
  17. Duarte-Mermoud M. A., Aguila-Camacho N., Gallegos J. A., Castro-Linares R.  Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems.  Communications in Nonlinear Science and Numerical Simulation. 22 (1–3), 650–659 (2015).
  18. Sabatier J., Moze M., Farges C.  LMI stability conditions for fractional order systems.  Computers and Mathematics with Applications. 59 (5), 1594–609 (2010).