1. MMC
  2. All Volumes and Issues
  3. Volume 8, Number 1, 2021
  4. Control synthesis by full state vector in systems with fractional-order derivatives using Caputo-Fabrizio operator

Control synthesis by full state vector in systems with fractional-order derivatives using Caputo-Fabrizio operator

MMC.
2021;
Volume 8, Number 1
: pp. 106–115
https://doi.org/10.23939/mmc2021.01.106
Received: October 05, 2020
Revised: January 11, 2021
Accepted: January 15, 2021

Mathematical Modeling and Computing, Vol. 8, No. 1, pp. 106–115 (2021)

Authors:
  1. Andriy Lozynskyy
  2. Orest Lozynskyy
  3. Lidiya Kasha
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University

In the paper, the control system synthesis by means of the full state vector is considered when using fractional derivatives in the description of this system.  To conduct research in the synthesized system with fractional derivatives in the Caputo--Fabrizio representation, a fundamental matrix of the system is formed, which also allows us to analyze the influence of initial conditions on the processes within the system.  In particular, the finding of the fundamental matrix of the system in the case of multiple roots of a characteristic polynomial, which are obtained by transforming the synthesized system to the binomial form, is demonstrated.  The influence of the fractional derivative index and the location of the roots of the characteristic polynomial transformed to the binomial form on the system operation is analyzed.

fractional-order derivative
fundamental matrix
modal control
  1. Khalil H. K.  Nonlinear Systems. Prentice Hall, New York (2002).
  2. Isidori A.  Nonlinear control systems.  Springer–Verlag (1995).
  3. Lozynskyy A., Demkiv L., Vantsevich V.  Enhancement of dynamical characteristics of a fuzzy control system by using unstable subsystem.  2018 IEEE International Conference on Fuzzy Systems (FUZZ–IEEE).  Rio de Janeiro, Brazil, July 08-13. 1240–1247 (2018).
  4. Petráš I.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation.  Springer–Verlag, Berlin, Heidelberg (2011).
  5. Monje C. A., Chen Y., Vinagre B. M., Xue D., Feliu-Batlle V.  Fractional-order Systems and Controls. Fundamentals and Applications.  Springer–Verlag, London (2010).
  6. Asadollahi M., Rikhtegar ghiasi A., Dehghani H.  Excitation control of a synchronous generator using a novel fractional-order controller.  IET Generation, Transmission & Distribution. 9 (15), 2255–2260 (2015).
  7. Tytiuk V., Chornyi O., Baranovskaya M., Serhiienko S., Zachepa I., Tsvirkun L., Kuznetsov V., Tryputen N.  Synthesis of a fractional-order  PIλDµ-controller for a closed system of switched reluctance motor control.  Eastern-European Journal of Enterprise Technologies. 2 (2), 35–42 (2019).
  8. Busher V., Melnikova L., Horoshko V.  Synthesis and implementation of fractional-order controllers in a current curcuit of the motor with series excitation.  Eastern-European Journal of Enterprise Technologies. 2 (2), 63–72 (2019).
  9. Soukkou A., Belhour M. C., Leulmi S.  Review design optimization and stability analysis of fractional-order PID controller.  Intelligent Systems and Applications. 8 (7), 73–96 (2016).
  10. Birs I., Muresan C., Nascu I., Ionescu C.  A Survey of Recent Advances in Fractional Order Control for Time Delay Systems.  IEEE Access. 7, 30951–30965 (2019).
  11. Shah P., Agashe S.  Review of fractional PID controller.  Mechatronics. 38, 29–41 (2016).
  12. Badri V., Tavazoei M. S.  Some Analytical Results on Tuning Fractional-Order [Proportional–Integral] Controllers for Fractional-Order Systems.  IEEE Transactions on Control Systems Technology. 24 (3), 1059–1066 (2016).
  13. Lozynskyy O., Lozynskyy A., Marushchak Y., Kopchak B., Kalenyuk P., Paranchuk Y.  Synthesis and research of electromechanical systems described by fractional order transfer functions.  2017 International Conference on Modern Electrical and Energy Systems (MEES). Kremenchuk, Ukraine. 16–19 (2017).
  14. Zeng F., Shu H., Zhu T., Swe T., Yang B.  Fractional-order Feedback Linearization Sliding-mode Control Design for Grid-connected PV Inverters.  2019 IEEE 3rd International Electrical and Energy Conference (CIEEC). Beijing, China. 874–878 (2019).
  15. Jiacai H., Hongsheng L., Fulin T., Di L.  Fractional order sliding mode controller for the speed control of a permanent magnet synchronous motor.  2012 24th Chinese Control and Decision Conference (CCDC). Taiyuan, China. 1203–1208 (2012).
  16. Liu H., Pan Y., Li S., Chen Y.  Adaptive Fuzzy Backstepping Control of Fractional-Order Nonlinear Systems.   IEEE Transactions on Systems, Man, and Cybernetics: Systems. 47 (8), 2209–2217 (2017).
  17. Nikdel N., Badamchizadeh M., Azimirad V., Nazari M. A.  Fractional-Order Adaptive Backstepping Control of Robotic Manipulators in the Presence of Model Uncertainties and External Disturbances.  IEEE Transactions on Industrial Electronics. 63 (10),  6249–6256 (2016).
  18. Aguila-Camacho N., Duarte-Mermoud M. A., Gallegos J. A.  Lyapunov functions for fractional order systems.  Communications in Nonlinear Science and Numerical Simulation. 19 (9),  2951–2957 (2014).
  19. Zhao Y., Wang Y., Li H.  State feedback control for a class of fractional order nonlinear systems.  IEEE/CAA Journal of Automatica Sinica. 3 (4), 483–488 (2016).
  20. Zhao Z., Zhang X., Wang Q.  Output Feedback Stabilization of Uncertain Rectangular Descriptor Fractional Order Systems With 0<alpha<1.  IEEE Access. 7, 108948–108956 (2019).
  21. Zhang X., Zhao Z., Wang Q. Static and dynamic output feedback stabilisation of descriptor fractional order systems.  IET Control Theory & Applications. 14 (2), 324–333 (2020).
  22. Luo Junhai.  State-Feedback Control for Fractional-Order Nonlinear Systems Subject to Input Saturation.  Mathematical Problems in Engineering. Special Issue: Chaos-Fractals Theories and Applications. 2014, Article ID 891639, 8 pages (2014).
  23. Caputo M., Fabrizio M.  A new Definition of Fractional Derivative without Singular Kernel.  Progress in Fractional Differentiation and Applications. An International Journal. 1 (2), 73–85 (2015).
  24. Diethelm K., Garrappa R., Giusti A., Stynes M.  Why fractional derivatives with nonsingular kernels should not be used.  Fractional Calculus and Applied Analysis. 23 (3), 610–634 (2020).
  25. Li H., Cheng J., Li H.-B., Zhong S.-M.  Stability Analysis of a Fractional-Order Linear System Described by the Caputo-Fabrizio Derivative.  Mathematics. 7 (2), 200 (2019).
  26. Ortigueira M. D., Machado J. T.  A critical analysis of the Caputo–Fabrizio operator.  Communications in Nonlinear Science and Numerical Simulation. 59, 608–611 (2018).
  27. Lozynsky А. О., Biletsky Yu. O., Lozynsky O. Yu., Moroz V. I.  Formation of the fundamental matrix of an open electromechanical system and its application for the calculation of time processes of state variables.  Energy Engineering and Control Systems. 6 (2), (2020), (under review), (in Ukrainian).
  28. Edwards C. H., Penney D. E.  Differential Equations and Linear Algebra. Pearson (2008).
  29. Marushchak Ya. Yu., Lozynsky A. O., Kushnir A. P.   Dynamics of two-mass mode stabilization systems in electric arc furnaces.  Lviv, Lviv Polytechnic Publishing House (2011), (in Ukrainian).