Vol 20 No 1 (2026): BAREKENG: Journal of Mathematics and Its Application
Articles

CHAOS CONTROL IN PERMANENT MAGNET SYNCHRONOUS MOTOR BY SLIDING MODEL CONTROLLER WITH LYAPUNOV OBSERVER UNDER UNKNOWN INPUTS

PDF
  • Seyed Mohamad Hamidzadeh,
  • Amiral Aziz,
  • Mohamad Afendee Mohamed,
  • Sundarapandian Vaidyanathan,
  • Muhamad Deni Johansyah
Seyed Mohamad Hamidzadeh
Department of Electrical Engineering, Ferdowsi University of Mashhad, Iran
Amiral Aziz
Research Center for Conversion and Conservation Energy-National Research and Innovation, Agency Republic of Indonesia (BRIN), Indonesia
Mohamad Afendee Mohamed
Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin (UniSZA), Malaysia
Sundarapandian Vaidyanathan
Centre for Control Systems, Vel Tech University, India
Muhamad Deni Johansyah
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Indonesia
DOI: https://doi.org/10.30598/barekengvol20iss1pp0541-0556
Published November 24, 2025
Keywords
  • Chaos,
  • PMSM,
  • SMC,
  • Nonlinear system,
  • Lyapunov exponent
How to Cite
[1]
S. M. Hamidzadeh, A. Aziz, M. A. Mohamed, S. Vaidyanathan, and M. D. Johansyah, “CHAOS CONTROL IN PERMANENT MAGNET SYNCHRONOUS MOTOR BY SLIDING MODEL CONTROLLER WITH LYAPUNOV OBSERVER UNDER UNKNOWN INPUTS”, BAREKENG: J. Math. & App., vol. 20, no. 1, pp. 0541-0556, Nov. 2025.

Copyright (c) 2025 Seyed Mohamad Hamidzadeh, Amiral Aziz, Mohamad Afendee Mohamed, Sundarapandian Vaidyanathan, Muhamad Deni Johansyah

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Abstract

The control of chaotic and hyper-chaotic systems represents a crucial area of research in the field of nonlinear dynamic systems. In this study, we focus on applying chaos control techniques to a permanent magnet synchronous motor (PMSM), a system known to exhibit chaotic behavior under certain conditions. To achieve this, a sliding mode control (SMC) strategy integrated with a Lyapunov-based observer is proposed. The core concept involves designing a candidate Lyapunov function that governs the application of the control law, ensuring system stability while effectively suppressing chaotic dynamics. Through numerical simulations, the proposed sliding mode controller demonstrates its effectiveness in rapidly eliminating chaotic behavior and stabilizing the PMSM system toward a predefined reference trajectory. Notably, the system achieves error convergence within approximately 0.7 seconds under full control (four channels). When control channels are reduced to two, the system still maintains stability, showing flexibility and cost efficiency. In a further simulation, the chaotic PMSM is subjected to two unknown external disturbances, and the proposed controller continues to maintain stability with only a slight increase in convergence time. These quantitative results affirm the robustness, accuracy, and practicality of the proposed control method. This research confirms that integrating sliding mode control with a Lyapunov observer is an effective approach for chaos suppression in PMSMs, offering promising insights for the development of advanced control strategies in nonlinear electromechanical systems.

PDF

Downloads

Download data is not yet available.

References

  1. E. N. Lorenz, “DETERMINISTIC NONPERIODIC FLOW,” J. Atmos. Sci., vol. 20, no. 2, pp. 130–141, 1963. doi: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
  2. K. Benkouider, et al., “A COMPREHENSIVE STUDY OF THE NOVEL 4D HYPERCHAOTIC SYSTEM WITH SELF-EXITED MULTISTABILITY AND APPLICATION IN THE VOICE ENCRYPTION,” Sci. Rep., vol. 14, no. 1, pp. 1–15, 2024. doi: https://doi.org/10.1038/s41598-024-63779-1
  3. G. Biban, R. Chugh, and A. Panwar, “IMAGE ENCRYPTION BASED ON 8D HYPERCHAOTIC SYSTEM USING FIBONACCI Q-MATRIX,” Chaos, Solitons and Fractals, vol. 170, no. March, p. 113396, 2023. doi: https://doi.org/10.1016/j.chaos.2023.113396
  4. L. Yin, Z. Deng, B. Huo, and Y. Xia, “FINITE-TIME SYNCHRONIZATION FOR CHAOTIC GYROS SYSTEMS WITH TERMINAL SLIDING MODE CONTROL,” IEEE Trans. Syst. Man, Cybern. Syst., vol. 49, no. 6, pp. 1–10, 2017. doi: https://doi.org/10.1109/TSMC.2017.2736521
  5. K. S. Oyeleke, O. I. Olusola, U. E. Vincent, D. Ghosh, and P. V. E. Mcclintock, “PARAMETRIC VIBRATIONAL RESONANCE IN A GYROSCOPE DRIVEN BY DUAL-FREQUENCY FORCES PARAMETRIC VIBRATIONAL RESONANCE IN A GYROSCOPE DRIVEN BY DUAL-FREQUENCY FORCES,” Phys. Lett. A, vol. 387, no. November, p. 127040, 2020. doi: https://doi.org/10.1016/j.physleta.2020.127040
  6. S. Farzami, R. E. Aval, and M. Afshari, “Chaotic satellite synchronization using neural and nonlinear controllers ,” Chaos Solitons & Fractals, vol. 97, no. October 2018, pp. 19–27, 2017. doi: https://doi.org/10.1016/j.chaos.2017.02.002
  7. S. M. Hamidzadeh and R. E. Aval, “CONTROL AND SYNCHRONIZTION CHAOTIC SATELLITE USING ACTIVE CONTROL CONTROL AND SYNCHRONIZATION CHAOTIC SATELLITE USING ACTIVE CONTROL,” Int. J. Comput. Appl., vol. 94, no. 10, pp. 29–33, 2022. doi: https://doi.org/10.5120/16380-5887
  8. A. Sambas, W. S. M. Sanjaya, M. Mamat, and R. P. Prastio, “MATHEMATICAL MODELLING OF CHAOTIC JERK CIRCUIT AND ITS APPLICATION IN SECURE COMMUNICATION SYSTEM,” Adv. chaos theory Intell. Control, vol. 337, pp. 133–153, 2016. doi: https://doi.org/10.1007/978-3-319-30340-6
  9. A. Sambas, W. S. M. Sanjaya, and M. Mamat, “DESIGN AND NUMERICAL SIMULATION OF UNDIRECTIONAL CHAOTIC SYNCHRONIZATION AND ITS APPLICATION IN SECURE COMMUNICATION SYSTEM,” J. Eng. Sci. Technol., vol. 6, no. 4, pp. 66–73, 2013.
  10. A. Ouannas, A. Karouma, G. Grassi, and V. Pham, “A NOVEL SECURE COMMUNICATIONS SCHEME BASED ON CHAOTIC MODULATION , RECURSIVE ENCRYPTION AND CHAOTIC MASKING,” Alexandria Eng. J., vol. 60, no. 1, pp. 1873–1884, 2021. doi: https://doi.org/10.1016/j.aej.2020.11.035
  11. Y. Luo, S. Tang, J. Liu, L. Cao, and S. Qiu, “IMAGE ENCRYPTION SCHEME BY COMBINING THE HYPER-CHAOTIC SYSTEM WITH QUANTUM CODING,” Opt. Lasers Eng., vol. 124, p. 105836, 2020. doi: https://doi.org/10.1016/j.optlaseng.2019.105836
  12. A. Sambas, M. Mujiarto, M. Mamat, and W. S. M. Sanjaya, “NUMERICAL SIMULATION AND CIRCUIT IMPLEMENTATION FOR A SPROTT CHAOTIC SYSTEM WITH ONE HYPERBOLIC SINUSOIDAL NONLINEARITY,” Far East J. Math. Sci., vol. 102, no. 6, pp. 1165–1177, 2017. doi: https://doi.org/10.17654/MS102061165
  13. X. Wang, Z. Lu, and C. Song, “CHAOTIC THRESHOLD FOR A CLASS OF POWER SYSTEM MODEL,” Shock Vib., vol. 2019, no. 1, p. 3479239, 2019. doi: https://doi.org/10.1155/2019/3479239
  14. A. Sambas, M. Mamat, and W. S. M. Sanjaya, “DYNAMICS, ADAPTIVE BACKSTEPPING CONTROL AND CIRCUIT IMPLEMENTATION OF SPROTT MO 5 CHAOTIC SYSTEM,” Int. J. Control Thery Appl., vol. 9, pp. 365–382, 2016.
  15. A. Sambas , W. S. M. Sanjaya, and M. Mamat, “BIDIRECTIONAL COUPLING SCHEME OF CHAOTIC SYSTEMS AND ITS APPLICATION IN SECURE COMMUNICATION SYSTEM,” J. Eng. Sci. Technol., vol. 8, no. 2, pp. 89–95, 2015.
  16. Y. Hajnorouzali, M. Malekzadeh, and M. Ataei, “FINITE-TIME CONTROL OF SPACECRAFT AND HARDWARE IN THE LOOP TUNING,” Asian J. Control, vol. 25, no. 1, pp. 262–270, 2023. doi: https://doi.org/10.1002/asjc.2770
  17. M. Asadollahi, N. Padar, A. Fathollahzadeh, M. Javad, and M. Ehsan, “FIXED-TIME TERMINAL SLIDING MODE CONTROL WITH ARBITRARY CONVERGENCE TIME FOR A CLASS OF CHAOTIC SYSTEMS APPLIED TO A NONLINEAR FINANCE MODEL,” Int. J. Dyn. Control, vol. 12, pp. 1874–1887, 2024. doi: https://doi.org/10.1007/s40435-023-01319-x
  18. D. Guegan, “CHAOS IN ECONOMICS AND FINANCE DOMINIQUE GUEGAN HAL ID : HALSHS-00187885,” Annu. Rev. Control, vol. 33, no. 1, pp. 89–93, 2018. doi: https://doi.org/10.1016/j.arcontrol.2009.01.002
  19. L. M. Pecora and T. L. Carroll, “SYNCHRONIZATION IN CHAOTIC SYSTEM,” Phys. Rev. Lett., vol. 64, no. 8, p. 821, 1990. doi: https://doi.org/10.1063/1.4917383
  20. E. Ott and J. A. Yorke, “CONTROLLING CHAOS,” Phys. Rev. Lett., vol. 64, no. 11, pp. 1196–1199, 1990. doi: https://doi.org/10.1002/asjc.2865
  21. W. Souhail and H. Khammari, “SENSORLESS ANTI-CONTROL AND SYNCHRONIZATION OF CHAOS OF BRUSHLESS DC MOTOR DRIVER,” Sci. reports, pp. 1–19, 2025. doi: https://doi.org/10.1038/s41598-024-80962-6
  22. I. Ahmad and M. Shafiq, “ROBUST ADAPTIVE ANTI-SYNCHRONIZATION CONTROL OF MULTIPLE UNCERTAIN CHAOTIC SYSTEMS OF DIFFERENT ORDERS,” Automika, vol. 61, no. 3, pp. 396–414, 2020. doi: https://doi.org/10.1080/00051144.2020.1765115
  23. T. Bonny, W. A. Nassan, and A. Sambas, “COMPARATIVE ANALYSIS AND FPGA REALIZATION OF DIFFERENT CONTROL SYNCHRONIZATION APPROACHES FOR CHAOS-BASED SECURED COMMUNICATION SYSTEMS,” PLoS One, vol. 19, no. 1, p. e0291714, 2024. doi: https://doi.org/10.1371/journal.pone.0291714
  24. L. Nie and X. Wang, “NEURAL-NETWORK-BASED ADAPTIVE CONTROL OF STRICT-FEEDBACK NONLINEAR SYSTEMS WITH ACTUATOR FAULTS : EVENT-TRIGGERED COMMUNICATIONS STRATEGY,” Chaos, Solitons and Fractals, vol. 181, no. January, p. 114626, 2024. doi: https://doi.org/10.1016/j.chaos.2024.114626
  25. M. Asadollahi, A. R. Ghiasi, and M. A. Badamchizadeh, “ADAPTIVE SYNCHRONIZATION OF CHAOTIC SYSTEMS WITH HYSTERESIS QUANTIZER INPUT,” ISA Trans., vol. 98, pp. 137–148, 2020. doi: https://doi.org/10.1016/j.isatra.2019.08.043
  26. W. Ao, T. Ma, R. Sanchez, and H. Gan, “FINITE-TIME AND FIXED-TIME IMPULSIVE SYNCHRONIZATION OF CHAOTIC SYSTEMS,” Sci. Direct, vol. 357, pp. 11545–11557, 2020. doi: https://doi.org/10.1016/j.jfranklin.2019.07.023
  27. Z. Xu, D. Peng, and X. Li, “SYNCHRONIZATION OF CHAOTIC NEURAL NETWORKS WITH TIME DELAY VIA DISTRIBUTED DELAYED IMPULSIVE CONTROL,” Neural Networks, vol. 118, pp. 332–337, 2019. doi: https://doi.org/10.1016/j.neunet.2019.07.002
  28. J. Yu, J. Lei, and L. Wang, “BACKSTEPPING SYNCHRONIZATION OF CHAOTIC SYSTEM BASED ON EQUIVALENT TRANSFER FUNCTION METHOD,” Optik (Stuttg)., vol. 130, pp. 900–913, 2017. doi: https://doi.org/10.1016/j.ijleo.2016.11.007
  29. American Psychiatric Association, DIAGNOSTIC AND STATISTICAL MANUAL OF MENTAL DISORDER. Washington DC: American Psychiatric Association, 2005.
  30. R. Luo and Y. Zeng, “THE CONTROL OF CHAOTIC SYSTEMS WITH UNKNOWN PARAMETERS AND EXTERNAL DISTURBANCE VIA BACKSTEPPING LIKE SCHEME,” Complexity, vol. 21, no. S1, pp. 573–583, 2016. doi: https://doi.org/10.1002/cplx.21771
  31. K. A. Alattas, J. Mostafaee, A. Sambas, A. K. Alanazi, and S. Mobayen, “NONSINGULAR INTEGRAL-TYPE DYNAMIC FINITE-TIME SYNCHRONIZATION FOR HYPER-CHAOTIC SYSTEMS,” Mathematics, vol. 10, no. 1, p. 115, 2022. doi: https://doi.org/10.3390/math10010115
  32. I. Ahmad, A. Bin, A. Binti, and M. Shahzad, “THE SYNCHRONIZATION OF CHAOTIC SYSTEMS WITH DIFFERENT DIMENSIONS BY A ROBUST GENERALIZED ACTIVE CONTROL,” Optik (Stuttg)., vol. 127, p. 94915, 2016. doi: https://doi.org/10.1016/j.ijleo.2015.12.134
  33. H. Gritli, “ROBUST MASTER-SLAVE SYNCHRONIZATION OF CHAOS IN A ONE-SIDED 1-DOF IMPACT MECHANICAL OSCILLATOR SUBJECT TO PARAMETRIC UNCERTAINTIES AND DISTURBANCES,” Mech. Mach. Theory, vol. 142, p. 103610, 2019. doi: https://doi.org/10.1016/j.mechmachtheory.2019.103610
  34. Q. Yao, “SYNCHRONIZATION OF SECOND ORDER CHAOTIC SYSTEMS WITH UNCERTAINTIES AND DISTURBANCES USING FIXED-TIME ADAPTIVE SLIDING MODE CONTROL,” Chaos, Solitons and Fractals, vol. 142, p. 110372, 2020. doi: https://doi.org/10.1016/j.chaos.2020.110372
  35. R. Chen, Z. Wang, and W. Che, “ADAPTIVE SLIDING MODE ATTITUDE-TRACKING CONTROL OF SPACECRAFT WITH PRESCRIBED TIME PERFORMANCE,” Mathematics, 2022. doi: https://doi.org/10.3390/math10030401
  36. J. Wu, “ROBUST ADAPTIVE FINITE-TIME AND FIXED-TIME SYNCHRONIZATION OF CHAOTIC SYSTEMS WITH SMOOTH CONTROL,” Int. J. Robust Nonlinear Control, no. July 2020, pp. 1–15, 2021. doi: https://doi.org/10.1002/rnc.5750
  37. Y. N. Golouje and seyyed M. Abtahi, “CHAOTIC DYNAMICS OF THE VERTICAL MODEL IN VEHICLES AND CHAOS CONTROL OF ACTIVE SUSPENSION SYSTEM VIA THE FUZZY FAST TERMINAL SLIDING MODE CONTROL,” J. Mech. Sci. Technol., vol. 35, no. 1, pp. 31–43, 2021. doi: https://doi.org/10.1007/s12206-020-1203-3
  38. M. Zhong, C. Huang, J. Cao, and H. Liu, “ADAPTIVE FUZZY ECHO STATE NETWORK OPTIMAL SYNCHRONIZATION CONTROL OF HYBRID–ORDER CHAOTIC SYSTEMS VIA REINFORCEMENT LEARNING,” Chaos, Solitons and Fractals, vol. 181, no. February, p. 114665, 2024. doi: https://doi.org/10.1016/j.chaos.2024.114665
  39. Z. Li, J. B. Park, Y. H. Joo, and B. Zhang, “BIFURCATIONS AND CHAOS IN A PERMANENT-MAGNET SYNCHRONOUS MOTOR,” IEEE Trans. Circuits Syst. I Fundam. Theory Appl., vol. 49, no. 3, pp. 383–387, 2002. doi: https://doi.org/10.1109/81.989176
  40. Z. Jing, C. Yu, and G. Chen, “COMPLEX DYNAMICS IN A PERMANENT-MAGNET SYNCHRONOUS MOTOR MODEL Q,” Chaos, Solitons and Fractals, vol. 22, no. 4, pp. 831–848, 2004. doi: https://doi.org/10.1016/j.chaos.2004.02.054
  41. Q. I. Dong-lian, W. Jia-jun, and Z. Guang-zhou, “PASSIVE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR CHAOTIC SYSTEMS,” J. Zhejiang Univ. Sci. Phys. Eng., vol. 6, no. 7, pp. 728–732, 2005. doi: https://doi.org/10.1631/jzus.2005.A0728
  42. M. Ataei, A. Kiyoumarsi, and B. Ghorbani, “CONTROL OF CHAOS IN PERMANENT MAGNET SYNCHRONOUS MOTOR BY USING OPTIMAL LYAPUNOV EXPONENTS PLACEMENT,” Phys. Lett. A, vol. 374, no. 41, pp. 4226–4230, 2010. doi: https://doi.org/10.1016/j.physleta.2010.08.047
  43. H. H. Choi, “ADAPTIVE CONTROL OF A CHAOTIC PERMANENT MAGNET,” Nonlinear Dyn., vol. 69, pp. 1311–1322, 2012. doi: https://doi.org/10.1007/s11071-012-0349-7
  44. J. Hu, Y. Qiu, and H. Lu, “ADAPTIVE ROBUST NONLINEAR FEEDBACK CONTROL OF CHAOS IN PMSM SYSTEM WITH MODELING UNCERTAINTY,” Appl. Math. Model., vol. 40, no. 19–20, pp. 8265–8275, 2016, doi: 10.1016/j.apm.2016.04.019. doi: https://doi.org/10.1016/j.apm.2016.04.019
  45. A. Iqbal and G. K. Singh, “CHAOS CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR USING SIMPLE CONTROLLERS,” Trans. Inst. Meas. Control, vol. 41, no. 8, pp. 2352–2364, 2018. doi: https://doi.org/10.1177/0142331218799830
  46. Z. Sun, L. Si, Z. Shang, and J. Lei, “FINITE-TIME SYNCHRONIZATION OF CHAOTIC PMSM SYSTEMS FOR SECURE COMMUNICATION AND PARAMETERS IDENTIFICATION,” Optik (Stuttg)., vol. 157, pp. 43–55, 2018. doi: https://doi.org/10.1016/j.ijleo.2017.09.057
  47. M. Zribi, A. Oteafy, and N. Smaoui, “CONTROLLING CHAOS IN THE PERMANENT MAGNET SYNCHRONOUS MOTOR,” Chaos, Solitons and Fractals, vol. 41, no. 3, pp. 1266–1276, 2009. doi: https://doi.org/10.1016/j.chaos.2008.05.019
  48. J. Zhang, J. Sun, H. Gu, H. Poloei, and A. Karami, “CONTROL OF PMSM CHAOS USING BACKSTEPPING-BASED ADAPTIVE FUZZY METHOD IN THE,” Syst. Sci. Control Eng., vol. 9, no. 1, pp. 673–688, 2021. doi: https://doi.org/10.1080/21642583.2021.1980130.