Control of Lorenz Chaotic System with Linear Feedback, High Gain, High Frequency and Model Predictive Control

Abstract

There are many control methods for systems that have been developed and continued to be developed, linear or non-linear, time-varying or invariant. In this study, in the control of the nonlinear Lorenz Chaotic system, model predictive control method with integrator action, which has not been applied to this system before, is applied. In addition to this method, different nonlinear control methods were applied to the same system and the control of the Lorenz Chaotic system was realized. Feedback control, high gain control, high frequency control and model predictive control techniques are used in the selected control methods. In addition, the methods used were obtained mathematically and their advantages and disadvantages were revealed. As a result, it has been shown that this type of nonlinear control methods can control the Lorenz chaotic system, and then the advantages and disadvantages are discussed in the results section and information about future studies is given.

Keywords

• Chaotic system
• feedback control
• high gain control
• high frequency control
• model predictive control

Lorenz Kaotik Sisteminin Doğrusal Geri Beslemeli, Yüksek Kazanç, Yüksek Frekans ve Model Öngörülü Kontrol ile Denetlenmesi

Abstract

Günümüze kadar gelişen ve geliştirilmeye devam edilen, doğrusal veya doğrusal olmayan, zamanla değişen veya zamanla değişmeyen sistemler için birçok kontrol yöntemleri bulunmaktadır. Bu çalışmada doğrusal olmayan Lorenz Kaotik sisteminin kontrolünde daha önceden bu sisteme uygulanmamış olan integratör içeren model öngörülü kontrol yöntemi uygulamıştır. Bu yöntemin yanı sıra aynı sisteme farklı doğrusal olmayan kontrol yöntemleri de uygulanarak Lorenz Kaotik sisteminin kontrolü gerçekleştirilmiştir. Seçilen kontrol yöntemlerinde geri beslemeli kontrol, yüksek kazanç kontrol, yüksek frekans kontrol ve model öngörülü kontrol teknikleri kullanılmıştır. Ayrıca kullanılan yöntemler matematiksel olarak elde edilmiş, avantaj ve dezavantajlarını ortaya konulmuştur. Sonuçta doğrusal olmayan bu tip kontrol yöntemlerinin Lorenz kaotik sisteminin kontrol edebildiği gösterilmiş ardından avantaj ve dezavantajları sonuçlar bölümünde tartışılmış ve ileriye yönelik çalışmalar hakkında bilgiler verilmiştir.

Keywords

• Kaotik sistem
• geri beslemeli kontrol
• yüksek kazanç kontrolü
• yüksek frekans kontrolü
• model öngörülü kontrol

References

1. [1] V. G. Ivancevic and T. T. Ivancevic, High-dimensional chaotic and attractor systems: a comprehensive introduction. Springer Science & Business Media, 2007.
2. [2] W. J. Burroughs, Climate change in prehistory: The end of the reign of chaos. Cambridge University Press, 2005.
3. [3] S. Solhjoo, A. M. Nasrabadi, and M. R. H. Golpayegani, "Classification of chaotic signals using HMM classifiers: EEG-based mental task classification," in 2005 13th European Signal Processing Conference, 2005: IEEE, pp. 1-4.
4. [4] G.-Q. Wu et al., "Chaotic signatures of heart rate variability and its power spectrum in health, aging and heart failure," PloS one, vol. 4, no. 2, 2009.
5. [5] D. S. Gutzler, "Ecological Climatology: Concepts and Applications," ed: JSTOR, 2003.
6. [6] R. Steinitz, "Music, maths & chaos," The Musical Times, vol. 137, no. 1837, pp. 14-20, 1996.
7. [7] J. Harley, "Generative processes in algorithmic composition: Chaos and music," Leonardo, vol. 28, no. 3, pp. 221-224, 1995.
8. [8] M. Baptista, "Cryptography with chaos," Physics letters A, vol. 240, no. 1-2, pp. 50-54, 1998.

9. [9] M. Sharafi, F. Fotouhi-Ghazvini, M. Shirali, and M. Ghassemian, "A low power cryptography solution based on chaos theory in wireless sensor nodes," IEEE Access, vol. 7, pp. 8737-8753, 2019.
10. [10] J. S. Khan and J. Ahmad, "Chaos based efficient selective image encryption," Multidimensional Systems and Signal Processing, vol. 30, no. 2, pp. 943-961, 2019.
11. [11] M. E. Cimen, Z. B. Garip, M. A. Pala, A. F. Boz, and A. Akgul, "Modelling of a Chaotic System Motion in Video with Artiıficial Neural Networks," Chaos Theory and Applications, vol. 1, no. 1, pp. 38-50.
12. [12] M. E. Çimen, S. Kaçar, E. GÜLERYÜZ, B. Gürevin, and A. Akgül, "Kaotik bir hareket videosunun yapay sinir ağları ile modellenmesi," Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 20, no. 3, pp. 23-35, 2018.
13. [13] E. OTT, C. GREBOGI, and J. YORKE, "Physical Review etters 64, 1196} 1199," Controlling chaos, 1990.
14. [14] T. L. V. J. YU, "Control of a Chaotic System," Dynamics and Control vol. 1, pp. 35-52, 1990.
15. [15] A. Mohammadbagheri and M. Yaghoobi, "Lorenz-Type Chaotic attitude control of satellite through predictive control," in 2011 Third International Conference on Computational Intelligence, Modelling & Simulation, 2011: IEEE, pp. 147-152.
16. [16] Lu, Xia. "A Financial Chaotic System Control Method Based on Intermittent Controller." Mathematical Problems in Engineering, 2020, doi:10.1155/2020/5810707.
17. [17] T. Yeap and N. Ahmed, "Feedback control of chaotic systems," Dynamics and Control, vol. 4, no. 1, pp. 97-114, 1994.
18. [18] Y. Zeng and S. N. Singh, "Adaptive control of chaos in Lorenz system," Dynamics and Control, vol. 7, no. 2, pp. 143-154, 1997.
19. [19] S.-K. Y. C.-L. C. H.-T. Yau, "Control of chaos in Lorenz system," Chaos, Solitons and Fractals vol. 13, pp. 767-780, (2002).
20. [20] P. P. Cardenas Alzate, G. C. Velez, and F. Mesa, "Chaos control for the Lorenz system," Advanced Studies in Theoretical Physics, vol. 12, no. 4, pp. 181-188, 2018, doi: 10.12988/astp.2018.8413.
21. [21] H.-T. Yau and J.-J. Yan, "Design of sliding mode controller for Lorenz chaotic system with nonlinear input," Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 891-898, 2004, doi: 10.1016/s0960-0779(03)00255-8.
22. [22] M. Chen, D. Zhou, and Y. Shang, "Nonlinear feedback control of Lorenz system," Chaos, Solitons & Fractals, vol. 21, no. 2, pp. 295-304, 2004, doi: 10.1016/j.chaos.2003.12.066.
23. [23] M. T. Yassen, "Chaos control of chaotic dynamical systems using backstepping design," Chaos, Solitons & Fractals, vol. 27, no. 2, pp. 537-548, 2006, doi: 10.1016/j.chaos.2005.03.046.
24. [24] L. Run-Zi, "Impulsive control and synchronization of a new chaotic system," Physics Letters A, vol. 372, no. 5, pp. 648-653, 2008, doi: 10.1016/j.physleta.2007.08.010.
25. [25] E. Daş, “Güdümlü Bir Mühimmat Kanatçık Tahrik Sistemi İçin İki Döngülü Kontrol Sistemi Tasarımı,” Fen Bilimleri Enstitüsü, İTÜ, 2014.
26. [26] E. Köse and A. Mühürcü, "Comparative Controlling of the Lorenz Chaotic System Using the SMC and APP Methods," Mathematical Problems in Engineering, vol. 2018, pp. 1-9, 2018, doi: 10.1155/2018/9612749.
27. [27] V.-T. Pham, S. T. Kingni, C. Volos, S. Jafari, and T. Kapitaniak, "A simple three-dimensional fractional-order chaotic system without equilibrium: Dynamics, circuitry implementation, chaos control and synchronization," AEU-international Journal of Electronics and Communications, vol. 78, pp. 220-227, 2017.
28. [28] L. Zhang and Y. Yan, "Discrete active model predictive control of continuous unified chaotic system," in 2019 Chinese Control And Decision Conference (CCDC), 2019: IEEE, pp. 3390-3394.
29. [29] K.-S. Park, J.-M. Joo, J.-B. Park, Y.-H. Choi, and T.-S. Yoon, "Control of discrete-time chaotic systems using generalized predictive control," in Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS'97, 1997, vol. 2: IEEE, pp. 789-792.
30. [30] S. M. Tabatabaei, S. Kamali, M. R. Jahed-Motlagh, and M. B. Yazdi, "Practical Explicit Model Predictive Control for a Class of Noise-embedded Chaotic Hybrid Systems," International Journal of Control, Automation and Systems, vol. 17, no. 4, pp. 857-866, 2019.