Feedback Control of the T Chaotic System

Abstract

In this study, the chaotic behavior of the T system is shown by analyzing Lyapunov exponents, Kaplan-Yorke dimension and equilibrium points. Also, the chaos control of T system showing chaotic behavior under certain parameters is investigated via linear feedback control. Routh-Hurwitz criterion is used to provide the condition of stability of the controlled system around the equilibrium points. The feedback gain is calculated to control the T chaotic system with single state feedback, then the same calculations repeated for the control of the system with two and three state feedback. The controller designed by calculating the feedback coefficients using the Routh-Hurwitz criterion is added to the T chaotic system. It is shown by phase portraits how the change of the feedback gains affects the control of the system. The design and implementation of the controller is quite simple. Due to the simple structure of the controller, the controller hardware is low and realized conveniently. Simulation results show that the controlled system with the application of feedback control converges to zero and other equilibrium points.

Keywords

• Chaotic T system
• chaos control
• feedback control
• Routh-Hurwitz Method

T Kaotik Sisteminin Geri Besleme ile Kontrolü

Abstract

Bu çalışmada, T sisteminin kaotik davranışı Lyapunov üstelleri, Kaplan-Yorke boyutu ve denge noktaları analiz edilerek gösterilmiştir. Daha sonra belirli parametreler altında kaotik davranış gösteren T sisteminin doğrusal geri beslemeli kontrol yardımıyla kontrolü araştırılmıştır. Denge noktaları etrafında kontrol edilen sistemin kararlılığını sağlamak için Routh-Hurwitz kriteri kullanılmıştır. T kaotik sistemini tek durum değişkeni geri besleme ile kontrol etmek için geri besleme katsayısı hesaplanmış daha sonra iki ve üç durum geri besleme ile sistemin kontrolü için aynı işlemler tekrarlanmıştır. Routh-Hurwitz kriteri kullanılarak geri besleme katsayıları hesaplanarak tasarlanan kontrolör T kaotik sistemine eklenmiştir. Kontrollü sistemde geri besleme katsayısı değişiminin sistemin kontrolünü nasıl etkilediği faz portreleri ile gösterilmiştir. Doğrusal geri beslemeli kontrolörün tasarlanması ve uygulanması oldukça basittir. Kontrolör basit yapıya sahip olduğundan donanımsal olarak maliyeti düşüktür ve pratik olarak gerçeklenmesi kolaydır. Geri beslemeli kontrolün uygulanmasıyla kontrollü sistemin sıfır ve diğer denge noktalarına yakınsadığı bilgisayar benzetim çalışmalarıyla gösterilmiştir.

Keywords

• Kaotik T sistemi
• kaos kontrolü
• geri beslemeli kontrol
• Routh-Hurwitz kriteri

References

1. Chen G, Ueta T, 1999. Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 9 (7): 1465-1466.
2. Chithra, A., Raja Mohamed, I., 2017. Synchronization and chaotic communication in nonlinear circuits with nonlinear coupling. Journal of Computational Electronics, 16(3), 833–844.
3. Emiroglu, S., Uyaroglu, Y., 2017. Time Delay Feedback Control based Chaos Stabilization in Current Mode Controlled DC Drive System, International Journal of Engineering and Applied Sciences, 4(10), pp. 100-103.
4. Fu, S., Liu, Y., Ma, H. Du, Y., 2020. Control chaos to different stable states for a piecewise linear circuit system by a simple linear control, Chaos, Solitons and Fractals, 130, 109431.
5. Gholamin, P., Refahi Sheikhani, A. H., 2017. A new three-dimensional chaotic system: Dynamical properties and simulation, Chinese Journal of Physics, 55(4), pp. 1300-1309.
6. Greiner, W., 2010. Lyapunov Exponents and Chaos, Classical Mechanics, pp.503-516.
7. Hu, Z., Chan C-K., 2018. A 7-D Hyperchaotic System-Based Encryption Scheme for Secure Fast-OFDM-PON, Journal of Lightwave Technology, 36(16), pp. 3373-3381.
8. Jian H, Yang Q, Hui L, 2016. Adaptive robust nonlinear feedback control of chaos in PMSM system with modeling uncertainty. Applied Mathematical Modelling, 40 (19):8265-8275.

9. Joshi, M., Ranjan, A., 2019, New simple chaotic and hyperchaotic system with an unstable node, AEU- International Journal of Electronics and Communications, 108, pp. 1-9.
10. Kocamaz UE, Uyaroğlu Y, Kızmaz H, 2017. Controlling hyperchaotic Rabinovich system with single state controllers: Comparison of linear feedback, sliding mode, and passive control methods. Optik, 130: 159-167.
11. Kocamaz, UE; Cicek, S; Uyaroglu, Y., 2018. Secure Communication with Chaos and Electronic Circuit Design Using Passivity-Based Synchronization, 27(4), 1850057.
12. Liu,CX, Tao L Ling L, Kai L, 2004. A new chaotic attractor. Chaos, Solitons and Fractals, 22 (5): 1031–1038.
13. Lorenz, EN, 1963. Deterministic non-periodic flows. Journal of the Atmospheric Sciences, 20: 130–141.
14. Murali K, Lakshmanan M, Chua LO, 1995. Controlling and Synchronization of Chaos in the Simplest Dissipative Nonautonomous Circuit. International Journal of Bifurcation and Chaos, 5 (2): 563-571.
15. Nazzal JM, Natsheh AN, 2007. Chaos control using sliding-mode theory. Chaos, Solitons and Fractals, 33 (2): 695-702.
16. Ott E, Grebogi C, York JA, 1990. Controlling chaos. Physical Review Letters, 64 (11): 1196-1199.
17. Rössler OE, 1976, An equation for continuous chaos, Physical Review Letters A, 57 (5): 397–398.
18. Sabaghian, A., Balochian S., Yaghoobi, M., 2020. Synchronisation of 6D hyper-chaotic system with unknown parameters in the presence ofdisturbance and parametric uncertainty with unknown bounds, Connection Science, pp.1-22.
19. Shivamoggi, B.K.,2014. Chaos in Dissipative Systems. In: Nonlinear Dynamics and Chaotic Phenomena: An Introduction. Fluid Mechanics and Its Applications, vol 103. Springer, Dordrecht, pp. 189.
20. Singh, J.P., Roy, B. K., 2018. Five new 4-D autonomous conservative chaotic systems with various type of non-hyperbolic and lines of equilibria, Chaos, Solitons and Fractals, 114, pp.81-91.
21. Singh, J.P., Rajagopal, K., Roy, B. K, 2018. A new 5D hyperchaotic system with stable equilibrium point, transient chaotic behaviour and its fractional-order form, Pramana, 91 (33), pp.1-10.
22. Sprott, J.C., 2003. Chaos and time series-analysis, Oxford: Oxford University Press.
23. Tigan Gh, 2005. Analysis of a dynamical system derived from the Lorenz system. Scientific Bulletin of the Politehnica University of Timisoara, 50 (64): 61-72.
24. Tigan Gh, 2008. Analysis of a 3D chaotic system. Chaos Soliton and Fractals, 36 (5): 1315-1319.
25. Ullah, M. Z., Mallawi, F., Baleanu D, Alshomrani, A. S., 2020. A new fractional study on the chaotic vibration and state-feedback control of a nonlinear suspension system, Chaos, Solitons and Fractals, 132, 109530.
26. Uyaroğlu Y, 2006. Kaotik Lorenz Sisteminin Yarı-Ayna Yapısı. Journal of İstanbul Kültür University, 3: 141-146.
27. Yassen MT, 2005. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons and Fractals, 26 (3): 913–920.
28. Zhang, H., Liu, D., Wang, Z.,2009. Controlling Chaos: Suppression, Synchronization and Chaotification, Springer-Verlag, London.