Anti-Control of Chaos of Linear Continuous-Time Systems with Unknown Parameters Using Adaptive Control

Abstract:

In this paper, a feedback control method is proposed for the anti-control of chaos of linear controllable systems based on model-matching. First, it is considered that the linear system is completely known and an anti-control method is designed. Then, the parameters of the linear controllable system in companion form are assumed to be unknown. The chaotification is achieved choosing an appropriate control law and a parametric updating law based on Lyapunov stability theory, which provides the stability of the resulting adaptive system and the convergence of the tracking errors to zero. The proposed method is applied to anti-control of chaos of a linear system, while the Rössler chaotic system is the reference model. The numerical simulation results show the effectiveness of the proposed method.

• Periodical: Advanced Materials Research (Volumes 403-408)
• Pages: 4806-4813
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.403-408.4806
• Online since: November 2011
• Authors: Farzaneh Akhgari, Zahra Rahmani, Behrooz Rezaie
• Keywords: Adaptive Control, Anti-Control of Chaos, Linear Controllable Systems, Rössler System

References

[1] X. Wu, J. Lu, H. Iu and S. Wong, Suppression and generation of chaos for a three-dimensional autonomous system using parametric perturbations, Chaos Solitons Fractals, 2007, 31: 811–819.

DOI: 10.1016/j.chaos.2005.10.050

[2] Z. M. Ge , C. M. Chang and Y. S. Chen, Anti-control of chaos of single time-scale brushless dc motors and chaos synchronization of different order systems, Chaos, Solitons and Fractals, 2006, 27, 1298–1315.

DOI: 10.1016/j.chaos.2005.04.095

[3] R. T. Yang, Y. G. Hong, H. S. Qin and G. R. Chen, Anticontrol of chaos for dynamic systems in p-normal form: a homogeneity-based approach, Chaos Solitons Fractals, 2005, 25: 687–697.

DOI: 10.1016/j.chaos.2004.11.090

[4] H. Huijberts, H. Nijmeijer and R. Willems, System identification in communication with chaotic systems, IEEE Trans. Circuits Syst I, 2000, 47(6): 800–808.

DOI: 10.1109/81.852932

[5] S. Sinha and W. L. Ditto, Computing with distributed chaos, Phys Rev E, 1999, 60(1): 363–376.

[6] S. J. Schiff, K. D. Jerger, H. Duong, T. Chang, M. L. Spano and W. L. Ditto, Controlling chaos in the brain, Nature, 1994, 370: 615–620.

DOI: 10.1038/370615a0

[7] W. Yang, M. Ding, A. J. Mandell and E. Ott, Preserving chaos: control strategies to preserve complex dynamics with potential relevance to biological disorders, Phys Rev E, 1995, 51: 102–110.

DOI: 10.1103/physreve.51.102

[8] G. Chen, J. Q. Fang, Y. Hong and H. S. Qin, Introduction to chaos control and anti-control, In: Leung TP, Qin HS (eds) Advanced topics in nonlinear control systems, Chap 6. World Scientific, Singapore, 2001, p.193–245.

DOI: 10.1142/9789812798541_0006

[9] G. Chen and X. Dong, From chaos to order: methodologies, perspectives and applications, Singapore: World Scientific, (1998).

[10] I. T. Georgiou and I. B. Schwartz, Dynamics of large scale coupled structural/mechanical systems: A singular perturbation/proper orthogonal decomposition approach, SIAM J. Appl. Math., 1999, 59: 1178–1207.

DOI: 10.1137/s0036139997299802

[11] X. F. Wang, G. R. Chen and K. F. Man, Making a continuous-time minimum-phase system chaotic by using time-delay feedback, IEEE Trans Circuits Syst I, 2001, 48: 641–645.

DOI: 10.1109/81.922469

[12] L. Yang, Z. Liu and G. R. Chen, Chaotifying a continuous-time system via impulsive input, Int J Bifurc Chaos, 2002, 12: 1121–1128.

DOI: 10.1142/s0218127402004954

[13] H. G. Zhang, Z. L. Wang and D. Liu, Chaotifying fuzzy hyperbolic model using impulsive and nonlinear feedback control approaches, Int J Bifurc Chaos, 2005, 15: 2603–2610.

DOI: 10.1142/s021812740501354x

[14] D. Chen, H. Wang and G. Chen, Anti-control of Hopf bifurcation, IEEE Trans Circuits Syst I, 2001, 48: 661–672.

DOI: 10.1109/81.928149

[15] J. G. Lu, Chaotic behavior in sampled-data control systems with saturating control, Chaos Solitons Fractals, 2006, 30: 147–155.

DOI: 10.1016/j.chaos.2005.08.191

[16] Q. F. Chen, Q. H. Zhong, Y. G. Hong and G. R. Chen, Generation and control of spherical circular attractors using switching schemes, Int J Bifurc Chaos, 2007, 17: 243–253.

DOI: 10.1142/s021812740701729x

[17] Z. Li, J. B. Park, G. R. Chen, Y. H. Joo and Y. H. Choi, Generating chaos via feedback control from a stable TS fuzzy system through a sinusoidal nonlinearity, Int J Bifurc Chaos, 2002, 12: 2283–2291.

DOI: 10.1142/s0218127402005844

[18] G. Chen and X. Yu, Chaos control, LNCIS 292, pp.179-204, 2003, Springer-Verlag Berlin Heidelberg.

[19] M. W. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, New York: Academic Press, (1974).

[20] J. C. Sprott, Some simple chaotic jerk functions, Amer. J. Phys., 1997, 65: 537–543.

DOI: 10.1119/1.18585

[21] T. Kailath, Linear systems, Englewood Cliffs, NJ: Prentice Hall, (1980).

[22] J. E. Slotine and W. Li, Applied nonlinear control, Prentice Hall, (1991).

[23] J. Lu, D. Zhang, Y. Sun and Y. Wu, Application of normal form in chaotic synchronization, American Control Conference, USA, (2005).

[24] J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, (2003).