zbMATH — the first resource for mathematics

On nonlinear control design for autonomous chaotic systems of integer and fractional orders. (English) Zbl 1073.93027
The authors present via numerical simulations the viability of the “backstepping” method as a design methodology for nonlinear chaos control. Using the “backstepping” method, the authors derive nonlinear controllers for the two chaotic models in this study. The controllers act in such a way as to drive the chaotic output trajectories to the nearest equilibrium points in the basins of attraction. Moreover, the derived controllers show robustness against total system order reduction arising from the use of fractional integrators in the models.

93C10 Nonlinear systems in control theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI
[1] Cuomo, K.; Oppenheim, A.; Strogatz, S., Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE trans. circ. syst. II, 40, 10, (1993)
[2] Huang, X.; Xu, J., Realization of the chaotic secure communication based on interval synchronization, J. xi’an jiatong univ., 33, 56-58, (1999)
[3] Bai, E.; Lonngren, K.; Sprott, J., On the synchronization of a class of electronic circuits that exhibit chaos, Chaos, solitons & fractals, 13, 1515-1521, (2002) · Zbl 1005.34041
[4] Al-Assaf Y, Ahmad W. Parameter identification of chaotic systems using waveltes and neural networks, Int J Bifurc Chaos, in press · Zbl 1084.37504
[5] Abed, E.H.; Fu, J.-H, Local feedback stabilization and bifurcation control. I. Hopf bifurcation, Syst. control lett., 7, 11-17, (1986) · Zbl 0587.93049
[6] Wang HO, Abed EH. Control of nonlinear phenomena at the inception of voltage collapse. In: Proceedings of the 1993 American Control Conference, San Fransisco, June 1993, p. 2071-5
[7] Harb, A.M.; Nayfeh, A.H.; Chin, C.; Mili, L., On the effects of machine saturation on subsynchronous oscillations in power systems, Electr. Mach. power syst. J., 28, 11, (1999)
[8] Abed, E.H.; Varaiya, P.P., Nonlinear oscillations in power systems, Int. J. electr. power energy syst., 6, 37-43, (1989)
[9] Abed EH, Alexander JC, Wang H, Hamdan AH, Lee HC. Dynamic bifurcation in a power system model exhibiting voltage collapse. In: Proceedings of the 1992 IEEE International Symposium on Circuit and Systems, San Diego, CA, 1992
[10] Abed, E.H.; Fu, J.H., Local feedback stabilization and bifurcation control. I. Hopf bifurcation, Syst. control lett., 7, 11-17, (1986) · Zbl 0587.93049
[11] Abed, E.H.; Fu, J.H., Local feedback stabilization and bifurcation control. II. stationary bifurcation, Syst. control lett., 8, 467-473, (1987) · Zbl 0626.93058
[12] Nayfeh, A.H.; Balachandran, B., Applied nonlinear dynamics, (1994), John Wiley New York
[13] Khalil, H., Nonlinear systems, (1996), Prentice Hall
[14] Harb A, Ahmad W. Control of chaotic oscillators using a nonlinear recursive backstepping controller. In: IASTED Conference on Applied Simulations and Modeling, Crete, Greece, June 2002, p. 451-3
[15] Krstic, M.; Kanellakopoulus, I.; Kokotovic, P., Nonlinear and adaptive control design, (1995), John Wiley & Sons Inc
[16] Zaher A, Zohdy M. Robust control of biped robots. In: Proceedings of ACC, Chicago IL, USA, June 2000, p. 1473-8
[17] Zaher A, Zohdy M, Areed F, Soliman K. Real-time model-reference control of non-linear processes. In: 2nd International Conference on Computers in Industry, Bahrain, November 2000
[18] Sprott, J.C., Simple chaotic systems and circuits, Am. J. phys., 68, 758-763, (2000)
[19] Zaher A, Zohdy M, Areed F, Soliman K. Robust model-reference control for a class of non-linear and piece-wise linear systems. In: Proceedings of ACC, Arlington VA, USA, June 2001, p. 4514-9
[20] Charef, A.; Sun, H.H.; Tsao, Y.Y.; Onaral, B., Fractal system as represented by singularity function, IEEE trans. autom. control, 37, 9, (1992) · Zbl 0825.58027
[21] Ahmad, W.; Sprott, C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons & fractals, 16, 2, 339-351, (2003) · Zbl 1033.37019
[22] Elwakil, A.S.; Kennedy, M.P., Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices, IEEE trans. circ. syst., 48, 3, (2001) · Zbl 0998.94048
[23] Ikhouane, F.; Krstic, M., Robustness of the tuning functions adaptive backstepping design for linear systems, IEEE trans. autom. control, 43, 3, 431-437, (1998) · Zbl 0908.93038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.