In this paper we propose a new adaptive feedback controller for linearizable chaotic systems with uncertainties. Based on the Lyapunov stability theory, the adaptation law is determined to tune the controller gains in order to track a predetermined linearizing feedback control. More importantly, this control method has a simple controller structure, high robustness against uncertainties and strong rejection of external disturbances. A remarkable feature of the proposed approach is that it can be used for chaos control as well as chaos synchronization. Numerical simulations of two well-known chaotic systems are illustrated to show the effectiveness and robustness of the proposed adaptive control strategy.
Awrejcewich, J. and Lamarque, C. H. , 2003, Bifurcation and Chaos in Non-smooth Mechanical Systems, World Scientific, Singapore .
2.
Awrejcewich, J. , Tomczak, K. , and Lamarque, C. H. , 1999, “Controlling systems with impacts,”International Journal of Bifurcation and Chaos9, 547-553 .
3.
Bocalleti, S. , Grebogi, C. , Lai, Y-C. , Maricini, H. , and Maza, D. , 2000, “The control of chaos: Theory and application,”Physical Reports329, 103-197 .
4.
Brocket, R. W. , 1978, “Feedback invariants for nonlinear systems,” in Proceedings of the IFAC World Congress, Helsinki, Finland, pp. 1115-1120 .
5.
Castro, R. , Alvarez, J. , and Alvarez, J. , 1990, “Nonlinear disturbance decoupling control of a binary distillation column,”Automatica26, 567-572 .
6.
Chua, L. O. , Yang, T. , Zhong, G. Q. , and Wu, C. W. , 1996, “Adaptive synchronization of Chua oscillator,”International Journal of Bifurcation and Chaos6, 455-462 .
7.
Di Bernado, M. , 1996, “An adaptive approach to the control and synchronization of continuous-time systems,”International Journal of Bifurcation and Chaos6, 557-568 .
8.
Doyle, F. J. III , Packard, A. K. , and Morari, M. , 1989, “Robust controller design for a nonlinear CSTR,”Chemical Engineering Science44, 192-1947 .
9.
Feki, M. , 2003, “An adaptive feedback control of linearizable chaotic systems,”Chaos, Solitons and Fractals15, 883-890 .
10.
Femat, R. , Alvarez-Ramirez, J. , Castillo-Toledo, B. , and Gonzalez, J. , 1999, “On robust chaos suppression in a class of non-driven oscillators: Application to Chua’s circuit,”IEEE Transactions on Circuits and Systems: Fundamental Theory and Applications46, 1150-1152 .
11.
Henson, M. and Seborg, D. E. , 1991, “Critique of exact linearization strategies for process control,”Journal of Process Control1, 122-139 .
12.
Hunt, L. R. , Su, R. , and Meyer, G. , 1983, “Global transformation of nonlinear systems,” in Differential Geometric Control Theory, R. W. Brocket , R. S. Millman , and H. J. Sussman , eds., Birkhauser, Boston .
13.
Isidori, A. , 1995, Nonlinear Control Systems, Springer-Verlag, Berlin .
14.
Jiang, Z. P. and Praly, L. , 1998, “Design of robust adaptive controllers for nonlinear systems with dynamics uncertainties,”Automatica34, 825-832 .
15.
Kapitaniak, T. , 1996, Controlling Chaos, Academic Press, London .
16.
Khalil, H. K. , 1995, Nonlinear Systems, 3rd edition, Springer-Verlag, Berlin .
17.
Kravaris, C. and Palanki, S. , 1988, “Robust nonlinear state feedback under structured uncertainty,”A.I.Ch.E. Journal47, 1119-1127 .
18.
Lévine, J. and Rouchon, P. , 1991, “Quality control of binary distillation columns via nonlinear aggregated models,”Automatica27, 463-480 .
19.
Liao, T-L. , Fu, L-Ch. , and Hsu, Ch-F. , 1992, “Output tracking control of nonlinear uncertain systems with mismatched uncertainties,”Systems Control Letters18, 39-47 .
20.
Marino, R. and Tomei, P. , 1995, Nonlinear Control Design: Geometric, Adaptive and Robust, Prentice-Hall, Englewood Cliffs, NJ .
21.
Moukam, F. , Bowong, S. , and Tchawoua, C. , 2001, “Exponential stabilization of two coupled nonlinearly oscillators by an estimated state feedback,”Physica Scripta6, 1-7 .
22.
Moukam, F. , Bowong, S. , Tchawoua, C. , and Kaptouom, E. , 2004, “Chaos control and synchronization of a 6-Van der Pol oscillator,”Physics Letters A322, 305-323 .
23.
Nijmeijer, H. and Berghuis, H. , 1995, “On Lyapunov control of the Duffing equation,”IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications42, 473-482 .
24.
Ott, E. , Grebogi, C. , and Yorke, J. A. , 1990, “Controlling chaos,”Physical Review Letters65, 1196-1199 .
25.
Susmann, H. J. , and Kokotovic, P. V. , 1991, “The peaking phenomenon and the global stabilization of nonlinear systems,”IEEE Transactions on Automatic and Control36, 461-476 .
26.
Yalçn, M. E. , Suykens, J. A. , and Vanderwalle, J. , 2000, “Experimental confirmation of 3 and 5 scroll attractors from generalized Chua’s circuit,”IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(3), 425-429 .
