Robust chaotic control of Lorenz system by backstepping design Chao-Chung Peng, Chieh-Li Chen * Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan Accepted 7 September 2006 Communicated by Prof. Ji-Huan He Abstract This work presents a robust chaotic control strategy for the Lorenz chaos via backstepping design. Backstepping technique is a systematic tool of control law design to provide Lyapunov stability. The concept of extended system is used such that a continuous sliding mode control (SMC) effort is generated using backstepping scheme. In the pro- posed control algorithm, an adaptation law is applied to estimate the system parameter and the SMC offers the robust- ness to model uncertainties and external disturbances so that the asymptotical convergence of tracking error can be achieved. Regarding the SMC, an equivalent control algorithm is chosen based on the selection of Lyapunov stability criterion during backstepping approach. The converging rate of error state is relative to the corresponding dynamics of sliding surface. Numerical simulations demonstrate its advantages to a regulation problem and an orbit tracking prob- lem of the Lorenz chaos. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction The chaotic behavior is a very interesting nonlinear phenomenon which has been intensively studied during the last two decades. The effect of chaotic system is usually undesirable in practice due to its sensitivity to initial conditions, unpredictable behavior and thereby restricts the operation of physical plants. Because of the difficulty of accurate pre- diction of a chaotic system behavior, chaos may cause system instability or degradation in performance, and it should be eliminated in many cases. Regarding the field of chaotic analysis and control, the Lorenz system is often taken as a paradigm, since it captures many of the features of chaotic dynamics. For example, the Lorenz system describes some of the unpredictable behavior which associates with the weather. It also formulates an incompressible fluid between two parallel horizontal bound- aries, with the lower boundary at a higher temperature than the upper boundary. Many approaches and techniques have been proposed for the control of chaos such as OGY method [1], bang–bang control [2], optimal control [3], intelligent control base on neural network [4], feedback linearization [5], differential 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.057 * Corresponding author. Tel.: +886 6 2389121/4; fax: +886 6 2389940. E-mail address: chiehli@mail.ncku.edu.tw (C.-L. Chen). Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 598–608 www.elsevier.com/locate/chaos