Nonlinear Dyn (2011) 63: 253–262 DOI 10.1007/s11071-010-9801-8 ORIGINAL PAPER Variable structure based robust backstepping controller design for nonlinear systems Chao-Chung Peng · Albert Wen-Jeng Hsue · Chieh-Li Chen Received: 13 January 2010 / Accepted: 6 August 2010 / Published online: 5 September 2010 © Springer Science+Business Media B.V. 2010 Abstract This study presents robust control architec- ture in the sense of variable structure control via a backstepping design. By using systematic backstep- ping design techniques, closed-loop behavior of an n-order nonlinear system can be transformed into a stability and convergence problem of a fast switched 2nd order system. There are two main parts contained within the proposed control algorithm; one is a nom- inal control effort generated according to the Lya- punov stability criterion during recursive backstepping processes, and the other belongs to a smooth robust control law designed to eliminate the effects of un- known lumped perturbations. Finally, a Genesio sys- tem is used as an illustrated example to demonstrate the robustness of the control algorithm. The feasibility and properties of the proposed method are given by numerical simulations. Keywords Chattering · Backstepping · Variable structure · Nonlinear system · Chaotic C.-C. Peng · C.-L. Chen ( ) Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan, R.O.C. e-mail: chiehli@mail.ncku.edu.tw A.W.-J. Hsue Department of Mechatronic Technology, Dahan Institute of Technology, HualienCounty, Taiwan, R.O.C. 1 Introduction A chaotic dynamic is a highly complex nonlinear phe- nomenon which exhibits a number of interesting char- acteristics, including but not limited to unpredictable behavior and excessive sensitivity to different initial conditions. The behavior of a chaotic system is some- times undesirable, however, owing to its powerful ap- plications in engineering (e.g., chemical reactions, bi- ological systems, and secure communications, etc.); controlling these nonlinear chaotic dynamics for ex- tensive application fields has become an attractive area of study. Recently, the backstepping design technique has been widely used to stabilize and synchronize a va- riety of chaotic systems [1–5]. However, rejection due to model uncertainties or disturbances has not been ad- dressed in these studies. In realistic, complete knowl- edge of the system parameters is not an easy task es- pecially for control practices. In regard to chaotic sys- tems subjected to model uncertainties, several adap- tation laws have been developed to estimate the un- certain parameters [6–9]. On the other hand, when the system is affected by unknown perturbations, vari- able structure control (VSC) theory [10–12] is a good choice to handle these changes in chaotic systems ow- ing to its inherent advantages, which include fast re- sponse, good performance and insensitivity to para- meters (e.g., deviation and exogenous disturbances). Especially, it is simple to implement. Unfortunately,