Stabilizing unstable fixed points of discrete chaotic systems via quasi-sliding mode method Hassan Salarieh * , Seyyed Mostafa Mostafavi Kashani, Gholamreza Vossoughi, Aria Alasty Center of Excellence in Design, Robotics and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, 11365-9567 Tehran, Iran Received 14 August 2007; accepted 17 October 2007 Available online 4 November 2007 Abstract The problem of stabilizing unstable fixed points of nonlinear discrete chaotic systems, subjected to bounded model uncertainties, is investigated in this article. The theory is then generalized to include any dth-order fixed point of the sys- tem. To design a suitable controller, the theory of quasi-sliding mode control is modified and applied. Sufficient conditions for the convergence of the control algorithm are theoretically derived and it is shown that the error trajectories converge toward a bounded region around zero where the measure of the steady-state error band depends on the magnitude of the system uncertainties. As a case study, the proposed method is applied to the Henon map to stabilize its first, second, and fourth-order unstable fixed points. Simulation results show the high performance of the control technique in quenching the chaos in the presence of uncertainties. Ó 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Gg; 05.45.Ac Keywords: Chaos; Quasi-sliding mode; Discrete system; Unstable fixed point 1. Introduction Control of nonlinear systems has always been of great interest due to their applications in real mechanical and electrical systems. Several control theories are developed right now for nonlinear continuous systems [1,2], while discrete systems are not studied that much. Of course, there exists some material in nonlinear control texts, but usually linear models are studied therein [3,4]. The fact that most of the continuous control schemes are implemented on a digital device has made many scientists study the behavior of discrete nonlinear systems. On the other hand, recently many economic and social systems are being discussed which are discrete in their nature. In addition, there exist some cases in 1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.10.012 * Corresponding author. Tel.: +98 21 6616 5586; fax: +98 21 6600 0021. E-mail address: salarieh@mech.sharif.edu (H. Salarieh). Available online at www.sciencedirect.com Communications in Nonlinear Science and Numerical Simulation 14 (2009) 839–849 www.elsevier.com/locate/cnsns