A fuzzy adaptive variable-structure control scheme for uncertain chaotic MIMO systems with sector nonlinearities and dead-zones A. Boulkroune a,⇑ , M. M’Saad b a Department of Automatic, Faculty of Engineering Sciences, University of Jijel, BP. 98, OuledAissa, 18000 Jijel, Algeria b GREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN, 6 Bd Maréchal Juin, 14050 Caen Cedex, France article info Keywords: Adaptive control Fuzzy system Variable-structure control MIMO chaotic systems Sector nonlinearities Dead-zone abstract In this paper, a fuzzy adaptive variable-structure controller is investigated for a class of uncertain multi- input multi-output (MIMO) chaotic systems with both sector nonlinearities and dead-zones. A suitable adaptive fuzzy system is used to reasonably approximate the uncertain functions. A Lyapunov approach is employed to derive the parameter adaptation laws and prove the boundedness of all signals of the closed-loop system as well as the exponential convergence of the closed-loop errors to an adjustable region. The proposed controller can be applied to the systems with or without sector nonlinearities and/or dead-zones in the input. The effectiveness of the proposed fuzzy adaptive controller is illustrated throughout simulation results. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Chaotic system is a very complex dynamical nonlinear system and its response exhibits some specific features such as excessive sensitivity to initial conditions, broad Fourier transform spectrums, and irregular identities of the motion in phase plane (Chen & Ueta, 1999; Vanecek & Celikovsky, 1996). Chaos control problem was firstly considered by Ott, Grebogi, and Yorke (1990). Since then, it has been extensively investigated in the past two decades (Chen & Dong, 1998). Several (linear and nonlinear) control techniques have been successfully applied for the control of chaotic systems (for synchronisation, tracking, or stabilisation purposes) such that: PID control (Ghezi & Peccardi, 1997), adaptive feedback control (Feki, 2003; Hua, Guan, & Shi, 2005), observer-based control (Boulkroune, Chekireb, Tadjine, & Bouatmane, 2006; Boulkroune, Chekireb, Tadjine, & M’Saad, 2006; Boulkroune, Chekireb, Tadjine, & Bouatmane, 2007), sliding-mode control (Ablay, 2009; Nazzal & Natsheh, 2007; Yau, Chen, & Chen, 2000), adaptive backstepping control (Ge & Wang, 2000; Wang & Ge, 2001), adaptive fuzzy con- trol (Boulkroune et al., 2006; Chang, 2001; Liu & Zheng, 2009; Poursamad & Markazi, 2009; Roopaei & Jahromi, 2008; Roopaei, Jahromi, & Jafari, 2009), adaptive neural control (Ge & Wang, 2002), etc. A key assumption in all previous control schemes (Ablay, 2009; Boulkroune, Chekireb, Tadjine, & Bouatmane, 2006; Boulkroune, Chekireb, Tadjine, & M’Saad, 2006; Boulkroune et al., 2007; Chang, 2001; Feki, 2003; Ghezi and Peccardi, 1997; Ge and Wang, 2000, 2002; Hua et al., 2005; Liu and Zheng, 2009; Nazzal and Natsheh, 2007; Poursamad and Markazi, 2009; Roopaei and Jahromi, 2008; Roopaei et al., 2009; Wang and Ge, 2001; Yau et al., 2000) is that the chaotic system has linear inputs. The control problem of uncertain nonlinear (chaotic or non- chaotic) systems with nonlinear inputs has received a great interest because of the input nonlinearities, such as saturation, backlash, dead-zones, and so on, naturally originate from physical limita- tions in system realization (Boulkroune, M’Saad, Tadjine, & Farza, 2008; Hsu, Wang, & Lin, 2004). It is worth mentioning that the existence of input nonlinearities may leads to notable performance degradations or even instability of the control system. It is thereby more advisable to take into account the effects of the input nonlin- earities in the control design as well as the stability analysis. Re- cently, some control schemes have been proposed (Chang, 2007; Chiang, Hung, Yan, Yang, & Chang, 2007; Hung, Yan, & Liao, 2008; Yan, Shyu, & Lin, 2005) for a class of chaotic systems with in- put sector nonlinearities and/or dead-zones. However, these underly- ing results suffer from some fundamental limitations. Firstly, the class considered of the chaotic systems is relatively simple (i.e. the chaotic systems considered in these works is not of type MIMO). Secondly, the so-called gain reduction tolerances of the in- put nonlinearities and upper bounds of the model uncertainties are required to be known or partially known. Motived by works in Yan et al. (2005), Chiang et al. (2007), Chang (2007) and Hung et al. (2008), one aims at designing a fuzzy adaptive variable-structure controller for a class of uncertain cha- otic MIMO systems containing both sector nonlinearities and dead- zones. Bearing in mind the available results (Chang, 2007; Chiang et al., 2007; Hung et al., 2008; Yan et al., 2005), the main contribu- tions of this paper are: 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.05.006 ⇑ Corresponding author. E-mail addresses: boulkroune2002@yahoo.fr (A. Boulkroune), msaad@greyc.en sicaen.fr (M. M’Saad). Expert Systems with Applications 38 (2011) 14744–14750 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa