Adaptive fuzzy tracking control for a class of MIMO nonaffine uncertain systems A. Boulkroune a,n , M. M’Saad b , M. Farza b a Department of Automatic, Faculty of Engineering Sciences, University of Jijel, BP. 98, OuledAissa, 18000 Jijel, Algeria b GREYC, UMR 6072 CNRS, Universite´ de Caen, ENSICAEN, 6 Bd, Mare´chal Juin, 14050 Caen Cedex, France article info Article history: Received 18 December 2011 Received in revised form 31 March 2012 Accepted 25 April 2012 Communicated by Prof. H. Zhang Available online 11 May 2012 Keywords: MIMO nonaffine systems Fuzzy control Adaptive control Unknown control direction Nussbaum function abstract In this paper, a novel fuzzy adaptive controller is investigated for a class of multi-input multi-output (MIMO) nonaffine systems with unknown control direction. An equivalent model in affine-like form is first derived for the original nonaffine system by using a Taylor series expansion. Then, a fuzzy adaptive control is designed based on the affine-like equivalent model. The adaptive fuzzy systems are used to appropriately approximate the unknown nonlinearities, while the lack of knowledge of the control direction being closely related to the sign of control gain matrix is handled by incorporating in the control law a Nussbaum-type function. A decomposition property of the control gain matrix is used in the controller design and the stability analysis. The effectiveness of the proposed fuzzy adaptive controller is illustrated through simulation results. Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. 1. Introduction Control process problems are more and more complex as the involved systems are multivariable in nature and exhibit uncertain nonlinear behaviors. This explains the fact that only few engi- neering solutions are available. Thanks to the universal approx- imation theorem [1], some adaptive fuzzy control systems [2–14] have been developed for a class of multivariable nonlinear uncertain systems. The stability of the underlying control systems has been investigated using a Lyapunov approach. The robustness issues with respect to the approximation error and external disturbances have been enhanced by appropriately modifying the available adaptive fuzzy controllers. The corner stone of such a modification consists in a robust compensator which is con- ceived using a sliding mode control design [3–5,7,9,11–14] or an H N based robust control design [4,6,8,10]. In the fuzzy indirect adaptive scheme [2–5,8,10], the singularity problem occurring when determining the inverse of the estimated control gain matrix has been particularly solved thanks to a suitable projection inside the parameter space up to an priori knowledge on the system under control, namely a feasible set in which the singu- larity problem does not happen [2,3,8,10]. A genuine procedure involving an appropriate regularization of the estimated control gain matrix has been also used in [4,5] up to an admissible tracking performance reduction. The key modeling assumption in these above fuzzy adaptive control schemes is that the systems considered are characterized by inputs appearing linearly in the system equation, i.e., the systems considered are affine-in- control. To the authors’ best knowledge, there are few works in the fuzzy control literature which were devoted to the control problem of the nonaffine multivariable systems. In practice, there are many nonlinear systems with nonaffine structure, such as chemical reactors [15], biochemical process [16], some aircraft dynamics [17], dynamic model in pendulum control [18], etc. Some remarkable results for nonaffine mono- variable systems have been obtained [19–31]. It is worth noting that affine systems are a special case of the nonaffine systems. Thus, all schemes in [19–31] can be applied directly to affine systems in which the control input appears in a linear fashion. In the literature, one can find five methods dealing with nonaffine problem, such as: (i) Method using Taylor series expansion in order to get an affine system (seen in [19,20]). (ii) Method using implicit function theorem (seen in [21–24]). (iii) Method exploit- ing the mean value theorem in order to obtain an affine form (seen in [25–28]). (iv) By differentiating the original system equation so that, in the augmented resulting model, the time derivative of the control input appears linearly and the latter can be used as a new control variable (seen in [29]). (v) By using a local inversion of the Takagi–Sugeno (TS) fuzzy affine model (seen in [30,31]). Note that there are two common modeling assumptions in these above adaptive control schemes [19–31] Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing 0925-2312/$ - see front matter Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.04.006 n Corresponding author. E-mail addresses: boulkroune2002@yahoo.fr (A. Boulkroune), msaad@greyc.ensicaen.fr (M. M’Saad), mfarza@greyc.ensicaen.fr (M. Farza). Neurocomputing 93 (2012) 48–55