816 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 3, MAY 2019 Fuzzy Adaptive Control of a Fractional Order Chaotic System With Unknown Control Gain Sign Using a Fractional Order Nussbaum Gain Khatir Khettab, Samir Ladaci, Member, IEEE, and Yassine Bensafia Abstract—In this paper we propose an improved fuzzy adaptive control strategy, for a class of nonlinear chaotic fractional order (SISO) systems with unknown control gain sign. The online control algorithm uses fuzzy logic sets for the identification of the fractional order chaotic system, whereas the lack of a priori knowledge on the control directions is solved by introducing a fractional order Nussbaum gain. Based on Lyapunov stability theorem, stability analysis is performed for the proposed control method for an acceptable synchronization error level. In this work, the Gr¨ unwald-Letnikov method is used for numerical approximation of the fractional order systems. A simulation example is given to illustrate the effectiveness of the proposed control scheme. Index Terms—Adaptive fuzzy control, chaos synchronization, fractional order Nussbaum function, Lyapunov stability, nonlin- ear fractional order systems. I. I NTRODUCTION F RACTIONAL order chaotic systems are gathering an important research effort because of their powerful prop- erties and potential applications in secure communication and control processing. Many mathematical models have been de- veloped in literature such as the fractional-order Chua system [1], the fractional-order Duffing system [2], the fractional- order L¨ u system and the fractional order Chen system [3]. Since the work of Deng and Li [4] who investigated the syn- chronization problem of fractional order chaotic L¨ u systems, Manuscript received September 1, 2015; revised December 29, 2015; accepted February 28, 2016. This work was supported by the Algerian Ministry of Higher Education and Scientific Research (MESRS) for CNEPRU Research Project (A01L08UN210120110001). Recommended by Associate Editor Dingyu Xue. (Corresponding author: S. Ladaci.) Citation: K. Khettab, S. Ladaci, and Y. Bensafia, “Fuzzy adaptive control of fractional order chaotic systems with unknown control gain sign using a fractional order Nussbaum gain,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 3, pp. 816-823, May 2019. K. Khettab is with the Electrical Engineering Department, University of Mohamed Boudiaf, M’sila 28000, Algeria, and also with the Electrical Engineering Department, University of Skikda, Skikda 21000, Algeria (e- mail: zoubirhh@yahoo.fr). Samir Ladaci is with the EEA Department, National Polytechnic School of Constantine, B. P. 75 A, Ali Mendjli 25000 Constantine, Algeria, and also with SP-Lab Laboratory, Department of Electronics, Mentouri University, Route de Ain Elbey, Constantine 25000, Algeria (e-mail: samir ladaci@yahoo.fr). Yassine Bensafia is with the Electrical Engineering Department, University of Bouira, 10000 Bouira, Algeria (e-mail: bensafiay@yahoo.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2016.7510169 many studies were focused on the control and synchronization of fractional order chaos [5], [6]. In this paper we are concerned by fractional adaptive control of nonlinear fractional order systems using fuzzy logic identification technique. Fractional adaptive systems have been largely investigated for a decade as they showed an improved behavior comparatively to classical adaptive control for par- tially unknown plants [7]−[10]. Based on the universal approximation theorem, adaptive fuzzy control systems present an effective control solution for a large class of nonlinear systems [11]. The adaptive controller is synthesized from a collection of fuzzy IF-THEN rules and the parameters of the membership functions characterizing the linguistic terms in the IF-THEN rules change according to some adaptive law for the purpose of controlling a plant to track a reference trajectory [12], [13]. A particular class of such nonlinear plants pose the chal- lenging control problem with unknown control directions [14], [15]. The Nussbaum function approach was introduced in the 1980’s [16]. This technique was used for adaptive control of first-order nonlinear systems in [17]. Later, many studies of adaptive control schemes with Nussbaum function were successfully carried out for different classes of nonlinear systems [18]−[20]. The main contribution of this study is the introduction of a Nussbaum function in the fuzzy adaptive control scheme for nonlinear fractional systems with unknown control gain sign. Stability analysis of the proposed adaptive fuzzy control system is performed using Lyapunov stability theory. More- over, the influence of the approximation error and external disturbance on the tracking error can be attenuated to an arbitrarily prescribed level via the proposed design technique. The fuzzy adaptive control design with Nussbaum function is applied for nonlinear fractional order chaotic systems with a large uncertainty or unknown variation in plant parameters and structures. The Gr¨ unwald-Letnikov technique is used for the numerical approximation of the fractional order chaotic system [21]. This paper is organized as follows: in Section II, basic definitions and preliminaries for fractional order systems are presented with the numerical approximation technique. A description of the Nussbaum-type function is given in Section III. Section IV presents the fuzzy adaptive control scheme with unknown control direction for uncertain fractional order