Abstract— In this paper, an adaptive fuzzy tracking control is presented for a class of SISO nonlinear strict-feedback systems with unknown time delays. The developed control algorithm uses the Mamdani-type fuzzy systems to approximate on-line the unknown nonlinear function. The Krasovskii-functional is constructed to compensate for the unknown delayed state. The proposed controller guarantees uniform ultimate boundedness of all signals in the closed-loop system. The designed control law is independent of the time delays and has a simple form with only one adaptive parameter, which is required to be updated on-line. Simulation results are presented to verify the effectiveness of the proposed approach. Index Terms— Adaptive fuzzy control, nonlinear time-delay systems, stability. I. INTRODUCTION ystems with delays frequently appear in engineering applications. Typical examples of time-delay systems are electrical networks, chemical processes, rolling mill systems, teleportation systems, underwater vehicles and so on. The existence of time delays may degrade the control performance and make the stabilization problem more difficult. So far, the stability analysis and robust control for these dynamic time- delay systems have attracted a number of researchers over the past years; see, for example, [1]-[2], and the references therein. Stability analysis and synthesis of time-delay systems are important issues addressed by many authors and for which surveys can be found in several articles [3]-[7]. Recently, by combining Lyapunov–Krasovskii functional and backstepping technique, an adaptive neural tracking control scheme was proposed in [8] for a class of strict- feedback nonlinear time-delay systems with unknown virtual control coefficients. The suggested controller guarantees the uniform ultimate boundedness of the adaptive closed-loop system, while the output tracking is achieved. Further improvements were given in [9, 10]. The robust stabilization methods were presented via the approximation capability of neural network [11, 12]. In [13], the adaptive H  control was addressed via backstepping and neural networks technique. The observer-based adaptive neural controller was designed H. Yousef is with the Department of Electrical Engineering, Faculty of Engineering, Alexandria University, Alexandria-21544, Egypt (e-mail: hyousef456@ yahoo.com). M. Hamdy is with the Department of Industrial Electronics and Control Engineering, Faculty of Electronic Engineering, Menofia University, Menof- 32952, Egypt (mhamdy72@ hotmail.com). for a special class of nonlinear time-delay systems [14]. However, these adaptive neural control methods [8]-[14] require a large number of neural weights to be adapted online simultaneously. This makes the learning time unacceptably large. In view of this disadvantage, several adaptive fuzzy control schemes have been developed in [15]-[18] for nonlinear delay-free systems. The advantage of these control schemes is that the proposed controllers require much less parameters to be updated online. Stability analysis and control synthesis of T-S fuzzy delayed systems were proposed in [19]-[22]. An approximation based adaptive control has also been addressed for nonlinear systems with time-delay. An attempt for using Mamdani-type fuzzy logic systems to design an adaptive fuzzy tracking controller by using the backstepping technique and Lyapunov– Krasovskii functional appeared in [23]. The main advantage of the results proposed in [23] is that the developed fuzzy controllers contain less adaptation laws. Based on the above observation, a novel systematic design procedure is developed for the synthesis of a stable adaptive fuzzy controller for a class of nonlinear time-delay systems. The Lyapunov-Krasovskii functional is constructed to compensate for the unknown delayed state uncertainties. Fuzzy logic systems are employed to approximate the unknown nonlinear function, and then, the adaptive law of adjustable parameters is obtained. Based on the Lyapunov stability theorem, the proposed adaptive fuzzy controller guarantees the semi-global uniform ultimate boundedness of all signals in the closed-loop system and achieves a good tracking performance. The main advantage of the proposed method is that for an nth order strict-feedback nonlinear system, only one parameter is needed to be estimated on-line regardless the number of fuzzy rule bases used. Therefore, the computation burden is significantly reduced and the algorithm is easily realized in practice. The paper is organized as follows. The problem under investigation and structure of fuzzy systems are introduced in section 2. An adaptive fuzzy control design using Lyapunov function approach is presented in section 3. The main result is presented in section 4. Simulation results are provided in section 5, with conclusions given in section 6. II. PROBLEM FORMULATION AND FUZZY SYSTEM Consider the SISO nonlinear time-delay dynamic systems in the following form: Adaptive Mamdani Fuzzy Control for a Class of Nonlinear Time-delays Systems H. Yousef and M. Hamdy S 978-1-4244-5844-8/09/$26.00 ©2009 IEEE 121