278 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 6, NO. 1, JANUARY 2019 Neural Network Based Adaptive Tracking Control for a Class of Pure Feedback Nonlinear Systems With Input Saturation Nassira Zerari, Mohamed Chemachema, and Najib Essounbouli Abstract—In this paper, an adaptive neural networks (NNs) tracking controller is proposed for a class of single-input/single- output (SISO) non-affine pure-feedback non-linear systems with input saturation. In the proposed approach, the original input saturated nonlinear system is augmented by a low pass filter. Then, new system states are introduced to implement states transformation of the augmented model. The resulting new model in affine Brunovsky form permits direct and simpler controller design by avoiding back-stepping technique and its complexity growing as done in existing methods in the literature. In controller design of the proposed approach, a state observer, based on the strictly positive real (SPR) theory, is introduced and designed to estimate the new system states, and only two neural networks are used to approximate the uncertain nonlinearities and compensate for the saturation nonlinearity of actuator. The proposed approach can not only provide a simple and effective way for construction of the controller in adaptive neural networks control of non-affine systems with input saturation, but also guarantee the tracking performance and the boundedness of all the signals in the closed-loop system. The stability of the control system is investigated by using the Lyapunov theory. Simulation examples are presented to show the effectiveness of the proposed controller. Index Terms—Adaptive control, input saturation, neural net- works systems (NNs), nonlinear pure-feedback. I. I NTRODUCTION T HE control design of nonlinear systems with parameter uncertainties using universal function approximators is an active research area that has received increasing attention along the past decades. Adaptive back-stepping becomes one of the most popular design methods in nonlinear control de- sign for synthesizing controllers of lower-triangular nonlinear systems. Many adaptive back-stepping design schemes have been reported for strict-feedback nonlinear systems with un- known nonlinear functions [1]−[6]. In [4], [5], adaptive neural Manuscript received March 2, 2018; revised April 24, 2018; accepted June 7, 2018. Recommended by Associate Editor Yanjun Liu. (Corresponding author: Nassira Zerari.) Citation: N. Zerari, M. Chemachema, and N. Essounbouli, “Neural network based adaptive tracking control for a class of pure feedback nonlinear systems with input saturation,” IEEE/CAA J. Autom. Sinica, vol. 6, no. 1, pp. 278-290, Jan. 2019. N. Zerari and M. Chemachema are with the Department of Electronics, Faculty of Technology, University of Constantine1, Compus A. Hamani, Route Aine El Bey, 2500 Constantine, Algeria (e-mail: zer.napg2010@gmail.com; m chemachema@yahoo.fr). N. Essounbouli is with CReSTIC Laboratory, University of Reims Cham- pagne Ardennes, France (e-mail: najib.essounbouli@univ-reims.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.2018.7511255 networks (NNs) or fuzzy control schemes were proposed to address the strict-feedback stochastic nonlinear systems with time-delays. In [6], adaptive neural output feedback controller was presented for strict-feedback nonlinear systems with un- modeled dynamics. However, the back-stepping technique was extended to deal with the more difficult control problem of larger class of uncertain pure-feedback nonlinear systems in which no affine appearance of the state variables was used as virtual control inputs. In [7]−[11], authors have investigated the adaptive back-stepping control problem for a class of pure- feedback nonlinear systems based on NN and fuzzy inference systems. Using the implicit function theorem, the mean value theorem and back-stepping technique, adaptive NN control schemes were proposed in [12]−[14], for the same class of uncertain non-linear pure-feedback systems. In [15], authors proposed an adaptive fuzzy neural networks control method for SISO stochastic nonlinear systems in pure-feedback form. A completely non-affine pure-feedback system is dealt with in [16] using the input-to-state stability analysis and the small gain theorem to develop an improved NN adaptive controller. A common problem in all cited papers is the drawback in- herent to the application of the back-stepping design technique known as the explosion of complexity. The latter problem is caused by the repeated differentiations of certain nonlinear functions, such as virtual control inputs [17]. To deal with the complexity growing problem, a dynamic surface control (DSC) technique was introduced in [18] for a class of strict- feedback nonlinear systems. In [19], by using a first-order filter at each recursive step, neural network based adaptive DSC was further developed for a class of pure-feedback nonlinear systems with unknown time-delay functions and uncertainties. By combining DSC technique with the back-stepping design, the pure-feedback nonlinear systems with unknown dead zone and uncertainties were considered in [20] with the restriction that the (n − 1)th and nth state equations were assumed to be affine such that the repeated differentiations of the virtual control inputs can be eliminated. However, from a point of view of practical applications, the methods developed in [20] are computationally expensive because the NN and fuzzy approximators were used at every design step to online approximate the unknown dynamics. Furthermore, the adaptive laws involved in DSC require a large number of parameters to updated online, which make the control law and stability analysis very complicated. Only few results were reported on the adaptive neural control for general nonlinear systems in pure-feedback or strict-feedback form without using back-