Output Constrained Adaptive Controller Design for Nonlinear Saturation Systems Yongliang Yang, Member, IEEE, Zhijie Liu, Member, IEEE, Qing Li, and Donald C. Wunsch, Fellow, IEEE Abstract—This paper considers the adaptive neuro-fuzzy control scheme to solve the output tracking problem for a class of strict-feedback nonlinear systems. Both asymmetric output constraints and input saturation are considered. An asymmetric barrier Lyapunov function with time-varying prescribed performance is presented to tackle the output-tracking error constraints. A high-gain observer is employed to relax the requirement of the Lipschitz continuity about the nonlinear dynamics. To avoid the “explosion of complexity”, the dynamic surface control (DSC) technique is employed to filter the virtual control signal of each subsystem. To deal with the actuator saturation, an additional auxiliary dynamical system is designed. It is theoretically investigated that the parameter estimation and output tracking error are semi-global uniformly ultimately bounded. Two simulation examples are conducted to verify the presented adaptive fuzzy controller design. Index Terms—Asymmetric barrier Lyapunov function, input saturation, Nussbaum function, time-varying prescribed performance.   I. Introduction A DAPTIVE control theory is widely applied to deal with the engineering applications, including robotic manipulators [1], [2], power systems [3]–[6], chemical process [7], [8], mechanical systems [9], circuit systems [10], [11], etc. Fuzzy logic systems (FLS) and neural networks (NN) can efficiently compensate for the typical nonlinearity and uncertainty in modeling and control, such as saturation [12]–[14], deadzones [15], and hysteresis backlash [16], to name a few. It is usually desirable to drive the plant to behave with predetermined behavior. Traditional neuro-adaptive control methods only consider the convergence of the parameter adaptation and the boundedness of the closed-loop signals. It is critical in real-world applications to guarantee the variables under consideration within some certain region. This paper investigates adaptive fuzzy control of uncertain systems with predefined transient tracking performance and input saturation simultaneously. The backstepping technique is widely used in adaptive control of strict feedback nonlinear systems, where the virtual control input of each subsystem is designed in each step [17]. The backstepping method suffers from an “explosion of complexity” due to the repeatedly differentiation of the virtual control input [18]. The complexity of external control input design grows drastically as the system order increases. To avoid this issue, the dynamic surface control (DSC) method with first-order and second-order filters are developed to obviate the virtual control differentiation [19]–[21]. Adaptive control designs with DSC with output consideration of constraints have been investigated in [22], [23]. However, these results are based on full-state measurement, which might be a strong requirement and impossible for some applications. Nonlinear observer design provides an efficient estimation of the full-state to obviate the requirement of full-state measurement, which is widely used in adaptive constrained output feedback design [24], [25]. Nevertheless, the input constraints are not considered in these literature. In addition, in existing observer-based neuro-adaptive controller design, [26]–[28], the nonlinear systems are restricted to be Lipschitz continuous captured by a Lipschitz constant. In this paper, we modified this requirement by presenting a weak Lipschitz condition, where the Lipschitz constant is relaxed to be a non- negative function. On this basis, we combine the high-gain observer with the DSC method to deal with the constraints on both input and output. In stability analysis, the Lyapunov function is usually not unique and the common selection for the Lyapunov candidate is in quadratic form [29]. However, only asymptotic behavior of the closed-loop system can be guaranteed using the quadratic Lyapunov candidate. To consider the transient behavior of dynamical systems, the concept of a “barrier function” in constrained optimization [30] has been adopted in Lyapunov candidate selection [31], which is referred to as the barrier Lyapunov function (BLF) [32]. BLF based adaptive control design has been applied to deal with multiple problems, such as constrained differential games [33], full- state constraints of a robotic manipulator [34] and pure- feedback systems [35], and the constrained tracking problem of nonlinear systems with unknown control direction [36]. A Manuscript received June 2, 2020; accepted October 1, 2020. This work was supported in part by the National Natural Science Foundation of China (61903028, 62073030), in part by the China Post-Doctoral Science Foundation (2019M660463), in part by the Fundamental Research Funds for the China Central Universities of University of Science and Technology Beijing (FRF-TP-18-031A1, FRF-BD-19-002A), and in part by the Postdoctor Research Foundation of Shunde Graduate School of University of Science and Technology Beijing (2020BH002). Recommended by Associate Editor Qinglai Wei. (Corresponding author: Zhijie Liu.) Citation: Y. Yang, Z. J. Liu, Q. Li, and D. C. Wunsch, “Output constrained adaptive controller design for nonlinear saturation systems,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 441–454, Feb. 2021. Y. Yang, Z. J. Liu and Q. Li are with the Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China (e-mail: yangyongliang@ieee. org; liuzhijie2012@gmail.com; liqing@ies.ustb.edu.cn). D. C. Wunsch is with the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: dwunsch@mst.edu). 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.2020.1003524 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 8, NO. 2, FEBRUARY 2021 441