# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 1 Introduction  
The uncertain disturbances widely reside in practical industrial control systems, such as hydro-turbine governing systems [1], autonomous surface vehicle control systems [2], robotic control systems [3], and quadrotor unmanned aerial vehicle control systems [4], which often degrade the system performance and even result in instability. Therefore, the effect of uncertain disturbances should be considered in the controller design, and there have been many interesting results about the controller design and its stability analysis for nonlinear systems with uncertain disturbances in [5, 6, 7]. For example, an asymptotic tracking control scheme is developed for nonlinear systems, where a newly designed disturbance estimator is introduced to effectively compensate for unknown disturbances [6]. For hydraulic turbine regulation systems, disturbance reconstruction techniques have been effectively implemented. Specifically, a nonlinear observation mechanism was constructed to precisely reconstruct disturbance components in hydro-turbine governing processes [1]. [PERSON] et al. further used a fixed-time observer to solve the problem of uncertain external disturbance in quadrotor unmanned aerial vehicle control systems [4]. A class of Gaussian RBF (radial basis function) neural networks was employed to approximate unknown dynamics and composite disturbances in robotic control systems [3]. An event-drivencontrol strategy was successfully implemented for marine autonomous vehicles through intelligent disturbance compensation. A fuzzy approximation mechanism was employed to address time-varying uncertainties in the closed-loop system [2]. The fuzzy logic system (FLS) or neural network is viewed as an effective approximation tool, so it has been widely used in the field of intelligent control to handle highly nonlinear functions or dynamic uncertainties [8; 9; 10; 11; 12; 13]. [PERSON] et al. developed an adaptive neural network controller for a class of switched nonlinear pure feedback systems [14]. An adaptive neural network controller is used to solve the tracking control problem of nonlinear stochastic systems [15]. A three-dimensional fuzzy PID controller was proposed for spacecraft orientation regulation in geostationary orbital operations. Enhanced positioning precision was achieved through this methodology, where proportional-integral-derivative parameters were dynamically adjusted using fuzzy logic principles [8]. [PERSON] et al. used fuzzy logic systems to identify unknown internal dynamics and mismatched disturbances in multi-agent control systems [11]. [PERSON] et al. used fuzzy logic systems to model unknown nonlinear dynamics when designing attitude control systems for spacecraft [12]. Thus, fuzzy logic systems show unique advantages in dealing with uncertain dynamics or disturbances. Therefore, fuzzy logic systems with Gaussian member functions will be used to deal with uncertain perturbations in control systems. In addition, this paper will design an adaptive update law with lower computational load, which will be used to update the weight parameters in fuzzy logic systems in real time. Non-affine pure-feedback systems represent a generalized class beyond conventional strict-feedback architectures [16]. The intrinsic structural complexity of these systems prevents explicit decoupling of affine components from control inputs, thereby complicating stabilization mechanism design [17; 18]. This topological constraint has motivated extensive investigations into advanced control methodologies for such nonlinear configurations [19; 20]. A distributed control framework has been developed for stochastic multi-agent systems with pure-feedback non-affine dynamics. Event-triggered communication protocols have been integrated to simultaneously address system uncertainties and minimize network resource utilization [21]. For dynamic systems requiring strict transient performance, non-overshoot control strategies have been systematically established.