# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 1 Introduction
Novel stability conditions with reduced conservatism have been derived through Lyapunov-based analysis techniques [22]. The small-gain theorem has been effectively utilized to resolve stabilization difficulties in uncertain nonlinear pure-feedback systems. This approach has proven particularly advantageous when combined with switched reference model architectures [14]. [PERSON] et al. used a class of positive control gain functions to solve the pure feedback problem in nonlinear control systems [17]. The differential mid-value theorem has been employed to convert the pure-feedback form of control system into the strict-feedback form [19]. In this paper, the non-affine purity feedback problem in the controlled system is solved by using the differential median theorem [23; 24]. The backstepping controller is viewed as a traditional control scheme and has been employed to control pure-feedback systems [25; 26] during the past years. For example, a fuzzy adaptive backstepping control scheme was presented for multi-agent systems with pure-feedback nonlinear dynamics [26]. [PERSON] et al. introduced observer technology into backstepping controllers to solve the tracking control problem of a class of pure-feedback systems [27]. However, backstepping technology cannot avoid the repeated differential operations of virtual control signals, which is called \"explosion of complexity\". Subsequently, dynamic surface control (DSC) has been come into being, whose core algorithm is that the output of the filter is used to approximate the intermediate control signal such that the shortcomings of the backstepping controller are overcome [15; 28; 29]. [PERSON] et al. introduced dynamic surface controllers into a class of strict feedback systems with output constraints [29]. The application scope of dynamic surface control algorithm extends from theoretical dynamic systems to practical applications, and it solves the formation control problem of quadrotor unmanned aerial vehicles [30]. An adaptive dynamic surface control algorithm was used to solve the tracking control problem of flexible manipulators, and the calculation efficiency of its weight parameters can be further improved [31]. Inspired by the abovementioned discussions, a fuzzy-based adaptive control scheme is proposed for non-affine pure-feedback systems subject to stochastic disturbances. Computational complexity issues arising from recursive differentiation procedures are resolved through dynamic surface control methodology. Unknown disturbances and state estimation errors are approximated via fuzzy logic mechanisms. Structural constraints inherent in non-affine system formulations are addressed using the differential mean-value theorem. Compared with the available research results, the design method of the proposed controller in this paper has the following contributions:  
1. Significant tracking performance improvements are achieved through the developed control scheme when compared with conventional dynamic surface control methodologies for completely unknown system dynamics [31]. Radial basis function networks are specifically employed to compensate for uncertain disturbances and state estimation errors. This structural simplification enables more efficient parameter adaptation processes while maintaining approximation accuracy. 2. In the design process of the controller, the solution to the nonlinear equations of the actual control signal is avoided by employing an adjusting parameter to replace the final gain function, which reduces the computational cost significantly. 3. Only one parameter is needed for the updating of the adaptive law, which also alleviates computational cost to a certain extent. The remainder of this paper is planned as follows. The system statement and the preliminaries are presented in Section 2. Section 3 provides the process of the controller design and its stability analysis. A comparison of the simulation results is presented to show the advantage of the proposed control approach in Section 4.