# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 2 Problem Formulation and Preliminaries
### Lemmas and Assumptions  
To achieve the design completion and stability verification of the proposed controller, standard assumptions and lemmas applicable to adaptive control frameworks will be incorporated. **Lemma 1** (see [33]). : _Consider a dynamical system described by the differential equation \(\dot{x}=g(x)\); if there exists a continuously differentiable Lyapunov function candidate \(V(x)\) satisfying_  
\[\dot{V}(x)\leq-qV(x)+\upsilon,\ t\geq 0 \tag{4}\]  
_where \(q\) and \(\upsilon\) are strictly positive constants, and \(t\) denotes the time variable, then the state trajectory \(x\) is semi-globally uniformly ultimately bounded (SQUUB)._  
**Lemma 2** (see [34]). : _Consider a fuzzy logic system represented by \(\bar{\kappa}^{T}\bar{W}(\bar{X})\), where \(f(\bar{X})\) denotes a bounded continuous function defined on the compact set \(\Omega\). Given a desired approximation accuracy \(\widetilde{\xi}>0\), when the total number of hidden nodes is sufficiently large, an optimal weight vector \(\bar{\kappa}^{*}\) can be determined through the following optimization problem:_  
\[\bar{\kappa}^{*}=\arg\ \left(\min_{\kappa\in\mathbb{R}^{k}}\left(\sup_{\bar{ \kappa}\in\Omega}\left|f(\bar{X})-\bar{\kappa}^{T}\bar{W}(\bar{X})\right| \right)\right) \tag{5}\]  
_where the optimal weight vector \(\bar{\kappa}^{*}\) is obtained by minimizing the supremum of the approximation error between the fuzzy logic system and the target function over the entire domain \(\Omega\)._  
**Remark 1**. : _During the controller design procedure for an \(n\)th-order pure-feedback system, \(n\) unknown optimal weight vectors \((\bar{\kappa}_{1}^{*},\bar{\kappa}_{2}^{*},\ldots,\bar{\kappa}_{n}^{*})\) are typically required. To reduce computational complexity, a single adaptive parameter \(\theta\) is introduced to estimate these weight vectors collectively, defined as_  
\[\theta=\max\biggl{\{}\left\|\vec{k}_{i}^{*}\right\|^{2}\biggr{\}}\;i=1,2,\ldots,n. \tag{6}\]  
_An adaptive law \(\hat{\theta}\) will be subsequently developed in Section 3 to provide online estimation of \(\theta\), thereby enabling the acquisition of each fuzzy logic system's optimal weights through the relation established in (6)._  
**Assumption 1** (see [28]). : _Within the framework of nonlinear control system design, the desired reference trajectory and its first-order time derivative are collectively represented by a bounded signal vector \([y_{1d},\dot{y}_{1d}]^{T}\). This vector is rigorously confined to a well-defined compact region \(\Omega_{1d}\) in the following two-dimensional Euclidean space:_  
\[\Omega_{1d}=\left\{[y_{1d},\dot{y}_{1d}]^{T}:y_{1d}^{2}+\dot{y}_{1d}^{2}\leq \widetilde{y}\right\}\subset\Re^{2} \tag{7}\]  
_where the bounding parameter \(\widetilde{y}\) serves as a strictly positive constant that quantitatively determines the maximum allowable magnitude of the reference signals. The compactness of \(\Omega_{1d}\) ensures that all possible trajectories of interest remain within this predetermined bounded region throughout the control process._  
**Remark 2**. : _The conventional backstepping control framework for nth-order nonlinear systems, as per the existing research in [35], imposes the requirement that both the desired trajectory \(y_{1d}\) and its complete set of \(n\) derivatives must be incorporated in the controller synthesis. In contrast, the developed control scheme is formulated to operate with only the reference signal and its first-order time derivative, i.e., \(y_{1d}\) and \(\dot{y}_{1d}\). The proposed approach is rendered particularly well suited for industrial applications due to the substantial reduction in information requirements. In such applications, the acquisition of higher-order derivatives is frequently found to be impractical or even unfeasible._  
**Assumption 2**. : _For ensuring the controllability of the system dynamics, an additional assumption is introduced regarding the boundedness of control gain variables.