# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis  
An adaptive fuzzy dynamic surface control (DSC) strategy with constant gain is developed for a class of non-affine pure-feedback systems described by (1). The proposed methodology integrates fuzzy logic systems (FLSs) with the mean-value theorem to achieve system stabilization, followed by a stability analysis demonstrating semi-global uniform ultimate boundedness (SGUUB) of the closed-loop signals. The DSC framework employs the following coordinate transformations at each design step:  
\[ \begin{array}{l}e_{1}=x_{1}-y_{1d},\ e_{m}=x_{m}-w_{m}\\ z_{m}=w_{m}-\alpha_{m-1},\ 2\leq m\leq n\end{array}  \tag{10}\]  
where \(e_{1}\) and \(e_{m}\) denote tracking errors, \(\alpha_{m-1}\) corresponds to either a virtual control law or the input of a first-order filter, and \(w_{m}\) represents the filtered output; \(z_{m}\) is the error between the filter output \(w_{m}\) and the virtual control signal \(\alpha_{m-1}\). The dynamics of the first-order filter are governed by [36]  
\[\dot{w}_{m}\Phi_{m}+w_{m}=\alpha_{m-1} \tag{11}\]  
where \(\Phi_{m}\) is a positive constant. The core component of the proposed control scheme is given as follows:  
\[\alpha_{1}=\frac{-a_{1}e_{1}}{\psi_{1}}-\frac{e_{1}\dot{\theta}\bar{W}_{1}^{T }\bar{W}_{1}}{\psi_{1}2b_{1}^{2}}-\frac{g_{1}}{\psi_{1}}-e_{2}+x_{2}^{0}+ \frac{\dot{y}_{1d}}{\psi_{1}} \tag{12}\]  
\[\alpha_{k}=\frac{-a_{k}e_{k}}{\psi_{k}}-\frac{e_{k}\dot{\theta}\bar{W}_{k}^{T }\bar{W}_{k}}{\psi_{k}2b_{k}^{2}}-\frac{g_{k}}{\psi_{k}}-e_{k+1}+x_{k+1}^{0}+ \frac{\dot{w}_{k}}{\psi_{k}} \tag{13}\]  
\[u=\frac{-a_{n}e_{n}}{\psi}-\frac{e_{n}\dot{\theta}\bar{W}_{n}^{T}\bar{W}_{n}}{ \psi 2b_{n}^{2}}+u^{0},\ k=2,3,\ldots,n-1 \tag{14}\]  
\[\dot{\theta}=\varsigma\sum_{i=1}^{n}\frac{e_{i}^{2}\bar{W}_{i}^{T}\bar{W}_{i}} {2b_{i}^{2}}-\varphi\dot{\theta} \tag{15}\]  
where \(a_{i}\in\Re^{+}\), \(b_{i}\in\Re^{+}\), \(i=1,2,\ldots,n\), \(\varsigma\in\Re^{+}\), \(\varphi\in\Re^{+}\), \(\psi\in\Re^{+}\) are design parameters. The vector \(\bar{W}_{i}\) corresponds to the basis function vector of the \(i\)th subsystem, and \(\dot{\theta}\) represents the estimated value of the adaptive parameter \(\theta\). Specially, the actual control input \(u\) is characterized by a constant gain \(\psi\), which is constrained such that \(0<\psi\leq\psi_{i}\leq\dot{\psi}\). In contrast, virtual control laws \(\alpha_{1}\), \(\alpha_{k}\) contain time-varying gain function \(\psi_{1}\) and \(\psi_{k}\), respectively. The entire closed-loop control system is shown in Figure 1.  
**Remark 3**. : _To enhance computational efficiency, the time-varying gain function \(\psi_{n}\) is substituted with a constant parameter \(\psi\). This modification eliminates the need to solve complex nonlinear equations associated with the control input \(u\), thereby substantially accelerating the algorithm's execution speed._  
**Theorem 1**. : _Assuming that the non-affine pure-feedback system (1) satisfies Assumptions 1-3, the proposed adaptive dynamic surface control system (it consists of three parts: intermediate control signals (12) and (13), the final control law (14), and the parameter adaptation mechanism (15)) can guarantee that all system states are semi-globally uniformly ultimately bounded (SGUIUB), with the tracking errors approach asymptotically to an arbitrarily small neighborhood of zero._  
Notice that the procedure of proof can be divided into \(n\) steps. **Step 1:** Following the coordinate transformations established in (10), the error dynamics can be expressed as  
\[e_{1}=x_{1}-y_{1d},\;z_{2}=w_{2}-\alpha_{1}. \tag{16}\]  
where \(e_{1}\) represents the output tracking error and \(z_{2}\) denotes the filter error.