# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis
Based on Assumption 3, we derive that \(d_{k}(t)\) is unknown bounded, so \(f_{k}(\bar{X}_{k})\) is also an unknown bounded function. Substituting (42) into (41) results in  
\[\dot{V}_{k}=e_{k}f_{k}-a_{k}e_{k}^{2}-\frac{e_{k}^{2}\hat{\theta}\tilde{W}_{k }^{T}\tilde{W}_{k}}{2b_{k}^{2}}+e_{k}\psi_{k}z_{k+1}-\frac{\widetilde{\psi} \acute{e}_{k}^{2}}{2}-\frac{e_{k}^{2}}{2}+z_{k+1}\dot{z}_{k+1}. \tag{43}\]  
It can be inferred from Lemma 2 that the FLS can be viewed as an approximator to model the unknown function \(f_{k}(\bar{X}_{k})\) such that for any accuracy \(\widetilde{\acute{\xi}}_{k}\in\Re^{*}\), \(f_{k}\) satisfies the following equation:  
\[f_{k}(\bar{X}_{k})=\kappa_{k}^{*T}\bar{W}_{k}(\bar{X}_{k})+\acute{\xi}_{k}( \bar{X}_{k}),\ |\acute{\xi}_{k}(\bar{X}_{k})|\leq\widetilde{\acute{\xi}}_{k} \tag{44}\]  
where \(\acute{\xi}_{k}(\bar{X}_{k})\) is error function with upper bound \(\widetilde{\acute{\xi}}_{k}\). According to the definition of (44), (43) can be further deformed as follows:  
\[\dot{V}_{k}= e_{k}\kappa_{k}^{*T}\bar{W}_{k}+e_{k}\xi_{k}-a_{k}e_{k}^{2}-\frac{ e_{k}^{2}\hat{\theta}\tilde{W}_{k}^{T}\tilde{W}_{k}}{2b_{k}^{2}} \tag{45}\] \[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]0  
with the help of the Young's inequality, we conclude that  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]1 \[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]2  
Substituting (46) into (45), one obtains  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]3The filter output \(w_{k+1}\) with design constant \(\Phi_{k+1}\) is defined as follows:  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]4  
According to the same analysis, whereby, from (34) to (38), we have  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]5  
where \(\varepsilon_{k}\) denotes \(\varepsilon_{k}(e_{1},\ldots,e_{k+1},z_{2},\ldots,z_{k+1},\hat{\theta},y_{1d},\hat{y}_{1d})\). Substituting (49) into (47), one obtains the following result:  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]6  
**Step \(n\):** The \(n\)th error surface is \(e_{n}=x_{n}-w_{n}\), and its time derivative can be described as  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]7  
Based on system (3) and (51), \(e_{n}\dot{e}_{n}\) can be represented as  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]8  
Substituting (14) into (52), one obtains  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]9  
According to the definition of \(\psi\), it can be inferred that the constant design parameter \(\psi\) is less than the minimum value of \(\psi_{n}\), so the following inequality holds:  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]0  
Establish a Lyapunov function candidate \(V_{n}=\frac{1}{2}\left(e_{n}^{2}+\frac{\tilde{\theta}^{2}}{\zeta}\right)\), and its own derivative can be derived as  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]1  
where \(\tilde{\theta}=\theta\ -\ \hat{\theta}\).