# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis
It can be inferred from (6) that \(\theta\) is a constant value whose derivative is \(0\); so, the following equation holds:  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]2  
Substituting (54) into (56) produces  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]3  
The input function of the \(n\)th FLS is constructed as  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]4with \(\bar{X}_{n}=[x_{1},\ldots,x_{n},y_{1d},w_{2},\ldots,w_{n},t]^{T}\). Based on Assumption 3, it can be derived that \(f_{n}(\bar{X}_{n})\) is an unknown bounded function. Substituting (58) into (57) results in  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]5  
Approximation of unknown function \(f_{n}(\bar{X}_{n})\) requires the FLS with given accuracy \(\widetilde{\xi}_{n}\in\Re^{+}\); therefore, \(f_{n}\) satisfies  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]6  
where \(\xi_{n}(\bar{X}_{n})\) is error function with upper bound \(\widetilde{\xi}_{n}\). Substituting (60) into (59) yields  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]7  
Based on [PERSON]'s inequality, we obtain  
\[ \begin{split}\frac{e_{n}\bar{\kappa}_{n}^{\star T}\bar{W}_{n}}{ b_{n}}&\leq\frac{e_{n}^{2}\hat{\theta}\bar{W}_{n}^{T}\bar{W}_{n}}{2b_ {n}^{2}}+\frac{b_{n}^{2}}{2}\\ e_{n}\xi_{n}&\leq\frac{e_{n}^{2}+\widetilde{\xi}_ {n}^{2}}{2}.\end{split}  \tag{62}\]  
Substituting (62) into (61), one obtains  
\[\dot{V}_{n}\leq\frac{e_{n}^{2}\hat{\theta}\bar{W}_{n}^{T}\bar{W}_{n}}{2b_{n} ^{2}}-a_{n}e_{n}^{2}+\frac{b_{n}^{2}+\widetilde{\xi}_{n}^{2}}{2}-\frac{\hat{ \theta}\hat{\theta}}{\varsigma}. \tag{63}\]  
Now, define the overall Lyapunov function for nonlinear control system (1) as follows:  
\[V{=}\sum_{i=1}^{n}V_{i}=\sum_{i=1}^{n}\frac{e_{i}^{2}}{2}+\sum_{m=2}^{n}\frac {z_{m}^{2}}{2}+\frac{\hat{\theta}^{2}}{2\varsigma}. \tag{64}\]  
Differentiating \(V\) with respect to time holds:  
\[\dot{V}{=}\sum_{i=1}^{n}\dot{V}_{i}=\sum_{i=1}^{n}e_{i}e_{i}+\sum_{m=2}^{n}z_{ m}\dot{z}_{m}-\frac{\hat{\theta}\hat{\theta}}{\varsigma}. \tag{65}\]  
By substituting (63), (50), and (38) into (65), it can be observed that \(\dot{V}\) satisfies  
\[\begin{split}\dot{V}\leq&\sum_{i=1}^{n}\frac{e_{  
Letting \(q=\min\left\{2a_{1},\ldots,2a_{n}\right\}\), we have the following inequality:  
\[-\sum_{i=1}^{n}a_{i}e_{i}^{2}\leq-q\sum_{i=1}^{n}\left(\frac{e_{i}^{2}}{2}\right). \tag{68}\]  
Assuming that \(\min\left(\frac{1}{\Phi_{m}}-1+\frac{\widehat{\psi}}{2}\right)\geq\frac{q}{2}, \ m=2,\ldots,n\), one can gain the following inequality:  
\[-\sum_{m=2}^{n}\left(\frac{1}{\Phi_{m}}-1+\frac{\widehat{\psi}}{2}\right)z_{m} ^{2}\leq-q\sum_{m=2}^{n}\left(\frac{z_{m}^{2}}{2}\right). \tag{69}\]  
**Remark 4**.