# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis
Then, (27) can be rewritten as  
\[\alpha_{k}=\frac{-a_{k}e_{k}}{\psi_{k}}-\frac{e_{k}\dot{\theta}\bar{W}_{k}^{T }\bar{W}_{k}}{\psi_{k}2b_{k}^{2}}-\frac{g_{k}}{\psi_{k}}-e_{k+1}+x_{k+1}^{0}+ \frac{\dot{w}_{k}}{\psi_{k}} \tag{13}\]0According to [PERSON]'s inequality, we can obtain  
\[ \begin{split}\frac{e_{1}\bar{\kappa}_{1}^{*T}\tilde{W}_{1}}{b_{1}}b_{1 }&\leq\frac{e_{1}^{2}\theta\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1}^{ 2}}+\frac{b_{1}^{2}}{2}\\ e_{1}\bar{\xi}_{1}&\leq\frac{e_{1}^{2}+\bar{\xi}_ {1}^{2}}{2}\\ e_{1}\psi_{1}z_{2}&\leq\frac{\psi_{1}\big{(}e_{1} ^{2}+z_{2}^{2}\big{)}}{2}\leq\frac{\widehat{\psi}e_{1}^{2}}{2}+\frac{\widehat{ \psi}x_{2}^{2}}{2}\end{split}  \tag{30}\]  
where \(\theta\) has been defined in (6). Substituting (30) into (29), one gains the following result:  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}+\frac{\widehat{ \psi}x_{2}^{2}}{2}+z_{2}\hat{z}_{2}. \tag{31}\]  
The first command control signal \(\alpha_{1}\) goes through a first-order filter; both the output derivative \(\dot{w}_{2}\) and the \(w_{2}\) are satisfied in the following equilibrium:  
\[\Phi_{2}\dot{w}_{2}+w_{2}=\alpha_{1} \tag{32}\]  
where \(\Phi_{2}\) is the filter positive constant and \(w_{2}(0)=\alpha_{1}(0)\). One of the conditions of which the dynamic system can be controlled is that the control signal is bounded; so, assume that \(|-\dot{\alpha}_{1}|\leq\varepsilon_{1}\), where \(\varepsilon_{1}\) is positive constant. Based on the definition of (12), we assume that there exists a bounded continues function \(\varepsilon_{1}\big{(}e_{1},\hat{\theta},e_{2},y_{1d},\dot{y}_{1d}\big{)}\) such that  
\[|-\dot{\alpha}_{1}|\leq\varepsilon_{1}\big{(}e_{1},\hat{\theta},e_{2},y_{1d}, \dot{y}_{1d}\big{)} \tag{33}\]  
where \(\varepsilon_{1}\big{(}e_{1},\hat{\theta},e_{2},y_{1d},\dot{y}_{1d}\big{)}\) can be simplified as \(\varepsilon_{1}\). According to (16) and (32), the time derivative of \(z_{2}\) is computed as  
\[\dot{z}_{2}=\frac{-z_{2}}{\Phi_{2}}-\dot{\alpha}_{1}. \tag{34}\]  
Based on the nature of the absolute value inequality, one has  
\[ \begin{split} z_{2}\hat{z}_{2}+\frac{z_{2}^{2}}{\Phi_{2}}& \leq|z_{2}|\Big{|}\dot{z}_{2}+\frac{z_{2}}{\Phi_{2}}\Big{|}\leq|z_{2}| \varepsilon_{1}\\ \Rightarrow\ z_{2}\dot{z}_{2}&\leq\frac{-z_{2}^{2}}{ \Phi_{2}}+|z_{2}|\varepsilon_{1}.\end{split}  \tag{35}\]  
Using the completion squares theorem, we have  
\[|z_{2}|\varepsilon_{1}\leq z_{2}^{2}+\frac{\varepsilon_{1}^{2}}{4}. \tag{36}\]  
Substituting (36) into (35), the following result holds:  
\[z_{2}\dot{z}_{2}\leq\frac{-z_{2}^{2}}{\Phi_{2}}+z_{2}^{2}+\frac{\varepsilon_{ 1}^{2}}{4}. \tag{37}\]  
Substituting (37) into (31) we can conclude that  
\[\dot{V}_{1}\leq\frac{e_{1}^{2}\bar{\theta}\tilde{W}_{1}^{T}\tilde{W}_{1}}{2b_{1 }^{2}}-a_{1}e_{1}^{2}+\frac{b_{1}^{2}+\bar{\xi}_{1}^{2}}{2}-\bigg{(}\frac{1}{ \Phi_{2}}-1+\frac{\widehat{\psi}}{\widehat{2}}\bigg{)}z_{2}^{2}+\frac{ \varepsilon_{1}^{2}}{4}. \tag{38}\]  
**Step \(k\) (\(k=2,3,\ldots,n-1\)):** Based on the similar derivation process (16)-(22), \(\acute{e}_{k}\) can be described as  
\[\acute{e}_{k}=-a_{k}e_{k}-\frac{e_{k}\hat{\theta}\tilde{W}_{k}^{T}\tilde{W}_{k}} {2b_{k}^{2}}+\psi_{k}z_{k+1}+d_{k}(t). \tag{39}\]  
Choose a smooth scalar function \(V_{k}=\frac{1}{2}\Big{(}e_{k}^{2}+z_{k+1}^{2}\Big{)}\), and its time derivative can be obtained:  
\[\dot{V}_{k}=e_{k}\acute{e}_{k}+z_{k+1}\dot{z}_{k+1}. \tag{40}\]  
Substituting (39) into (40) produces  
\[\dot{V}_{k}=-a_{k}e_{k}^{2}-\frac{e_{k}^{2}\hat{\theta}\tilde{W}_{k}^{T}\tilde {W}_{k}}{2b_{k}^{2}}+e_{k}\psi_{k}z_{k+1}+e_{k}d_{k}(t)+z_{k+1}\dot{z}_{k+1}. \tag{41}\]  
The input function of the \(k\)th FLS is defined as follows:  
\[f_{k}(\bar{X}_{k})=\frac{\widetilde{\psi}\acute{e}_{k}}{2}+\frac{e_{k}}{2}+d_ {k}(t) \tag{42}\]  
with \(\bar{X}_{k}=[x_{1},\ldots,x_{k},y_{1d},w_{2},\ldots,w_{k+1},t]^{T}\).