# Fuzzy Adaptive Dynamic Surface Control with Constant Gain for Non-Affine Pure-Feedback Systems
## 3 Controller Design and Stability Analysis
: _The constant parameter \(\Phi_{m}\) of each filter must be determined under the condition that the aforementioned inequality \(\min\left(\frac{1}{\Phi_{m}}-1+\frac{\widehat{\psi}}{2}\right)\geq\frac{q}{2}, \ m=2,\ldots,n\) is satisfied. When the bounds of \(\widehat{\psi}\) and \(q\) cannot be precisely estimated, a sufficiently small \(\Phi_{m}\) near zero may be selected to meet the required conditions. However, excessively reducing this parameter can introduce unbounded terms in the system dynamics differential equation, resulting in divergent solutions. To avoid this issue, the filter constant \(\Phi_{m}\) should be adjusted to a larger value._  
_As \(\hat{\theta}=\theta-\bar{\theta}\), one obtains_  
\[\frac{\varphi\bar{\theta}\hat{\theta}}{\xi}\!=\!\frac{\varphi\bar{\theta}( \theta-\bar{\theta})}{\xi}=\frac{\varphi\left(\hat{\theta}\theta-\bar{\theta}^ {2}\right)}{\xi}. \tag{70}\]  
_By using [PERSON]'s inequality, the following inequality holds:_  
\[\bar{\theta}\theta\leq\frac{\bar{\theta}^{2}}{2}+\frac{\theta^{2}}{2}. \tag{71}\]  
_Substituting (71) into (70) yields:_  
\[\frac{\varphi\bar{\theta}\hat{\theta}}{\xi}\leq\frac{\varphi\theta^{2}}{2\xi }-\frac{\varphi\bar{\theta}^{2}}{2\xi}. \tag{72}\]  
_By substituting (68), (69), and (72) into (68), one obtains_  
\[\dot{V}\leq-q\sum_{i=1}^{n}\left(\frac{e_{i}^{2}}{2}\right)-q\sum_{m=2}^{n} \left(\frac{z_{m}^{2}}{2}\right)-\frac{\varphi\bar{\theta}^{2}}{2\xi}\!+\! \frac{\varphi\theta^{2}}{2\xi}\!+\!\sum_{i=1}^{n}\frac{b_{i}^{2}+\xi_{i}^{2}} {2}\!+\!\sum_{j=1}^{n-1}\frac{\xi_{j}^{2}}{4} \tag{73}\]  
_Letting_  
\[v=\frac{\varphi\theta^{2}}{2\xi}+\sum_{i=1}^{n}\frac{b_{i}^{2}+\widetilde{ \xi}_{i}^{2}}{2}+\sum_{j=1}^{n-1}\frac{e_{j}^{2}}{4} \tag{74}\]  
_where \(\varphi\), \(\xi\), \(b_{i}\), \(\xi_{i}\), \(\widehat{\alpha}_{m}\) are positive constants, and \(\frac{\varphi\theta^{2}}{2\xi}>0\), so \(v>0\). Then, (73) can be simplified as_  
\[V{=}\sum_{i=1}^{n}V_{i}=\sum_{i=1}^{n}\frac{e_{i}^{2}}{2}+\sum_{m=2}^{n}\frac {z_{m}^{2}}{2}+\frac{\hat{\theta}^{2}}{2\varsigma}. \tag{64}\]0  
_Inequality (75) satisfies all the conditions mentioned in Lemma 1, so we conclude that all signals in the system (1) are SGUIUB [37]. The proof of Theorem 1 is completed. Furthermore, when time tends to infinity, based on (64), the following result can be obtained:_  
\[V{=}\sum_{i=1}^{n}V_{i}=\sum_{i=1}^{n}\frac{e_{i}^{2}}{2}+\sum_{m=2}^{n}\frac {z_{m}^{2}}{2}+\frac{\hat{\theta}^{2}}{2\varsigma}. \tag{64}\]1when the control system enters steady state, its tracking error satisfies the following inequality:_  
\[V{=}\sum_{i=1}^{n}V_{i}=\sum_{i=1}^{n}\frac{e_{i}^{2}}{2}+\sum_{m=2}^{n}\frac {z_{m}^{2}}{2}+\frac{\hat{\theta}^{2}}{2\varsigma}. \tag{64}\]2  
_Based on (77), it implies that arbitrarily given tracking accuracy can be achieved by determining adequate control parameters._  
**Remark 5**. : _By employing the framework of DSC [28], this research tackles with the fuzzy adaptive control problem for a class of pure-feedback systems by using only one adaptive updating law. Therefore, the proposed scheme not only is the extension of previous research achievements for strict-feedback systems [20], but it also reduces the computational burden in comparison of [28; 35]._