## 3 Main Results  
### Adaptive Output Feedback Control Design  
A newly adaptive fuzzy controller based on a switched observer was designed by incorporating the TD method into the backstepping technique. Firstly, we construct the following switched state observers:  
\[d\hat{x}_{1} =(\hat{x}_{2}-l_{k,1}(\hat{x}_{1}-y))dt,\] \[d\hat{x}_{2} =(u_{k}-l_{k,2}(\hat{x}_{1}-y))dt, \tag{11}\]  
where \(\hat{x}_{i}\) are the estimates of \(x_{i}\), and constants \(l_{k,i}>0\) (\(i=1,2\)) are chosen to make:  
\[A_{k}= \begin{bmatrix}-l_{k,1}&1\\ -l_{k,2}&0\end{bmatrix} \]  
strictly Hurwitz; hence, for \(\forall Q_{k}^{T}=Q_{k}>0\), positive definite matrices \(P_{k}\), \(A_{k}^{T}P_{k}+P_{k}A_{k}=-Q_{k}\) hold. Consider \(\hat{x}_{i}=x_{i}-\hat{x}_{i}\) as the errors, which satisfy (\(i=1,2\)):  
\[d\hat{x}=(A_{k}\tilde{x}+F_{k})dt+G_{k}d\omega, \tag{12}\]  
where \(F_{k}=[0,f_{k}]^{T},G_{k}=[0,\frac{1}{T}g_{k}^{T}]^{T}\) with \(\|G_{k}^{T}G_{k}\|\leq\sigma\), and \(\sigma\) is the bounded constant. The following tracking error variables are determined before the control design:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\] \[z_{2}= \hat{x}_{2}-\alpha_{2d},\] (14) \[\tilde{z}_{i}= z_{i}-\theta_{i},\quad i=1,2, \tag{15}\]  
where \(\tilde{z}_{i}\) is the new tracking error signal, and \(\theta_{i}\) and \(\alpha_{2d}\) will be shown in detail in the control design. _Step 0._ Consider the stochastic Lyapunov function below:  
\[V_{k0}=\frac{r}{2}(\tilde{x}^{T}P_{k}\tilde{x})^{2}, \tag{16}\]  
where \(r>0\) is a constant. By combining (12) and (16), we obtain:  
\[\mathcal{L}V_{k0}= -r(\tilde{x}^{T}P_{k}\tilde{x})\tilde{x}^{T}Q_{k}\tilde{x}+2r( \tilde{x}^{T}P_{k}\tilde{x})\tilde{x}^{T}P_{k}F_{k}\] \[+rTr\{G_{k}(2P_{k}\tilde{x}\tilde{x}^{T}P_{k}+\tilde{x}^{T}P_{k} \tilde{x}P_{k})G_{k}^{T}\}. \tag{17}\]  
Here, FLS \(W_{0}^{T}S_{0}\) is introduced to approximate \(F_{k}\), then:  
\[F_{k}= W_{0}^{T}S_{0}+\delta_{0},||\delta_{0}||\leq\varepsilon_{0}, \tag{18}\]  
where \(\varepsilon_{0}>0\). Under [PERSON]'s inequality, it yields:  
\[2r(\bar{x}^{T}P_{k}\bar{x})\bar{x}^{T}P_{k}F_{k}\leq \frac{3r}{2}\upsilon_{0}^{\frac{4}{3}}\|P_{k}\|_{\bar{x}}^{\frac{5} {3}}\|\bar{x}\|^{4}+\frac{r}{2\upsilon_{0}^{4}}(\theta^{2}+\varepsilon_{0}^{4})\] \[rTr\{G_{k}^{T}(2P_{k}\bar{x}\bar{x}^{T}P_{k}+\bar{x}^{T}P_{k} \bar{x}P_{k})G_{k}\}\] \[z_{1}= x_{1}-y_{d}, \tag{13}\]0 \[z_{1}= x_{1}-y_{d}, \tag{13}\]1  
where \(\upsilon_{0}\) is a positive designed scalar. According to (17) and (19), we have:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]2  
where \(\Xi=r\lambda_{\min}(P_{k})\lambda_{\min}(Q_{k})-\frac{3r}{2}\upsilon_{0}^{ \frac{4}{3}}\|P_{k}\|^{\frac{8}{3}}-\frac{3}{2}rn\sqrt{n}\upsilon_{0}\|P_{k}\| ^{4}\) and \(D_{0}=\frac{r}{2\upsilon_{0}^{4}}(\theta^{2}+\varepsilon_{0}^{4})+\frac{3}{2 \upsilon_{0}}rn\sqrt{n}\bar{\upsilon}^{2}\). _Step 1._ Consider the stochastic Lyapunov function below:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]3  
According to (13)-(15), we have:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]4  
By combining (21) and (22), we obtain:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]5  
Under [PERSON]'s inequality, we have:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]6 \[z_{1}= x_{1}-y_{d}, \tag{13}\]7  
where \(\upsilon_{1j}\,(j=1,2)\) are positive designed scalars. Substituting (24) and (25) into (23) leads to:  
\[z_{1}= x_{1}-y_{d}, \tag{13}\]8 \[z_{1}= x_{1}-y_{d}, \tag{13}\]9  
In order to eliminate the traditional problem of the \"explosion of terms\", we introduce the TD technique in [20] to help the control design. Next, we let the virtual control function \(\alpha_{1}\) pass through the following TD:  
\[\left\{ \begin{array}{l}\dot{\alpha}_{2d}=\chi_{2}\\ \dot{\chi}_{2}=-R_{2}^{2}[a_{21}\sinh(l_{21}(\alpha_{2d}-\alpha_{1}))+a_{22} \sinh(l_{22}\chi_{2}/R_{2})],\end{array} \right. \tag{27}\]  
where \(R_{2},a_{21},a_{22},l_{21},l_{22}\) are positive arguments; \(\alpha_{2d}\) is the estimate of \(\alpha_{1}\) with \(s_{2d}=\alpha_{2d}-\alpha_{1}\) denoting the generated estimation error.