## 5 Output Feedback Control Design  
In order to derive the adaptive control scheme based on output feedback, let us first consider the adaptive control design problem when \(u\) and \(c\) are known. Define \(\varsigma=u+c\). We perform the following coordinate transformation for system (1) as  
\[\left\{ \begin{aligned} \dot{x}&=\varsigma,\\ \dot{\varsigma}&=\frac{X_{u|u|}}{m-X_{u}}(\varsigma- c)|\varsigma-c|+\frac{1}{m-X_{u}}\Delta f(\varsigma-c)-\dot{c}+\frac{1-C_{n}}{m-X_{u}} \tau.\end{aligned} \right. \tag{33}\]  
Let \(z_{1}=x-x_{d}^{*}\) and \(z_{2}=\varsigma-\alpha\), where \(\alpha=\dot{x}_{d}^{*}-\frac{\iota}{2}(b_{z}^{2}-z_{1}^{2})z_{1}/b_{z}^{2}\). Similar to the derivation in (10)-(12), by calculating the time derivative of \(V_{1}=\frac{z_{1}^{2}}{2(b_{z}^{2}-z_{1}^{2})}\), we have  
\[\dot{V}_{1}= \sigma z_{1}(z_{2}+\alpha-\dot{x}_{d}^{*})=-\frac{\iota}{2}\frac{ z_{1}^{2}}{b_{z}^{2}-z_{1}^{2}}+\sigma z_{1}z_{2}, \tag{34}\]  
where \(\sigma=\frac{b_{z}^{2}}{(b_{z}^{2}-z_{1}^{2})^{2}}\). On the basis of (33) and the definition of \(z_{2}\), we obtain  
\[\dot{z}_{2}= \varsigma-\dot{\alpha}\] \[= \frac{X_{u|u|}}{m-X_{\dot{u}}}(\varsigma-c)|\varsigma-c|+\frac{1} {m-X_{\dot{u}}}\Delta f(\varsigma-c)-\dot{c}+\frac{1-C_{n}}{m-X_{\dot{u}}}\tau -\dot{\alpha} \tag{35}\]  
Define \(V_{2}=\frac{1}{2\partial}z_{2}^{2}\) with \(\varpi=\frac{1-C_{n}}{m-X_{\dot{u}}}\). According to (35), we obtain  
\[\dot{V}_{2}= z_{2}[\varpi^{-1}(\frac{X_{u|u|}}{m-X_{\dot{u}}}(\varsigma-c)| \varsigma-c|+\frac{1}{m-X_{\dot{u}}}\Delta f(\varsigma-c)-\dot{c})-\varpi^{-1 }\dot{\alpha}+\tau] \tag{36}\]  
Similar to (15), the following equation holds as  
\[\varpi^{-1}(\frac{X_{u|u|}}{m-X_{\dot{u}}}(\varsigma-c)|\varsigma-c|+\frac{1} {m-X_{\dot{u}}}\Delta f(\varsigma-c)-\dot{c})=\mathbf{\omega^{*}}^{T}\mathbf{\psi}( \mathbf{Y})+\epsilon(\mathbf{Y}), \tag{37}\]  
where \(|\epsilon(\mathbf{Y})|\leq\epsilon_{y,N}\), \(\mathbf{\psi}(\mathbf{Y})=\frac{\xi_{y,\rm t}}{\xi_{y,\rm t}\pm\exp(-\mathbf{Y}/\xi_{y, \rm t})}+\xi_{y,\rm t}\), \(\mathbf{Y}=[x,\varsigma,x_{d}^{*},\dot{x}_{d}^{*},\dot{x}_{d}^{*}]\), \(\xi_{y1}-\xi_{y,\rm t}\) are the design parameters and \(\epsilon_{y,N}\) is an unknown bounded constant. Combining (36) with (37) yields  
\[\dot{V}_{2}= z_{2}[\mathbf{\omega^{*}}^{T}\mathbf{\psi}(\mathbf{Y})+\epsilon(\mathbf{Y})+p\dot{ \alpha}+\tau], \tag{38}\]  
where \(p\) denotes \(-\varpi\). Based on (38), the following control law is designed as  
\[\tau= -\sigma z_{1}-\mu_{12}-\hat{\mathbf{\omega}}^{T}\mathbf{\psi}(\mathbf{Y})- \hat{p}\dot{\alpha} \tag{39}\] \[\hat{\mathbf{\omega}}= \mu_{2}z_{2}\mathbf{\psi}(\mathbf{Y})-k_{1}\mu_{3}\hat{\mathbf{\omega}},\] (40) \[\hat{p}= \mu_{3}z_{2}\hat{\mathbf{\omega}}-k_{2}\mu_{4}\hat{p}, \tag{41}\]  
On the basis of the above signal reconstruction, we can develop the output feedback barrier adaptive control law as follows:  
\[\tau= -\sigma z_{1}-\mu_{1}\hat{z}_{2}-\hat{\mathbf{\omega}}^{T}\mathbf{\psi}( \hat{\mathbf{Y}})-\hat{\mathbf{\theta}}\hat{\mathbf{x}} \tag{48}\] \[\dot{z}_{2}= \varsigma-\dot{\alpha}\]0 (49) \[\dot{z}_{2}= \varsigma-\dot{\alpha}\]1  
where  
\[\dot{z}_{2}= \varsigma-\dot{\alpha}\]2  
\(\hat{\mathbf{Y}}=[x,\xi,x_{d}^{*},x_{d}^{*},x_{d}^{*}]\), \(\xi_{g1}-\xi_{g4}\) are design parameters of the neural network, and the other variables and parameters are similar to the corresponding ones in (39)-(41). **Remark 4**: _Compared with the similar research results given in [13; 14; 15], the output feedback barrier adaptive control scheme (48)-(50) possesses the following characteristics:_  
1. _The proposed control scheme may be used in the cases that velocity signals are unmeasurable. Therefore, the applicable scope is extended._
2. _By incorporating the initial rectification technique into the barrier backstepping design, deferred output constraint adaptive control scheme is developed. So the robustness and safety of AUV systems have been improved._