## 3 Control Scheme Design
### Fixed-Time Disturbance Observer  
In practice, ROVs are affected by various complex environmental forces and modeling errors. After this, the dynamics cannot be restored to their actual state, so it is necessary to a design disturbance observer for online estimation of and compensation for uncertain disturbances.  
**Assumption 1**.: _External disturbance changes are relatively slow._  
**Assumption 2**.: _The aggregate uncertainty \(w_{\eta}\) of the system dynamics is bounded, namely, there exists a nonzero constant \(\rho\) satisfying \(||w_{\eta}||\leq\rho\leq\infty\)._  
**Assumption 3**.: _Each element in the parameter matrix \(Z_{2}\) is greater than \(\rho\)._  
We first define the error variable \(E\) as follows:  
\[E=\bar{M}v-\mathrm{Y} \tag{41}\]  
where \(v\) is the velocity variable of the ROVs, \(\mathrm{Y}\) is the dynamical auxiliary variable of the introduced disturbance observer with the same dimension as the velocity and \(\mathrm{Y}\) is as follows:  
\[\dot{\mathrm{Y}}=-\bar{C}(v)v-\bar{D}(v)v+\tau_{\eta}+Z_{1}E+Z_{2}sgnE+Z_{3} sig^{\delta}(E)+Z_{4}sig^{\zeta}(E) \tag{42}\]  
where \(Z_{i}\in\mathbb{R}^{6\times 1}(i=1,2,3,4)\) are positive definite diagonal matrices and \(\lambda_{min}(Z_{2})\geq\rho\). That is, every element in the diagonal matrix \(Z_{2}\) is greater than or equal to \(\rho\). \(\delta\) and \(\zeta\) are positive constants, with \(0<\delta<1\), \(\zeta>1\).  
The derivative of \(E\) is as follows:  
\[\dot{E} =\bar{M}\dot{v}-\dot{\mathrm{Y}} \tag{43}\] \[=-\bar{C}(v)v-\bar{D}(v)v+\tau_{\eta}+w_{\eta}-[-\bar{C}(v)v-\bar {D}(v)v+\tau_{\eta}\] \[+Z_{1}E+Z_{2}sgnE+Z_{3}sig^{\delta}(E)+Z_{4}sig^{\zeta}(E)]\] \[=w_{\eta}-Z_{1}E-Z_{2}sgnE-Z_{3}sig^{\delta}(E)-Z_{4}sig^{\zeta}(E)\]  
Designing a fixed-time disturbance observer \(\widehat{w}_{\eta}\) enables (43) to converge as follows:  
\[\widehat{w}_{\eta}=Z_{1}E+Z_{2}sgnE+Z_{3}sig^{\delta}(E)+Z_{4}sig^{\zeta}(E) \tag{44}\]  
To evaluate the performance of the fixed-time disturbance observer, we define the estimation error variable \(\widehat{w}_{\eta}\) as follows:  
\[\widehat{w}_{\eta}=w_{\eta}-\widehat{w}_{\eta}=w_{\eta}-Z_{1}E-Z_{2}sgnE-Z_{3} sig^{\delta}(E)-Z_{4}sig^{\zeta}(E) \tag{45}\]  
Under the action of (44), this estimation error will gradually converge to zero. The proof is as follows:  
From (45), if \(E\) converges, then its derivative \(\dot{E}\) also converges, and thus \(\widehat{w}_{\eta}\) can also converge.  
Define the Lyapunov function \(V_{E}\) as follows:  
\[V_{E}=\frac{1}{2}E^{T}E \tag{46}\]\[\dot{V}_{E} =E^{T}\dot{E} \tag{47}\] \[=E^{T}\Big{[}w_{\eta}-Z_{1}E-Z_{2}sgnE-Z_{3}sig^{\delta}(E)-Z_{4}sig^{ \zeta}(E)\Big{]}\] \[\leq\rho||E||-Z_{1}||E||^{2}-Z_{2}||E||-E^{T}\Big{(}Z_{3}E^{\delta} +Z_{4}E^{\zeta}\Big{)}\] \[\leq-Z_{3}\Big{(}E^{T}E\Big{)}^{\frac{\delta+1}{2}}-Z_{4}\Big{(}E ^{T}E\Big{)}^{\frac{\zeta+1}{2}}\] \[=-2^{\frac{\delta+1}{2}}\lambda_{min}(Z_{3})\left(\frac{1}{2}E^{ T}E\right)^{\frac{\delta+1}{2}}-2^{\frac{\zeta+1}{2}}\lambda_{min}(Z_{4}) \left(\frac{1}{2}E^{T}E\right)^{\frac{\zeta+1}{2}}\] \[=-2^{\frac{\delta+1}{2}}\lambda_{min}(Z_{3})V_{E}^{\frac{\delta+1 }{2}}-2^{\frac{\zeta+1}{2}}\lambda_{min}(Z_{4})V_{E}^{\frac{\zeta+1}{2}}\]  
where the function \(\lambda_{min}(\bullet)\) is the minimum eigenvalue of the matrix.  
By Lemma 1 it can be shown that \(E\) is fixed-time stable. The upper bound of the convergence time of the system is given by the following:  
\[T_{0}\leq T_{max}=\frac{2^{\frac{1-\delta}{2}}}{\lambda_{min}(Z_{3})(1-\delta) }+\frac{2^{\frac{1-\zeta}{2}}}{\lambda_{min}(Z_{4})(\zeta-1)} \tag{48}\]  
This shows that \(E=0\) when the system time \(t\geq T_{0}\), which leads to the conclusion that \(\bar{w}_{\eta}=0\) when \(t\geq T_{0}\).