## 4 Control Design
### Inertial Delay Control (IDC)  
Due to the underwater circumstances, the dynamic equations of the UVMS include unknown external disturbances and an amount of parameter uncertainties caused by identification errors. These lumped uncertainties and disturbances can be expressed as (21) with reference to (3) and (4).  
\[\delta=-(\Delta\mathbf{M}_{\mathbf{\zeta}}^{\hat{\mathbf{\xi}}}+\Delta\mathbf{C}\hat{\mathbf{\xi}} +\Delta\mathbf{D}\hat{\mathbf{\xi}})+\mathbf{\tau}_{dis} \tag{21}\]  
where \(\hat{\mathbf{\xi}}\) and \(\hat{\mathbf{\xi}}\) denote the estimates of the system states via EKF.  
Then, based on (3), (4) and (21), the acceleration vector of the system can be obtained as  
\[\hat{\mathbf{\xi}}{=}{-}(\mathbf{M}^{*})^{-1}(\mathbf{C}^{*}\hat{\mathbf{\xi}}{+}\mathbf{D}^{*} \hat{\mathbf{\xi}}{+}\mathbf{g}){+}(\mathbf{M}^{*})^{-1}(\mathbf{u}{+}\mathbf{\delta}) \tag{22}\]  
Substituting the proposed control law (20) in (22), dynamical equation of the tracking errors is  
\[\check{\epsilon}+\mathbf{k}_{2}\hat{\mathbf{e}}+\mathbf{k}_{1}\mathbf{e}+(\mathbf{M}^{*})^{-1}\bm {J}^{T}\mathbf{k}_{E}\epsilon_{E}=(\mathbf{M}^{*})^{-1}\mathbf{e}_{\delta} \tag{23}\]  
where \(\mathbf{e}_{\delta}=\delta-\hat{\delta}\) is the estimated error vector.  
It is assumed that a slow-varying signal can be approximated and estimated by a filter with appropriate bandwidth [37]. Based on this assumption, the uncertainty and disturbance estimator (UDE) is proposed for estimating slow-varying uncertainties [35; 37]. Then, the estimations of lumped uncertainties and disturbances \(\hat{\delta}\) can be given as  
\[\hat{\delta}=\mathbf{G}_{f}(\mathbf{s})\delta \tag{24}\]  
where \(\mathbf{G}_{f}(\mathbf{s})\) is a strictly proper low-pass filter possessing a uniform steady-state gain and a sufficiently large bandwidth. Based on (24), it is found that by passing the lumped uncertainties and disturbances \(\delta\) through a inertial filter \(\mathbf{G}_{f}(\mathbf{s})\), the estimation vector \(\hat{\delta}\) can be obtained. The UDE method is redefined as inertial delay control (IDC) [37], because it is analogous to the time delay control (TDC) which delays the plant signals in time to obtain the estimates.  
Based on (23) and (24), we can obtain  
\[\hat{\delta}=\mathbf{G}_{f}(\mathbf{s})[\mathbf{M}^{*}(\check{\mathbf{e}}+\mathbf{k}_{2}\hat{\mathbf{ e}}+\mathbf{k}_{1}\mathbf{e})+\mathbf{J}^{T}\mathbf{k}_{E}\epsilon_{E}+\hat{\delta}] \tag{25}\]  
A choice of \(\mathbf{G}_{f}(\mathbf{s})\) with first order is given by  
\[\mathbf{G}_{f}(\mathbf{s})=\frac{\mathbf{I}}{\mathbf{I}+\mathbf{T}\mathbf{s}} \tag{26}\]  
where \(\mathbf{T}\) is a diagonal matrix with small positive constant. \(\mathbf{I}\) is the identity matrix.  
Then (25) can be rewritten as  
\[\mathbf{T}\hat{\delta}+\hat{\delta}=\mathbf{M}^{*}(\check{\epsilon}+\mathbf{k}_{2}\hat{ \mathbf{e}}+\mathbf{k}_{1}\mathbf{e})+\mathbf{J}^{T}\mathbf{k}_{E}\epsilon_{E}+\hat{\delta} \tag{27}\]Therefore, the estimates of the lumped uncertainties and disturbances can be obtained as  
\[\hat{\delta}=\mathbf{T}^{-1}\mathbf{M}^{*}(\hat{e}+\mathbf{k}_{2}\mathbf{e}+\mathbf{k}_{1}\int_{0}^{t }\mathbf{e}dt)+\mathbf{T}^{-1}\mathbf{J}^{T}\mathbf{k}_{E}\int_{0}^{t}\mathbf{e}_{E}dt \tag{28}\]  
From (25) and (26), the equation of estimated errors can be written as  
\[\dot{\mathbf{e}}_{\delta}=-\mathbf{T}^{-1}\mathbf{e}_{\delta}+\hat{\delta} \tag{29}\]  
If the lumped uncertainties and disturbances \(\delta\) are slowly varying, then \(\dot{\delta}\) is small and \(\dot{\delta}\approx\mathbf{0}\). Therefore, the estimated errors (\(\mathbf{e}_{\delta}\)) go to zero asymptotically. If \(\dot{\delta}\) is not small, but \(\ddot{\delta}\) is small, \(\mathbf{e}_{\delta}\) is ultimately bounded and the estimated accuracy can be improved by estimating \(\delta\) and \(\dot{\delta}\).