# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 3 Ellipsoidal State Estimation Procedure
### Ellipsoidal State Prediction Step
**Step P2:**: Compute interval bounds for the term  
\[\mathbf{\mathrm{b}}_{k}=\left(\mathbf{\Phi}(\mathbf{\mathrm{p}})-\tilde{\mathbf{\Phi}}\right) \cdot\mathbf{\mu}_{k}\in[\mathbf{\mathrm{b}}_{k}]\;\;, \tag{34}\]  
which accounts for a non-zero ellipsoid midpoint with \(\mathbf{x}_{k}\), \(\mathbf{\Phi}\), and \(\mathbf{\mathrm{p}}\) defined according to (26), (28), and (29). Inflate the outer ellipsoid bound described by the shape matrix (31) with  
\[\mathbf{\mathrm{Q}}_{k+1}=(1+\rho_{\mathbf{\mathrm{O}},k+1})^{2}\cdot\tilde{\mathbf{ \mathrm{Q}}}_{k+1}\;\;\;\text{and}\;\;\;\;\rho_{\mathbf{\mathrm{O}},k+1}=\sup\Bigl{\{} \Bigl{\|}\alpha_{k+1}^{-1}\cdot\mathbf{\Gamma}_{k}^{-1}\cdot[\mathbf{\mathrm{b}}_{k}] \Bigr{\|}\Bigr{\}}\;\;. \tag{35}\]  
For a definition of the interval-valued generalization of the Euclidean norm operator, see [12]. **Step P3:**: Compute the updated ellipsoid midpoint as  
\[\mathbf{\mu}_{\mathbf{\Phi},k+1}=\tilde{\mathbf{\Phi}}\cdot\mathbf{\mu}_{k}\;\;. \tag{36}\]  
The outer ellipsoidal enclosure \(\mathcal{R}_{\mathbf{\Phi},k+1}\) of \(\mathbf{x}_{\mathbf{\Phi},k+1}\) at the time instant \(k+1\) then becomes  
\[\mathbf{x}_{\mathbf{\Phi},k+1}\in\mathcal{R}_{\mathbf{\Phi},k+1}(\mathbf{\mu}_{\mathbf{\Phi},k +1},\mathbf{\Gamma}_{\mathbf{\Phi},k+1})\;\;, \tag{37}\]  
where  
\[\mathbf{\Gamma}_{\mathbf{\Phi},k+1}=\alpha_{k+1}\cdot(1+\rho_{\mathbf{\mathrm{O}},k+1}) \cdot\tilde{\mathbf{\Phi}}\cdot\mathbf{\Gamma}_{k}\;. \tag{38}\]  
**Step P4:**: Determine an ellipsoidal enclosure (the aforementioned Lowner-John ellipsoid) for the summand  
\[\mathbf{x}_{\boldsymbol{\Psi},k+1}=\boldsymbol{\psi}(\mathbf{p}) \tag{39}\]  
according to  
\[\mathbf{x}_{\boldsymbol{\Psi},k+1}\in\mathscr{E}_{\boldsymbol{\Psi},k+1}( \boldsymbol{\mu}_{\boldsymbol{\Psi},k+1},\boldsymbol{\Gamma}_{\boldsymbol{\Psi},k+1})\enspace. \tag{40}\]  
To determine this ellipsoid from an interval vector \(\boldsymbol{\psi}(\mathbf{p})\) with the corresponding vertices \(\mathbf{x}_{\boldsymbol{\Psi}}^{(j)},j\in\{1,\ldots,L\}\), where  
\[\mathbf{Q}_{\boldsymbol{\Psi},k+1}=\boldsymbol{\Gamma}_{\boldsymbol{\Psi},k+ 1}\cdot\boldsymbol{\Gamma}_{\boldsymbol{\Psi},k+1}^{T} \tag{41}\]  
holds, the minimum-volume Lowner-John ellipsoid is determined by solving the linear matrix inequality constrained optimization problem  
\[\min_{\mathbf{Q}_{\boldsymbol{\Psi},k+1}\cdot\boldsymbol{\mu}_{ \boldsymbol{\Psi},k+1}}\bigl{(}\text{trace}\{\mathbf{Q}_{\boldsymbol{\Psi},k +1}\}\bigr{)}\] \[ \begin{bmatrix}1&\bigl{(}\mathbf{x}_{\boldsymbol{\Psi},k+1}^{(j)}- \boldsymbol{\mu}_{\boldsymbol{\Psi},k+1}\bigr{)}^{T}\\ \bigl{(}\mathbf{x}_{\boldsymbol{\Psi},k+1}^{(j)}-\boldsymbol{\mu}_{ \boldsymbol{\Psi},k+1}\bigr{)}&\mathbf{Q}_{\boldsymbol{\Psi},k+1}\end{bmatrix}  \succeq 0\quad\text{for all}\quad j\in\{1,\ldots,L\} \tag{42}\] \[\mathbf{Q}_{\boldsymbol{\Psi},k+1}\succeq 0\enspace.\]  
Further details concerning these inequality constraints, the choice of the number of vertices \(L\), and a simplified version purely based on interval analysis are discussed in [12]. Moreover, note that the exact volume minimization task would require the solution of an optimization task in which a complex determinant minimization task is involved. This task is replaced by the minimization of a matrix trace as described in Appendix C of [24]. Due to the linearity of the trace operator (and its close-to-optimal behavior), this version is used for the proposed ellipsoid prediction step when determining the bounds \(\mathscr{E}_{\boldsymbol{\Psi},k+1}\).