# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 3 Ellipsoidal State Estimation Procedure
### Ellipsoidal State Prediction Step  
The first three steps of the following state prediction procedure for discrete-time systems (21) were basically published and proven in [4]. The major difference is that the quoted publication made use of a thick ellipsoid state enclosure approach in which inner and outer ellipsoid bounds for the first term \(\mathbf{\Phi}(\mathbf{p})\cdot\mathbf{\mathrm{x}}_{k}\) in (21) were determined simultaneously. However, for the sake of a pure state estimation in the form of computing guaranteed outer enclosures, only the outer bounds of those so-called thick ellipsoids are of interest. For that reason, the approach from [4] is simplified by leaving out the computation of the associated inner state bounds. For finding outer ellipsoidal enclosures of the subexpression \(\mathbf{\Phi}(\mathbf{p})\cdot\mathbf{\mathrm{x}}_{k}\) in (21), we assume that this part of the system model is reformulated in the form  
\[\mathbf{\mathrm{x}}_{\mathbf{\Phi},k+1}=\mathbf{\Phi}(\mathbf{p})\cdot\mathbf{\mathrm{x}}_{k} =\mathbf{\Phi}(\mathbf{p})\cdot\tilde{\mathbf{\mathrm{x}}}_{k}+\tilde{\mathbf{\Phi}}\cdot \mathbf{\mu}_{k}+\left(\mathbf{\Phi}(\mathbf{p})-\tilde{\mathbf{\Phi}}\right)\cdot\mathbf{\mu}_ {k}\enspace. \tag{25}\]  
This allows for propagating an origin-centered ellipsoid with the help of the first term in (25), while the remaining terms account for the influence of the generally non-zero ellipsoid midpoint. For that purpose, introduce the following notation already applied in (25) and further used during the state prediction consisting of the **Steps P1-P5**:\[\mathcal{R}_{k}=\mathcal{R}_{k}(\mu_{L}\,\mathbf{\Gamma}_{k})\;\;\text{denotes the uncertainty on the non-origin centered states $\mathbf{x}_{k}$}\;\;, \tag{26}\] \[\breve{\mathcal{R}}_{k}=\breve{\mathcal{R}}_{k}(\mathbf{0},\mathbf{\Gamma}_ {k})\;\;\text{the uncertainty of $\breve{\mathbf{x}}_{k}$ after shifting the ellipsoid to the origin, and}\] (27) \[\mathbf{\Phi}=\mathbf{\Phi}(\text{mid}([\mathbf{p}]))\;\;\text{the midpoint approximation of the quasi-linear system matrix with}\] (28) \[\mathbf{\mathrm{p}}\in[\mathbf{\mathrm{p}}]=\left[\mathbf{\mathrm{p}}\;;\; \overline{\mathbf{\mathrm{p}}}\right],\;\;\text{and the interval midpoint mid}([\mathbf{\mathrm{p}}])=\frac{1}{2}\cdot\left(\mathbf{\mathrm{p}}+\overline{\mathbf{ \mathrm{p}}}\right)\;\;. \tag{29}\]  
**Step P1:**: Apply  
\[\breve{\mathbf{x}}_{k+1}=\mathbf{\Phi}(\mathbf{\mathrm{p}})\cdot\breve{\mathbf{x}}_{k} \tag{30}\]  
to the ellipsoid \(\breve{\mathcal{R}}_{k}\) in (27). The shape matrix of the outer ellipsoid enclosure of the image set is described by an ellipsoid with the shape matrix  
\[\tilde{\mathbf{\mathrm{Q}}}_{k+1}=\alpha_{k+1}^{2}\cdot\mathbf{\Gamma}_{k+1}\cdot\mathbf{ \Gamma}_{k+1}^{T}\;\;, \tag{31}\]  
where \(\alpha_{k+1}\geq 0\) is the smallest value for which the linear matrix inequality  
\[\mathcal{M}_{k+1}:=\mathbf{\Lambda} \begin{bmatrix}-\mathbf{\mathrm{Q}}_{k}^{-1}&\mathbf{ \Phi}^{T}(\mathbf{\mathrm{p}})\cdot\tilde{\mathbf{\Phi}}^{-T}\\ \tilde{\mathbf{\Phi}}^{-1}\cdot\mathbf{\Phi}(\mathbf{\mathrm{p}})&-\alpha_{k+1}^{2}\mathcal{ R}_{k}\end{bmatrix} \mathbf{\Lambda}\preceq 0\;,\;\;\mathbf{\mathrm{Q}}_{k}=\mathbf{\Gamma}_{k} \cdot\mathbf{\Gamma}_{k}^{T} \tag{32}\]  
is satisfied for all \(\mathbf{\mathrm{p}}\in[\mathbf{\mathrm{p}}]\) with  
\[\mathcal{R}_{k}:=\mathbf{\Gamma}_{k}\cdot\mathbf{\Gamma}_{k}^{T}\;\;. \tag{33}\]  
In (32), the preconditioning matrix \(\mathbf{\Lambda}=\text{blkdiag}\left(\beta\mathbf{\mathrm{I}},\beta^{-1}\mathbf{\mathrm{ I}}\right)\) is defined with the help of the identity matrix \(\mathbf{\mathrm{I}}\in\mathbb{R}^{n\times n}\) and the square root \(\beta=\sqrt{\min\{\lambda_{i}(\mathbf{\mathrm{Q}}_{k})\}}\) of the smallest eigenvalue of \(\mathbf{\mathrm{Q}}_{k}\) as described in [22]. This choice helps to prevent unnecessarily conservative state bounds in cases in which the norms of \(\tilde{\mathbf{\Phi}}^{-1}\cdot\mathbf{\Phi}(\mathbf{\mathrm{p}})\) and \(\mathbf{\mathrm{Q}}_{k}\) are significantly different.