# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 3 Ellipsoidal State Estimation Procedure
### Ellipsoidal Measurement Update Step  
To perform the ellipsoidal measurement update step, we employ the intersection technique for ellipsoids with different midpoints that was derived in [4]. This approach is an extension of the computation of Dikin ellipsoids, which is discussed in detail in [27].  
This extension consists of the following two steps:  
**Step C1:**: Determine the common center point for the bounds of the intersection, where the center point must be included _in all_ ellipsoids to be intersected;
**Step C2:**: Determine the shape matrices for the outer ellipsoid bound according to the computation of Dikin ellipsoids according to [27].  
Preliminary work in [4] has shown that an efficient heuristic approach for the computation of the common center point \(\tilde{\mathbf{\mu}}_{k+1}\) in **Step C1** of the two ellipsoids \(\mathcal{R}(\mathbf{\mu}_{k+1},\mathbf{\Gamma}_{k+1})\), \(\mathbf{Q}_{k+1}=\mathbf{\Gamma}_{k+1}\mathbf{\Gamma}_{k+1}^{T}\) and \(\mathcal{R}(\mathbf{\mu}_{m,k},\mathbf{\Gamma}_{m,k+1})\), \(\mathbf{Q}_{m,k+1}=\mathbf{\Gamma}_{m,k+1}\mathbf{\Gamma}_{m,k+1}^{T}\) is given by an approach motivated by the innovation step of a Kalman filter [14; 28].  
Here, the ellipsoid \(\mathcal{R}(\mathbf{\mu}_{m,k+1},\mathbf{\Gamma}_{m,k+1})\) characterizes the measurement model (23) with the output matrix \(\mathbf{C}\).  
With the help of this information, the Kalman gain matrix  
\[\mathbf{L}_{k+1}=\mathbf{Q}_{k+1}\cdot\mathbf{C}^{T}\cdot\left(\mathbf{C}\cdot \mathbf{Q}_{k+1}\cdot\mathbf{C}^{T}+\mathbf{Q}_{m,k+1}\right)^{-1} \tag{48}\]  
is computed. Then, the updated ellipsoid midpoint results in  
\[\tilde{\mathbf{\mu}}_{k+1}=\mathbf{\mu}_{k+1}+\mathbf{L}_{k+1}\cdot\left(\mathbf{\mu}_{m,k +1}-\mathbf{\mu}_{k+1}\right). \tag{49}\]  
Both ellipsoids to be intersected are first enclosed during **Step C2** by new ellipsoids centered at the midpoint \(\tilde{\mathbf{\mu}}_{k+1}\). For that purpose, the scaling factors  
\[\zeta_{k+1}=1+\left\|\mathbf{\Delta}\mathbf{\mu}_{1,k+1}\right\|\quad\text{with}\quad \mathbf{\Delta}\mathbf{\mu}_{1,k+1}=\mathbf{Q}_{k+1}^{-\frac{1}{2}}\cdot\left(\tilde{ \mathbf{\mu}}_{k+1}-\mathbf{\mu}_{k+1}\right) \tag{50}\]  
and  
\[\zeta_{m,k+1}=1+\left\|\mathbf{\Delta}\mathbf{\mu}_{m,k+1}\right\|\quad\text{with} \quad\mathbf{\Delta}\mathbf{\mu}_{m,k+1}=\left(\mathbf{P}_{m}^{\prime}\right)^{\frac {1}{2}}\cdot\left(\tilde{\mathbf{\mu}}_{k+1}-\mathbf{\mu}_{m,k+1}^{\prime}\right) \tag{51}\]  
are determined, which represent the maximum distances of the new midpoint computed in Equation (49) from the original ellipsoid surfaces. Here, \(\left\|\mathbf{\Delta}\mathbf{\mu}_{\cdot,k+1}\right\|\) is the Euclidean norm of the corresponding vector-valued argument.  
Second, the outer bound of the intersection of these two rescaled ellipsoids is given by Equation (53) in [4] by  
\[\mathcal{E}_{k+1}=\mathcal{E}_{k+1}\left(\tilde{\mathbf{\mu}}_{k+1},\mathbf{\Gamma}_{ k+1}^{\prime}\right)\quad\text{with}\quad\mathbf{\Gamma}_{k+1}^{\prime}=\left( \mathbf{Q}_{k+1}^{\prime}\right)^{\frac{1}{2}}\, \tag{52}\]  
where the shape matrix is determined with the closed-form expression  
\[\mathbf{Q}_{k+1}^{\prime}=4\cdot\left(2\cdot\left(\left(\zeta_{k+1}^{2}\cdot \mathbf{Q}_{k+1}\right)^{-1}+\zeta_{m,k+1}^{-2}\cdot\mathbf{P}_{m}^{\prime} \right)\right)^{-1} \tag{53}\]  
with \(\mathbf{Q}_{k+1}\) from (35). As shown in [4], this approach is applicable also in cases in which not all components of the state vector are measured, i.e., if the matrix \(\mathbf{P}_{m}^{\prime}\) introduced in (24) represents a degenerate ellipsoid with bounds that may be infinitely wide in some components of the state space. Although this procedure may be less accurate than the technique presented in [8; 29], it is preferred in this paper due to its simplified closed-form solution representation and because it avoids the online solution of linear matrix inequalities.