## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Dikin Ellipsoids for the Intersection of Ellipsoids
#### 3.2.2 Generalization to the Intersection of Ellipsoids with Different Midpoints  
An extension of the technique for computing Dikin ellipsoids to a scenario in which two ellipsoids with different centers are intersected, is based on the following three-stage procedure:  
**Step 1**: Determine the common center point for the desired inner and outer bounds of the intersection that must be included _in all_ ellipsoids to be intersected;
**Step 2**: Determine initial approximations of the shape matrices for the inner and outer bounds according to Section 3.2.1;
**Step 3**: For non-empty inner bounds, correct the outer enclosure so that the inner and outer ellipsoid surfaces become parallel to each other and, thus, form a thick ellipsoid (\(\mathcal{E}\)). **Step 3***: As an alternative to **Step 3**, the initial outer enclosure remains fixed and the inner ellipsoid surface is adapted to become parallel to each other to form a thick ellipsoid (\(\mathcal{E}\)). As a heuristic approach for the computation of the common center point \(\bar{\mathbf{\mu}}_{k}\) in **Step 1** of two ellipsoids that are described by the midpoints \(\mathbf{\mu}_{1,k}\) and \(\mathbf{\mu}_{2,k}\) as well as the shape matrices \(\mathbf{Q}_{1,k}\) and \(\mathbf{Q}_{2,k}\), respectively, the first one is interpreted as the prior knowledge in the innovation step of a Kalman filter [25; 55] and the second one as the corresponding information of the state measurement. Under this assumption, one obtains the Kalman gain matrix  
\[\mathbf{L}_{k}=\mathbf{Q}_{1,k}\cdot\left(\mathbf{Q}_{1,k}+\mathbf{Q}_{2,k} \right)^{-1} \tag{42}\]  
with the updated ellipsoid midpoint  
\[\bar{\mathbf{\mu}}_{k}=\mathbf{\mu}_{1,k}+\mathbf{L}_{k}\cdot\left(\mathbf{\mu}_{2,k}-\bm {\mu}_{1,k}\right) \tag{43}\]  
that is now used for the solution of the **Steps 2** and **3** listed above. For both ellipsoids \(i\in\{1,2\}\) to be intersected, scaling factors \(\xi_{i}>\) and \(\zeta_{i}>\) are determined which represent the minimum and maximum distances of the new midpoint from the ellipsoid surface according to  
\[\xi_{i,k}=1-\left\|\mathbf{\Delta}\mathbf{\mu}_{i,k}\right\|\quad\text{and}\quad\zeta _{i}=1+\left\|\mathbf{\Delta}\mathbf{\mu}_{i,k}\right\|\, \tag{44}\]  
with  
\[\mathbf{\Delta}\mathbf{\mu}_{i,k}=\mathbf{Q}_{i,k}^{-\frac{1}{2}}\cdot\left(\bar\mathbf{{\mu}}_{k}-\mathbf{\mu}_{i,k}\right) \tag{45}\]  
and \(\left\|\mathbf{\Delta}\mathbf{\mu}_{i,k}\right\|\) as the Euclidean norm of the vector-valued argument. In such a way, both ellipsoids to be intersected can be enclosed by thick ellipsoids  
\[\left(\mathcal{E}\right)_{i,k}=\left(\mathcal{E}\right)_{i,k}\left(\bar{\mathbf{ \mu}}_{k},\mathbf{\Gamma}_{i,k},\left[\xi_{i,k}:\zeta_{i,k}\right]\right)\quad \text{with}\quad\mathbf{\Gamma}_{i,k}=\mathbf{Q}_{ijk}^{\frac{1}{2}}. \tag{46}\]The initial inner approximation of the shape matrix of the resulting intersection is given by  
\[\mathbf{Q}_{k}^{\mathrm{I}}=2\cdot\left(2\cdot\left(\left(\xi_{1,k}^{2}\cdot \mathbf{Q}_{1,k}\right)^{-1}+\left(\xi_{2,k}^{2}\cdot\mathbf{Q}_{2,k}\right)^{-1 }\right)\right)^{-1} \tag{47}\]  
according to the previous subsection, while its outer counterpart results in  
\[\mathbf{Q}_{k}^{\mathrm{O}^{\prime}}=4\cdot\left(2\cdot\left(\left(\zeta_{1,k}^ {2}\cdot\mathbf{Q}_{1,k}\right)^{-1}+\left(\xi_{2,k}^{2}\cdot\mathbf{Q}_{2,k} \right)^{-1}\right)\right)^{-1}. \tag{48}\]  
Usually, these two shape matrices are not proportional to each other in an element-wise sense. Hence, the inner shape matrix \(\mathbf{Q}_{k}^{\mathrm{I}}\) is inflated in **Step 3** according to the following theorem so that it contains the outer ellipsoid with certainty. **Theorem 5** (Thick interval representation of the Dikin intersection of ellipsoids with different midpoints).