## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Dikin Ellipsoids for the Intersection of Ellipsoids
#### 3.2.3 Illustrating Example  
In this subsection, various examples for the intersection of ellipsoids and the result interpretation by means of the thick ellipsoids \(\left(\mathcal{E}\right)_{\mathrm{D},k}\) and \(\left(\mathcal{E}\right)_{\mathrm{D},k}^{*}\), respectively, are summarized. Figure 2(a) deals with the application of the intersection procedure of Section 3.2.1 for two ellipsoids identically centered at the origin with the shape matrices  
\[\mathbf{Q}_{1,k}= \begin{bmatrix}1&0\\ 0&8\end{bmatrix} \quad\text{and}\quad\mathbf{Q}_{2,k}= \begin{bmatrix}4&-5\\ -5&8\end{bmatrix} . \tag{55}\]  
Here, both enclosures \(\left(\mathcal{E}\right)_{\mathrm{D},k}\) and \(\left(\mathcal{E}\right)_{\mathrm{D},k}^{*}\) are identical. Moreover, the fact that the ellipsoid midpoints are identical results in inner and outer enclosures \(\mathcal{E}_{\mathrm{D},k}^{\mathrm{I}}\) and \(\mathcal{E}_{\mathrm{D},k}^{\mathrm{O}}\) that are quite close to the optimal ellipsoidal result representation. To visualize the procedure of Section 3.2.2 for the case of two different midpoints  
\[\mu_{1,k}= \begin{bmatrix}0&0\end{bmatrix} ^{T}\quad\text{and}\quad\mu_{2,k}= \begin{bmatrix}1&2\end{bmatrix} ^{T} \tag{56}\]  
either with the shape matrices (55) or with the new shape matrices  
\[\mathbf{Q}_{1,k}= \begin{bmatrix}1&0\\ 0&8\end{bmatrix} \quad\text{and}\quad\mathbf{Q}_{2,k}=4\cdot \begin{bmatrix}4&-5 \\ -5&8\end{bmatrix} \, \tag{57}\]  
the results in Figure 2(c),d are presented. These results confirm the observation described in Remark 3.2 that the use of Equation (49) leads to tighter inner bounds, while Equation (53) possesses the better outer bounds. Depending on the application to detect states belonging to the solution set with certainty or proving in a guaranteed way that specific state domains are not reachable in the frame of a safety or feasibility test, either the first or the second option should be preferred. However, the observed larger distance between the resulting inner and outer bounds in comparison with Figure 2(a) gives rise to the assumption that further improvements of the solution could be possible. This could either be done by means of the procedures in [49] or by intersecting the newly obtained outer bounds of both \(\left(\mathcal{E}\right)_{\mathrm{D},k}\) and \(\left(\mathcal{E}\right)_{\mathrm{D},k}^{*}\). Note, also the choice of the common ellipsoid midpoint represents a degree of freedom that can be investigated in future work together with possibilities to intersect the enlarged dashed-dotted bounds (which correspond to the outer ellipsoid surfaces of (46)) in a recursive manner with both \(\left(\mathcal{E}\right)_{\mathrm{D},k}\) and \(\left(\mathcal{E}\right)_{\mathrm{D},k}^{*}\). Interestingly, the proposed intersection procedures can also be applied to the case where one of the original ellipsoids becomes degenerate. In practice, this is the case for a pure measurement of a single state variable. This case is considered in Figure 4, where in all subplots the matrices \(\mathbf{Q}_{1,k}\) from Equations (55) and (57) were replaced with \(\mathbf{Q}_{1,k}^{-1}= \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \), corresponding to \(\left|x_{1,k}\right|\leq 1\). Due to the fact that the basic Formula (40) for computing Dikin ellipsoids only makes use of the inverted shape matrices, the solution procedure remains applicable for the degenerate case. This is a crucial requirement to use this intersection approach in cases in which only a subset to the state variables appears in the system's output equation when considering state estimation procedures according to the following two sections. In Figure 4b,c, the new ellipsoid midpoint \(\tilde{\mu}_{k}\) results from the following adapted Kalman gain  
\[\mathbf{L}_{k}=\mathbf{Q}_{2,k}\cdot \begin{bmatrix}1\\ 0\end{bmatrix} \cdot\left( \begin{bmatrix}1&0\end{bmatrix} \cdot\mathbf{Q}_{2,k} \cdot \begin{bmatrix}1\\ 0\end{bmatrix} +\sigma_{1,k}^{2}\right)^{-1},\ \ \text{where}\ \ \mathbf{Q}_{1,k}^{-1}= \begin{bmatrix} \sigma_{1,k}^{-2}&0\\ 0&0\end{bmatrix}