## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Mapping of Thick Ellipsoidal Domains via (Quasi-)Linear System Models
#### 3.1.1 Outer Bounds
The extremal realizations can be obtained by computing an interval extension of the state- and parameterdependent system matrix \(\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\) with the help of toolboxes such as IntLab[42] for Matlab and determining all matrix entries that have a non-zero diameter. Then, it is possible to represent (18) by the convex polytope_  
\[\mathcal{D}=\left\{\mathbf{\mathcal{M}}_{k+1}\Big{|}\mathbf{\mathcal{M}}_{k+1}(\xi)= \sum_{v=1}^{n_{v}}\xi_{v}\cdot\mathbf{\mathcal{M}}_{v,k+1}\ ;\ \ \sum_{v=1}^{n_{v}}\xi_{v}=1\ ;\ \ \xi_{v}\geq 0\right\}\, \tag{20}\]  
_which is only used in the following example for the sake of comparison with the proposed method, where for the list of all \(n_{v}=2^{n^{2}}\) interval vertices formed of \(\mathbf{\underline{\Lambda}}_{k}\) and \(\mathbf{\overline{\Lambda}}_{k}\) the corresponding extremal realizations_  
\[\mathbf{\mathcal{M}}_{v,k+1}= \begin{bmatrix}-\mathbf{Q}_{k}^{-1}&\mathbf{A}_{v,k}^ {T}\cdot\tilde{\mathbf{A}}_{k}^{-T}\\ \tilde{\mathbf{A}}_{k}^{-1}\cdot\mathbf{A}_{v,k}&-\mathbf{P}_{k+1}^{\mathsf{O}} \end{bmatrix} \,\ \ v\in\{1,\ldots,n_{v}\}\, \tag{21}\]  
_are defined and need to be negative semi-definite with \(\mathbf{\mathcal{M}}_{v,k+1}\leq 0\). Note, this formulation of all (21) directly accounts for the structural symmetry of (18). A (nearly) optimal solution can be obtained by minimizing \(\operatorname{trace}\left\{\mathbf{\mathcal{P}}_{k+1}^{\mathsf{O}}\right\}\), cf. [43; 44], which can be employed as a substitute for the optimization task involving the logarithm of the shape matrix determinant according to ([45], Appendix C)._  
To avoid the necessity to solve linear matrix inequalities in each temporal discretization step of a dynamic system, the following Theorem 2 describes a relaxed version of the solution procedure after a-priori fixing the matrix \(\mathbf{\mathcal{R}}_{k}\), for example, by setting \(\mathbf{\mathcal{R}}_{k}=\mathbf{Q}_{k}\). **Theorem 2** (Eigenvalue criterion for outer ellipsoidal bounds). : _For a fixed combination of the shape and stretch parameters \(\mathbf{\mathcal{R}}_{k}\) and \(\alpha_{\mathsf{O},k+1}\), respectively, \(\mathcal{E}_{k+1}^{\mathsf{O}}\) is a guaranteed outer enclosure of the solution set \(\mathcal{E}_{k+1}^{*}\), if_  
\[\lambda_{i}(\operatorname{mid}([\mathbf{\mathcal{M}}_{k+1}]))+\rho(\operatorname{ rad}([\mathbf{\mathcal{M}}_{k+1}]))\leq 0 \tag{22}\]  
_holds for all \(i\in\{1,\ldots,2n\}\), where \(\lambda_{i}(\operatorname{mid}([\mathbf{\mathcal{M}}_{k+1}]))\) is the \(i\)-th eigenvalue of \(\frac{1}{2}\cdot(\mathbf{\underline{\mathcal{M}}}_{k+1}+\overline{\mathbf{\mathcal{M}} }_{k+1})\) and \(\rho(\operatorname{rad}([\mathbf{\mathcal{M}}_{k+1}]))\) the spectral radius of \(\frac{1}{2}\cdot(\mathbf{\overline{\mathcal{M}}}_{k+1}-\underline{\mathbf{\mathcal{M} }}_{k+1})\)._  
Proof. : Due to the structural symmetry of \([\mathbf{\mathcal{M}}_{k+1}]\), all possible eigenvalues \(\lambda_{i}(\mathbf{\mathcal{M}}_{k+1})\) of \(\mathbf{\mathcal{M}}_{k+1}\in[\mathbf{\mathcal{M}}_{k+1}]\) are guaranteed to be real-valued. According to [PERSON] [46; 47], interval bounds for the range of each eigenvalue are given by the enclosures  
\[\lambda_{i}(\mathbf{\mathcal{M}}_{k+1})\in[\lambda_{i,\mathbf{\mathcal{M}}}]=\lambda_ {i}(\operatorname{mid}([\mathbf{\mathcal{M}}_{k+1}]))+[-1\ ;\ 1]\cdot\rho( \operatorname{rad}([\mathbf{\mathcal{M}}_{k+1}])) \tag{23}\]  
located on the real axis of the complex plane. If \(\sup([\lambda_{i,\mathbf{\mathcal{M}}}])\leq 0\) holds for all \(i\in\{1,\ldots,2n\}\), Theorem 1 is satisfied due to the negative semi-definiteness of the interval matrix (18). **Remark 3**.