## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Mapping of Thick Ellipsoidal Domains via (Quasi-)Linear System Models
#### 3.1.1 Outer Bounds
: Without any assumptions on the regularity of \(\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\mathbf{Q}_{k}\cdot\mathbf{A}^{ T}(\mathbf{x}_{k},\mathbf{p}_{k})\), the relation \(\mathcal{E}_{k+1}^{*}\subseteq\mathcal{E}_{k+1}^{\mathrm{O}}\) is satisfied, if mapping the domain \(\mathcal{E}_{k+1}^{\mathrm{O}}\) one time step back leads to a domain \(\mathcal{S}_{k}\) that is a superset of the ellipsoid \(\mathcal{E}_{k}\) according to \(\mathcal{S}_{k}\supseteq\mathcal{E}_{k}\) with  
\[\mathcal{S}_{k}= \Big{\{}\mathbf{x}_{k}\in\mathbb{R}^{n}\;\big{|}\;\mathbf{x}_{k} ^{T}\cdot\mathbf{A}^{T}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\frac{1}{a_{\mathrm{ O},k+1}^{2}}\cdot\Big{(}\bar{\mathbf{A}}_{k}\cdot\mathbf{\mathcal{R}}_{k}\cdot\bar{ \mathbf{A}}_{k}^{T}\Big{)}^{-1}\cdot\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k}) \cdot\mathbf{x}_{k}\leq 1\, \tag{14}\] \[\qquad\text{where}\;\;\mathbf{x}_{k}\in\mathcal{E}_{k}\,\;\mathbf{p}_{k}\in[ \mathbf{p}_{k}]\Big{\}}\.\]  
Geometrically speaking, this means that the level curve of height 1 of the quadratic form in (8), which corresponds to the bound of the ellipsoid \(\mathcal{E}_{k}\), must be an inner bound for the level curve of identical height of \(\mathcal{S}_{k}\). This condition corresponds to the scalar inequality  
\[\mathbf{x}_{k}^{T}\cdot\mathbf{A}^{T}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot \frac{1}{a_{\mathrm{O},k+1}^{2}}\cdot\Big{(}\bar{\mathbf{A}}_{k}\cdot\mathbf{ \mathcal{R}}_{k}\cdot\bar{\mathbf{A}}_{k}^{T}\Big{)}^{-1}\cdot\mathbf{A}( \mathbf{x}_{k},\mathbf{p}_{k})\cdot\mathbf{x}_{k}\leq\mathbf{x}_{k}^{T}\cdot \mathbf{Q}_{k}^{-1}\cdot\mathbf{x}_{k} \tag{15}\]  
for all \(\mathbf{x}_{k}\in\mathcal{E}_{k}\) and \(\mathbf{p}_{k}\in[\mathbf{p}_{k}]\). Factoring out the vector \(\mathbf{x}_{k}\) to the left and right of (15) turns this scalar inequality into the matrix inequality  
\[\mathbf{A}^{T}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\frac{1}{a_{\mathrm{O},k+1}^ {2}}\cdot\Big{(}\bar{\mathbf{A}}_{k}\cdot\mathbf{\mathcal{R}}_{k}\cdot\bar{ \mathbf{A}}_{k}^{T}\Big{)}^{-1}\cdot\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})- \mathbf{Q}_{k}^{-1}\leq 0\ \, \tag{16}\]  
which (for the choice of a regular matrix \(\bar{\mathbf{A}}_{k}\)) is equivalent to the condition  
\[\mathbf{A}^{T}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\bar{\mathbf{A}}_{k}^{-T} \cdot\Big{(}a_{\mathrm{O},k+1}^{2}\mathbf{\mathcal{R}}_{k}\Big{)}^{-1}\cdot\bar{ \mathbf{A}}_{k}^{-1}\cdot\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})-\mathbf{Q}_ {k}^{-1}\leq 0\ . \tag{17}\]  
Removing the difference of the two matrices in (17) with the help of the Schur complement formula [36; 37] leads to Equation (13) which completes the proof of Theorem 1. **Remark 2**. : _As stated before, the quasi-linear system matrix \(\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\) is bounded by an element-wise defined interval matrix \(\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\in[\mathbf{A}_{k}]=\big{[}\underline {\mathbf{A}}_{k}\ ;\ \overline{\mathbf{A}}_{k}\big{]}\). Therefore, a conservative solution to the matrix inequality in Theorem 1 can be determined by searching for a state- and parameter-independent, strictly positive definite matrix \(a_{\mathrm{O},k+1}^{2}\mathbf{\mathcal{R}}_{k}\) that satisfies the interval-valued matrix inequality_  
\[\mathbf{\mathcal{M}}_{k+1}\in[\mathbf{\mathcal{M}}_{k+1}]= \begin{bmatrix}-\mathbf{Q}_{k }^{-1}&[\mathbf{A}_{k}]^{T}\cdot\bar{\mathbf{A}}_{k}^{-T}\\ \bar{\mathbf{A}}_{k}^{-1}\cdot[\mathbf{A}_{k}]&-a_{\mathrm{O},k+1}^{2}\mathbf{ \mathcal{R}}_{k}\end{bmatrix} \leq 0\ . \tag{18}\]  
_This matrix inequality can be cast into a linear one by the substitution_  
\[\mathbf{\mathcal{P}}_{k+1}^{\mathrm{O}}=\Big{(}\mathbf{\mathcal{P}}_{k+1}^{\mathrm{O}} \Big{)}^{T}=a_{\mathrm{O},k+1}^{2}\mathbf{\mathcal{R}}_{k}>0 \tag{19}\]  
_to make standard routines for linear matrix inequalities such as the Matlab toolbox SeDuMi in combination with Yalmip[38; 39] applicable if (18) is further replaced by the convex combination of a finite number of extremal realizations with the help of the ideas given in [40; 41].