## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Mapping of Thick Ellipsoidal Domains via (Quasi-)Linear System Models
#### 3.1.1 Outer Bounds  
Given an ellipsoid  
\[\mathcal{E}_{k}=\left\{\mathbf{x}_{k}\in\mathbb{R}^{n}\ \big{|}\ \mathbf{x}_{k}^{T}\cdot \mathbf{Q}_{k}^{-1}\cdot\mathbf{x}_{k}\leq 1\right\} \tag{8}\]  
that is described by a positive definite shape matrix \(\mathbf{Q}_{k}=\mathbf{Q}_{k}^{T}>0\) and centered at the origin \(\mathbf{x}_{k}=\mathbf{0}\) of the state space, it is desired to enclose the exact (generally non-ellipsoidal) solution set of the mapping (7) in terms of a guaranteed outer ellipsoidal bound  
\[\mathcal{E}_{k+1}^{\mathrm{O}}=\left\{\mathbf{x}_{k+1}\in\mathbb{R}^{n}\ \Big{|}\ \frac{1}{\alpha_{ \mathrm{O},k+1}^{2}}\cdot\mathbf{x}_{k+1}^{T}\cdot\mathbf{Q}_{k+1}^{-1}\cdot \mathbf{x}_{k+1}\leq 1\right\} \tag{9}\]  
with degrees of freedom concerning the choice of the shape matrix \(\mathbf{Q}_{k+1}\) and its stretch parameter \(\alpha_{\mathrm{O},k+1}>0\). The latter will be computed by means of a simple line-search optimization procedure to avoid the necessity for an online solution of linear matrix inequalities. With the help of the free matrix variable \(\mathbf{\mathcal{R}}_{k}\), the shape matrix of the predicted ellipsoid is described by  
\[\mathbf{Q}_{k+1}=\mathbf{Q}_{k+1}^{T}=\tilde{\mathbf{A}}_{k}\cdot\mathbf{\mathcal{ R}}_{k}\cdot\tilde{\mathbf{A}}_{k}^{T}>0\,\ \ \mathbf{\mathcal{R}}_{k}=\mathbf{\mathcal{R}}_{k}^{T}>0\, \tag{10}\]  
where \(\tilde{\mathbf{A}}_{k}\) is some invertible point matrix typically satisfying  
\[\tilde{\mathbf{A}}_{k}\in\left[\mathbf{A}_{k}\right]. \tag{11}\]  
This definition is inherited from the propagation of a representative point in the interior of \(\mathcal{E}_{k}\) which was chosen to be its midpoint in [20; 24]. **Remark 1**. : _Possible choices for the parameterization of the matrices \(\mathbf{\mathcal{R}}_{k}\) and \(\tilde{\mathbf{A}}_{k}\) introduced in (10) and (11) are discussed in the illustrating example in Section 3.1.3._  
Assuming regularity of \(\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\mathbf{Q}_{k}\cdot\mathbf{A}^ {T}(\mathbf{x}_{k},\mathbf{p}_{k})\) for all \(\mathbf{x}_{k}\in\mathcal{E}_{k},\mathbf{p}_{k}\in\left[\mathbf{p}_{k}\right]\), the exact solution of the mapping (7) is defined by  
\[\mathcal{E}_{k+1}^{*}= \left\{\mathbf{x}_{k+1}\in\mathbb{R}^{n}\ \big{|}\ \mathbf{x}_{k+1}^{T}\cdot\left(\mathbf{A}_{k}\cdot\mathbf{Q}_{k} \cdot\mathbf{A}_{k}^{T}\right)^{-1}\cdot\mathbf{x}_{k+1}\leq 1\,\right. \tag{12}\] \[\left.\text{where}\ \mathbf{A}_{k}=\mathbf{A}(\mathbf{x}_{k}, \mathbf{p}_{k})\ \ \text{with}\ \ \mathbf{x}_{k}\in\mathcal{E}_{k}\,\ \mathbf{p}_{k}\in\left[\mathbf{p}_{k}\right]\right\}\,\]  
where \(\mathcal{E}_{k+1}^{*}\subseteq\mathcal{E}_{k+1}^{\mathrm{O}}\) needs to be satisfied for (9) to represent a guaranteed outer bound of the solution domain. Note, even if this regularity assumption does not hold, the following Theorem 1 remains true. **Theorem 1** (Guaranteed outer bound). : _The ellipsoid \(\mathcal{E}_{k+1}^{\mathrm{O}}\) according to Equation (9) is a guaranteed outer bound for \(\mathcal{E}_{k+1}^{*}\) if the matrix inequality_  
\[\mathbf{\mathcal{M}}_{k+1}:= \begin{bmatrix}-\mathbf{Q}_{k}^{-1}&\mathbf{A}^{T}( \mathbf{x}_{k},\mathbf{p}_{k})\cdot\bar{\mathbf{A}}_{k}^{-T}\\ \bar{\mathbf{A}}_{k}^{-1}\cdot\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})&- \alpha_{\mathrm{O}k+1}^{2}\mathbf{\mathcal{R}}_{k}\end{bmatrix} \leq 0 \tag{13}\]  
_holds for all \(\mathbf{x}_{k}\in\mathcal{E}_{k}\) and \(\mathbf{p}_{k}\in[\mathbf{p}_{k}]\)._  
Proof.