## 2 Thick Ellipsoids  
Thick ellipsoids were introduced by the authors in [20; 24] to represent conservative outer bounds of the domains of reachable states of dynamic systems in parallel with inner bounds which represent those states that belong to the reachable solutions with absolute certainty. Geometrically, a thick ellipsoid can be understood as an ellipsoid with an uncertain boundary as illustrated in the left-hand side of Figure 1 that allows for enclosing a complexly shaped domain \(\mathcal{A}_{k}\) according to  
\[\mathcal{E}_{k}^{\mathrm{I}}\subseteq\mathcal{A}_{k}\subseteq\mathcal{E}_{k}^ {\mathrm{O}}\ . \tag{1}\]  
The previous publications [20; 24] focused on propagating this ellipsoidal domain via a nonlinear map  
\[\mathbf{x}_{k+1}=\mathbf{f}(\mathbf{x}_{k}) \tag{2}\]  
so that the true outputs \(\mathcal{A}_{k+1}\) after performing a state prediction according to (2) are again enclosed by a thick ellipsoid in a computationally cheap manner. For that reason, linear matrix inequalities are only employed in offline phases for the derivation of the proposed solution technique. Our procedure avoids the search for the globally optimal (i.e., tightest possible) solution of state prediction (and also the subsequent innovation stages) by a replacement of the online use of linear matrix inequalities or complex minmax optimization tasks by low-dimensional optimizations. Classically, the task of finding ellipsoidal state enclosures would be solved by using the aforementioned, often time-consuming techniques in each time step if the approaches published in [19; 30; 31; 32] were applied directly. However, for the goal of a reduction of the computational effort and, hence, for a simplification of real-time implementability, we propose low-dimensional optimization procedures which avoid the search for the typically large number of degrees of freedom if the optimal inner and outer ellipsoids \(\mathcal{E}_{k}^{\mathrm{I}}\) and \(\mathcal{E}_{k}^{\mathrm{O}}\), respectively, were computed. In the following, we recapitulate the mathematical definitions of thick ellipsoids together with binary operators and function extensions. Note, similar concepts were also introduced in [16] for so-called thick intervals (which represent scalar interval variables with uncertain lower and upper bounds) as well as thick boxes as the Cartesian product of thick intervals. A closely related concept is the so-called type-2 interval arithmetic discussed in detail in [15]. **Definition 1** (Thick ellipsoid). : _Define a thick ellipsoid \((\mathcal{E})=\big{(}\mathcal{E}\big{)}\Big{(}\boldsymbol{u},\boldsymbol{ \Gamma},\Big{[}\varrho\ ;\ \overline{\varrho}\Big{]}\Big{)}\) as a subset of the power set \(\mathcal{P}(\mathbb{R}^{n})\) so that_  
\[(\mathcal{E})=\Big{\{}\mathcal{A}\in\mathcal{P}(\mathbb{R}^{n})\ \big{|}\ \mathcal{E}^{ \mathrm{I}}\subseteq\mathcal{A}\subseteq\mathcal{E}^{\mathrm{O}}\Big{\}} \tag{3}\]  
Figure 1: Definition of a thick ellipsoid \((\mathcal{E})_{k}\) enclosing the domain \(\mathcal{A}_{k}\) and its mapping via the system model (2). _with_  
\[ \begin{split}\mathcal{E}^{\text{I}}&=\left\{\mathbf{x} \in\mathbb{R}^{n}\ \big{|}\ (\mathbf{x}-\mathbf{\mu})^{T}\Big{(}\underline{\rho}\mathbf{\Gamma} \Big{)}^{-T}\Big{(}\underline{\rho}\mathbf{\Gamma}\Big{)}^{-1}(\mathbf{x}-\mathbf{\mu} )\leq 1\right\},\\ \mathcal{E}^{\text{O}}&=\left\{\mathbf{x}\in\mathbb{R }^{n}\ \big{|}\ (\mathbf{x}-\mathbf{\mu})^{T}(\overline{\rho}\mathbf{\Gamma})^{-T}( \overline{\rho}\mathbf{\Gamma})^{-1}(\mathbf{x}-\mathbf{\mu})\leq 1\right\}\end{split}  \tag{4}\]  
_and \(0\leq\underline{\rho}\leq\overline{\rho}\)._  
**Definition 2**. : _(Thick ellipsoid binary operators and function extensions). A thick ellipsoid extension of the binary operators \(\diamond\in\{+,-,\,,/\,,/\,,\cup,\cap\}\) (For the case of division, the value zero is assumed not to belong to the denominator expression in analogy to classical interval arithmetic [11]) satisfies the relation_  
\[