## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Dikin Ellipsoids for the Intersection of Ellipsoids
#### 3.2.1 Intersection of Ellipsoids with Identical Midpoints  
Outer as well as inner ellipsoidal representations for the intersection of various ellipsoidal domains were investigated in numerous publications. Especially, the reference [49] gives a thorough overview of a large number of applicable techniques that can be cast into computationally feasible optimization problems. Basically, all of these routines are based on the use of linear matrix inequalities. At each intersection stage, such linear matrix inequalities need to be solved when computing so-called inner and outer bounds by means of a technique that is based on findings of [PERSON] and [PERSON] [50]. Analogously, this holds for the Nerimovski approximation [51] that is compared with the first option in [49].  
To avoid the necessity for solving linear matrix inequalities explicitly during online applications (note, these matrix inequalities are exploited for the offline proof of the validity of the following enclosure relations), a conservative approximation based on the so-called Dikin ellipsoids [49] is considered in this paper. These ellipsoids can be applied to describe inner and outer bounds for the intersection \(\mathcal{I}_{k}\) of \(m\) ellipsoids  
\[\mathcal{E}_{ijk}=\left\{\mathbf{x}_{k}\in\mathbb{R}^{n}\;\middle|\;\mathbf{x}_ {k}^{T}\cdot\mathbf{Q}_{ijk}^{-1}\cdot\mathbf{x}_{k}\leq 1\;,\;\;i\in\{1,\dots,m \}\right.\right\} \tag{37}\]  
with an identical center located in the origin of the state space. The following relations for the shape matrices hold equally for identical non-zero center points.  
Using the results of Lemma 5 in [49], the inner and outer bounds are given by  
\[\mathcal{E}_{\text{D,I}k}\subseteq\mathcal{I}_{k}\subseteq\mathcal{E}_{\text {D,O}k} \tag{38}\]  
with \(\mathcal{E}_{\text{D,I}k}=\mathcal{E}_{\text{D,}k}^{2}\) and \(\mathcal{E}_{\text{D,O}k}=\mathcal{E}_{\text{D,}k}^{2m}\), respectively, where  
\[\mathcal{E}_{\text{D,}k}^{r}=\left\{\mathbf{x}_{k}\in\mathbb{R}^{n}\;\middle| \;\mathbf{x}_{k}^{T}\cdot r^{-1}\cdot\mathbf{Q}_{\text{D,}k}^{-1}\cdot \mathbf{x}_{k}\leq 1\right\} \tag{39}\]  
is defined for the common shape matrix  
\[\mathbf{Q}_{\text{D,}k}^{-1}=2\sum_{i=1}^{m}\mathbf{Q}_{ijk}^{-1}\;\;. \tag{40}\]  
This definition of inner and outer ellipsoidal enclosures corresponds to a thick ellipsoid introduced in the Definitions 1 and 2 in the form  
\[(\mathcal{E})_{\text{D,}k}=(\mathcal{E})_{\text{D,}k}\Big{(}\mathbf{0},\mathbf{ \Gamma}_{\text{D,}k},\left[\sqrt{2}\;;\;\sqrt{2m}\right]\Big{)}\quad\text{ with}\quad\mathbf{\Gamma}_{\text{D,}k}=\mathbf{Q}_{\text{D,}k}^{\frac{1}{2}}\;\;. \tag{41}\]  
Although these enclosures are generally suboptimal, we restrict ourselves to these Dikin ellipsoids due to their computational inexpensive evaluation which is advantageous if real-time state estimation procedures are desired. A generalization to the case of ellipsoids with different midpoints is presented in the following subsection.  
Efficient techniques that allow for checking whether the intersection of two ellipsoids is empty were published in [52; 53]. This check is especially interesting if the thick ellipsoid state observer technique derived in the following section is applied to tasks in the area of fault detection. There, the case that a predicted state domain does not intersect with those domains that are compatible with sensor data typically indicates failures of system components, actuators, or sensors. This corresponds to the fact that the investigated (nominal) system model is proven to be infeasible.  
A further related technique, however, not only relying on ellipsoidal state enclosures was very recently derived in [54]. There, outer bounds for the domains of reachable states are characterized by ellipsoids, while disturbances are bounded by zonotopes that may become degenerate if one or more coordinates are unbounded. This property of degeneracy will be picked up again in Section 3.2.3, where an example is presented in which only partial state measurements are available in the thick ellipsoid framework.