## 2 Materials and Methods
### Observability of Uncertain Nonlinear Systems
In case it is possible to prove that all initial states in \(D_{x}\) are distinguishable from \(x_{a}\), the initial state \(x_{a}\) is distinguishable in \(D_{x}\). In case it is even possible to prove this for all \(x_{0}\in D_{x}\), the nonlinear system is globally observable according to Definition 2. To verify condition (9) with the observability function (8), at least \(n\) Lie derivatives are necessary. However, for a general nonlinear system, the amount of necessary Lie derivatives can exceed the system dimension \(n\). Therefore, the amount of calculated Lie derivatives in (8) is \(\kappa\geq n\). Further information and an example are provided in [15]. There exists no theorem to determine the necessary number of Lie derivatives for a particular nonlinear system to prove distinguishability in advance. How to address this problem will be discussed in Section 2.3. For now, it is assumed that \(\kappa\) is known. However, verifying condition (9) for all initial states \(x_{0}\in D_{x}\subseteq\mathbb{R}^{n}\) is not possible using a general numerical algorithm. Using intervals facilitates this verification due to the fact that \(x_{0}\in x_{0}^{I}\) is not a single point in \(D_{x}\) but an interval. The interval includes an infinite number of points. If the set \(D_{x}\) is divided into a finite number of multidimensional intervals, verifying distinguishability of all initial states \(x_{0}\) is possible. To this, condition (9) needs to be adjusted to multidimensional intervals, that is  
\[q^{I}\left(x_{a}^{I}\right)\cap q^{I}\left(x_{b}^{I}\right)=\emptyset \tag{10}\]  
with the interval vectors \(x_{a}^{I},x_{b}^{I}\subset D_{x}\). These intervals are distinguishable if they do not intersect in at least one of the vector components. If condition (10) holds for every combination of intervals, all initial states are distinguishable. Hence, the nonlinear system is globally observable. However, interval arithmetics is accompanied by overestimation. A possibility to reduce this overestimation is the bisection of the intervals. Therefore, it could be reasonable to bisect the set \(D_{x}\) up to the point where the overestimation is sufficiently reduced to prove distinguishability. Note that system uncertainties \(\delta\) defined in Equation (3) need to be bisected, as well. Then, condition (10) only needs to be verified if  
\[\delta_{a}^{I}\cap\delta_{b}^{I}\
eq\emptyset.\]  
However, proving global observability solely by distinguishability using intervals is not possible. The reason is the consistent intersecting edge between two neighboring intervals that leads to an intersection in the mapping. Figure 1 illustrates this issue exemplarily. The observability function \(q(x)\) is calculated for the illustrated neighboring intervals \(x_{a}^{I}\) and \(x_{b}^{I}\). The black line demonstrates the intersection in the mapping. Overestimation in the evaluation of the observability function leads to the fact that the interval enclosures \(q^{I}\left(x_{a}^{I}\right)\) and \(q^{I}\left(x_{b}^{I}\right)\) overlap mutually. The shaded area is the intersection of the two results. In the figure it can be observed that this issue even arises without overestimation. Figure 1: Overestimation of adjacent intervals in the mapping area. Common edge (illustrated in black) leads to an overlapping (illustrated as black shaded area). Using condition (10) to check distinguishability facilitates the reduction of the investigation of global observability to a local problem. The justification is that only directly neighboring intervals are indistinguishable due to overestimation. A local formulation of observability for a nonlinear system is given by the following definition. To utilize this definition, the Jacobian  
\[Q\left(x\right)=\frac{\partial q(x)}{\partial x}\]  
of the observability function is introduced, which is called observability matrix. **Definition 3** (Local observability of autonomous systems for \(x_{0}\)).