## 2 Materials and Methods
### Algorithm for Studying the Observability of Nonlinear Systems
#### 2.3.4 Applying the Local Condition  
After calculating the Lie derivatives \(q^{I}\left(x_{a}^{I}\right)\) and \(q^{I}\left(x_{b}^{I}\right)\), condition (10) is checked. If the algorithm is in any iteration, the original list \(\mathcal{L}_{i}\) is not empty. In the previous iterations it could already be ensured that \(x_{i}^{I}\) is indistinguishable from all intervals in the list. Therefore, \(x_{a}^{I}\) and \(x_{b}^{I}\) only need to be checked against the intervals located in the list. This results in lists \(\mathcal{L}_{a}\) and \(\mathcal{L}_{b}\). The hope here is that the bisection will reduce the overestimation, whereby areas can be distinguished from \(D_{x}\), which were previously indistinguishable. As already explained, the condition (10) alone cannot be used to numerically prove the observability of nonlinear systems. It is therefore necessary to refer to the local conditions. First it is necessary to form the hull of all intervals of the list \(\mathcal{L}_{j}\) with the interval \(x_{j}^{I}\) for \(j=a,b\). To check the local condition (12), the interval observability matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) must be calculated. This requires the Jacobian. By using the automatic differentiation provided in the program library _Eigen_, the value of the matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) can be determined without having to analytically set up the Jacobian. The rank is then checked. If the matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) has full rank, it can be proven that the states enclosed by the corresponding interval \(x_{j}^{I}\) are globally observable on \(D_{x}\). However, if the full rank of the matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) is not detectable for the currently considered hull \(\tilde{x}_{j}^{I}\), the corresponding interval \(x_{j}^{I}\) cannot be discarded yet. The interval observability matrix is a matrix of dimension \(\left(n\ \times\ \kappa p\right)\), where \(n\) represents the number of system states, \(\kappa\) the number of Lie derivatives and \(p\) the dimension of the output. To observe all states, \(\kappa\) is selected so that \(\kappa p\geq n\) applies. Consequently, the maximum rank of the interval observability matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) is equal to \(n\). The exact rank of the matrix is not of importance for the local condition (12), only whether the rank is full. A suitable method for checking the rank is to examine the estimated eigenvalues. For this purpose, the method from [PERSON] from [25] offers a possibility to enclose these eigenvalues as sharply as possible using an interval. The precondition is that the interval matrix is a square matrix. Accordingly, the quadratic interval matrix is defined with the dimension \(\left(\left(n+\kappa p\right)\times\left(n+\kappa p\right)\right)\) as  
\[V^{I}\coloneqq\left( \begin{array}{cc}0&Q^{I}\left(\tilde{x}_{j}^{I}\right) \\ Q^{I}\left(\tilde{x}_{j}^{I}\right)^{\text{T}}&0\end{array} \right).\]  
Using [PERSON]'s theorem from [25], the interval eigenvalues of the matrix \(V^{I}\) can be enclosed. The second theorem of [PERSON] from [25] states that the eigenvalues of an interval matrix are enclosed by  
\[\lambda_{i}^{I}\left(V^{I}\right)\subseteq\left[\lambda_{i}\left(\text{mid} \left(V^{I}\right)\right)-\rho\left(\text{rad}\left(V^{I}\right)\right), \lambda_{i}\left(\text{mid}\left(V^{I}\right)\right)+\rho\left(\text{rad} \left(V^{I}\right)\right)\right].\]Here, \(\rho\) is the spectral radius of the matrix \(\text{rad}\left(V^{I}\right)\), whose radius is determined for all entries according to Equation (4). The singular values of matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\) are related to the eigenvalues of matrix \(V^{I}\). Therefore, it is sufficient to verify that \(2n\) eigenvalues of matrix \(V^{I}\) are non-zero to demonstrate the full rank of the interval observability matrix \(Q^{I}\left(\tilde{x}_{j}^{I}\right)\).