## 2 Materials and Methods
### Algorithm for Studying the Observability of Nonlinear Systems
#### 2.3.2 Calculation of Lie Derivatives for Proving Distinguishability Using the Power Series  
As described above, Lie derivatives are required to prove global observability. In contrast to the classic derivation, the Lie derivative of the form \(L_{f}h\left(x\right)\) is the derivative of the function \(h\left(x\right)\) along \(f\left(x\right)\). Therefore, the number of Lie derivatives required to prove observability cannot be determined a priori. If an algorithm is used to determine and evaluate the Lie derivatives, it is necessary to establish a sufficient number of Lie derivatives in advance. This is hardly possible, since it can happen that more Lie derivatives are required than have been implemented. Thus, the determination of the observability can fail. Therefore, we reveal how an automatic determination of Lie derivatives can be performed in the following. An alternative to the Lie derivatives is the use of the power series. The basic calculation rules are first described. For this, the base object is defined as  
\[a\left(t\right)\coloneqq\sum_{k=0}^{\infty}\alpha_{k}t^{k}. \tag{14}\]  
Given two power series \(a(t)\) and \(b(t)\) according to definition (14), the calculation rules  
\[\left(a+b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\alpha_{k}+b_{k}\right)t^{k}, \tag{15}\] \[\left(a-b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\alpha_{k}-b_{k}\right)t^{k},\] (16) \[\left(a\cdot b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\sum_{l=0}^{k}a_{l}\cdot\mathbf{b}_{k-l }\right)t^{k},\] (17) \[\left(a/b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\frac{\alpha_{k}}{b_{0}}-\frac{1}{b_{0} }\sum_{l=0}^{k-1}\frac{\alpha_{0}}{b_{0}}b_{k-l}\right)t^{k}\]  
follow. The corresponding power series expressions can also be created for other mathematical functions, such as trigonometric functions. For simplicity, these are not described. To combine the power series with the Lie derivatives, the state \(x\left(t\right)\) is described as a power series  
\[x\left(t\right)=\sum_{k=0}^{\infty}\kappa_{k}t^{k}\]according to definition (14). Thus, it follows for the state equations  
\[f\left(x\left(t\right)\right)=\sum_{k=0}^{\infty}\hat{\eta}_{k}t^{k}. \tag{18}\]  
For the ordinary differential equation of the nonlinear system, it follows that  
\[\hat{x}\left(t\right)=\sum_{k=0}^{\infty}\hat{\eta}_{k+1}\left(k+1\right)t^{k}. \tag{19}\]  
Inserting (19) in (18) yields  
\[\hat{\tau}_{k+1}=\frac{1}{k+1}\hat{\eta}_{k}, \tag{20}\]  
with \(\hat{\eta}_{k}=f\left(\hat{\tau}_{0},\ \hat{\tau}_{1},\ \ldots,\hat{\tau}_{k}\right)\) and \(x\left(0\right)=\hat{\tau}_{0}\). Furthermore, if the output equation (2) is formulated by the power series  
\[y\left(t\right)=\sum_{k=0}^{\infty}\eta_{k}t^{k}, \tag{21}\]  
the Taylor series coefficients of equation (6) can be determined. Therefore, the coefficients \(\eta_{k}=\eta_{k}\) with \(\mathfrak{h}_{k}=h\left(\hat{\tau}_{0},\ \hat{\tau}_{1},\ \ldots,\ \hat{\tau}_{k}\right)\) are required. Consequently, the definition (21) is used together with the Taylor series from equation (6) and  
\[\left(a+b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\alpha_{k}+b_{k}\right)t^{k}, \tag{15}\]0  
holds for each of the coefficients. Inserting \(\mathfrak{\eta}_{k}=\mathfrak{\eta}_{k}\) into (22) yields the Lie derivatives  
\[\left(a+b\right)\left(t\right) =\sum_{k=0}^{\infty}\left(\alpha_{k}+b_{k}\right)t^{k}, \tag{15}\]1  
Thus, it is possible to determine the value of any Lie derivative without computing its analytic expression. This procedure can be implemented as an algorithm. Additionally, both automatic differentiation and interval arithmetic can be applied. Consequently, it becomes possible to calculate the solution set of Lie derivatives over a whole range of the values of \(x_{0}\). Simultaneously, the values of the Jacobian can also be determined by using automatic differentiation. Note that the calculation of the Lie derivatives as well as the interval observability matrix (11) is performed accordingly even if the nonlinear system has uncertainties.