## 2 Materials and Methods
### Observability of Uncertain Nonlinear Systems  
In order to determine the observability of a nonlinear system (1) with the corresponding output function (2), first a definition of global observability is needed. In our case we use the distinguishability to determine wherever the mapping of the trajectories of states can be distinguished. **Definition 1** (Distinguishability). : _Assume \(T>0\). Two states \(z^{1},z^{2}\in D_{x}\) are called indistinguishable on the interval \([0,T]\), if their output courses are identical, that is_  
\[\forall t\in[0,T]:\ \ h\left(\varphi\left(t,z^{1}\right)\right)=h\left( \varphi\left(t,z^{2}\right)\right). \tag{5}\]  
_Otherwise, the states are called distinguishable [1; 19]._  
Thereby, the flux \(\varphi(t,x)\) of the vector field \(f\) on the output function \(h\) can be formulated as the convergent Taylor series  
\[y(t)\ =\ h\left(\varphi\left(t,x\right)\right)\ =\ \sum_{k=0}^{\infty}\frac{t^{k}}{k!}L _{f}^{k}h(x). \tag{6}\]  
Definition 1 is used to check wherever two states are indistinguishable. To obtain a statement about the entire nonlinear system, the following definition is formulated. **Definition 2** (Global observability using distinguishability). : _A system defined on \(D_{x}\subseteq\mathbb{R}^{n}\) is globally observable if and only if all pairs \(z^{1}\) and \(z^{2}\) are distinguishable, that is if the sets_  
\[\mathcal{M} = \left\{\left( \begin{array}{c}z^{1}\\ z^{2}\end{array} \right)\in\mathbb{R}^{2n}\Bigg{|}h\left(\varphi\left(t,z^{1} \right)\right)=h\left(\varphi\left(t,z^{2}\right)\right)\Bigg{\}}\,,\] \[\mathcal{D} = \left\{\left( \begin{array}{c}z^{1}\\ z^{2}\end{array} \right)\in\mathbb{R}^{2n}\Bigg{|}z^{1}=z^{2}\right\}\]  
_are equal, hence \(\mathcal{M}=\mathcal{D}\)[5]._  
This definition is used to verify the observability of the nonlinear system on a given state space \(D_{x}\). For verifying the condition (5), the Taylor series (6) is utilized. The Taylor series (6) contains the Lie derivatives \(L_{f}h(x)\) and is therefore called Lie series. Using this series changes the mapping  
\[x\mapsto h\left(\varphi\left(t,x\right)\right)\]  
from the function space \(\mathbb{R}^{n}\) to a mapping  
\[x\mapsto\left(L_{f}^{k}h\left(x\right)\right)_{k=0}^{\infty} \tag{7}\]  
to the sequence space \(\mathbb{R}^{\infty}\). The domain \(D_{x}\), which is mentioned in Definitions 1 and 2, is a subset of the vector space \(\mathbb{R}^{n}\). Thus, the dimension of the image of the mapping (7) cannot exceed \(n\) topologically [19]. Consequently, for parameterizing the image of the mapping (7), the maximum number of required components \(L_{f}^{k}h\) is \(n\). The Lie derivatives  
\[L_{f}^{k}h\left(x\right)=\frac{\partial}{\partial x}\left(L_{f}^{k-1}h\left(x \right)\right)\ \cdot f\left(x\right)\]  
with \(k=1,\ldots\kappa\) in which \(\kappa\in\mathbb{N}\) and  
\[L_{f}^{0}h\left(x\right)=h\left(x\right)\]  
from the convergent Taylor series (6) are used to generate the vector  
\[q\left(x\right)=\left[ \begin{array}{c}L_{f}^{0}h\left(x\right)\\ L_{f}^{1}h\left(x\right)\\ \vdots\\ L_{f}^{\kappa-1}h\left(x\right)\end{array} \right]. \tag{8}\]  
The vector \(q\left(x\right)\) is called observability function. With  
\[z=\left[ \begin{array}{c}y\\ \dot{y}\\ \vdots\\ y^{\left(\kappa-1\right)}\end{array} \right]\]  
the correlation  
\[z=q\left(x\right)\]  
holds, which is used to verify Definition 1 and therefore Definition 2. Note that the nonlinear system (1) and the output function (2) are both real-analytic functions, therefore, it is possible to compute as many Lie derivatives as necessary. The vector \(q(x)\) is used to verify the distinguishability of states according to Definition 1. For this purpose, two random initial states \(x_{a},x_{b}\in D_{x}\) are considered. The initial states \(x_{a}\) and \(x_{b}\) are distinguishable if the condition  
\[q(x_{a})\
eq q(x_{b}) \tag{9}\]  
holds for \(x_{a}\
eq x_{b}\). That means, at least one of the Lie derivatives differs regarding the different initial states.