non-measurable states on the basis of the available information and provide them to a controller. Before an observer can be developed, it is necessary to verify the observability of the system. In the general case of nonlinear systems, however, this is not a binary decision. It is also possible that a subset \(\mathcal{B}\subset D_{x}\subseteq\mathbb{R}^{n}\) of the states is indistinguishable. Consequently, a nonlinear system can be observable in some areas and not observable in others. Furthermore, the observability of nonlinear systems can be classified into different categorizes such as global, local, weakly observable and others, see [1] for further information. Checking observability of nonlinear systems is an already well-researched domain, yet, had a limited applicability. Nonlinear systems, however, are not inhibited by any limitations that constrain linear systems regarding the precise description of the system dynamics. Nevertheless, the determination whether nonlinear systems are observable has always been a challenging task. In [2] for example, a nonlinear disturbance observer (DOB) is presented. The DOB is used to estimate the disturbances by only utilizing the measurements of the output. The internal uncertainties, external disturbances, parameter uncertainties as well as the modeled dynamics are brought together as a single disturbed term and estimated by the DOB. In order to determine whether this is possible, the nonlinear system needs to be observable. In the case of nonlinear systems [3; 4] (based on [1]) already introduced a possibility to prove local observability using Lie algebra. The set of control Lie algebra consisting of the vector field at position \(x(t)\) is examined. If a subset can be found whose elements multiplied by the Lie derivatives result in zero, then this subset is termed by the authors as unobservability Lie algebra. If the dimension of this subset equals \(N\), a sufficient condition for local observability at point \(x(t)\) can be stemmed. That is, if the subset is equal to the isotropic algebra of the original set. The work of [PERSON] [5] investigates further how algebraic methods can be used to demonstrate global observability. Therein, systems with polynomial functions are considered. All states that are not distinguishable from the vector \(x(t)\) are considered a ring. The variety of the ideal of this ring is used to prove the global observability of polynomial systems. The investigation of [PERSON] and [PERSON] [6] also deals with the observability of polynomial systems. The main difference is that [PERSON] and [PERSON] extend the method for non-autonomous systems and find a more general condition. Another example can be found in [7]. The authors examine the effect on observability if the initial conditions \(x(0)=x_{0}\) have a probability distribution. This is called ensemble observability problem [8]. The approach here is to determine the trajectories of the initial distribution to reconstruct the probability distribution of the initial states \(x_{0}\). Other works merely reconstruct the associated trajectories. In order to prove observability, the observability function must be injective. An algorithm for determining the injectivity of a function is described in [9; 10]. The method is based on interval arithmetics and determines the injectivity by examining whether the intervals intersect in the mapping for a given interval. This is done iteratively until the width of the intervals is smaller than the width of a comparison interval due to bisection, the hull of all intervals is formed. Afterwards, the resulting hull of intervals is substituted into the Jacobian and the rank determined. The investigated matrix is the interval Jacobian multiplied by the pseudo-inverse midpoint matrix to obtain a square matrix if the interval Jacobian has the dimension \((n\times p)\) with \(n\leq p\). The suggested methods to check the rank are to test whether the matrix is strictly diagonal dominant or to identify the rank by using the determinant.