## 1 Introduction  
Predictor-corrector approaches for the model-based estimation of state variables and disturbances were presented in several previous research activities [1, 2]. These approaches rely on predicting state enclosures for sets of ordinary differential equations either in an interval-based form [3, 4], perform Taylor series expansions with respect to time as well as uncertain parameters and initial conditions [5], or employ Taylor model-based approaches [6, 7]. However, these techniques are not directly capable of representing inner and outer bounds of the estimated domain in a unified manner as they were determined by means of affine sets [8], combinations of affine arithmetic with automatic differentiation [9], and data-driven overapproximation techniques for a reachability analysis [10] in current references. On the one hand, the predictor-corrector methods mentioned above provide guaranteed outer bounds of all state variables that are consistent with the state equations as well as with measured output information. Both, state equations and output models, the latter representing sensor characteristics with bounded noise, may include interval parameters to reflect limited knowledge and the influence of model simplifications. On the other hand, however, the lack of information concerning domains that belong to the exact sets of reachable states makes it impossible to distinguish between the phenomena of wide state enclosures as a consequence of uncertainty or the result of wide bounds due to pessimism in numerical evaluations. The latter aspect is well-known in the frame of interval analysis and denoted as _overestimation_ that can be traced back to the so-called dependency effect (numerous variables are treated as independent despite underlying physical or mathematical correlations) or to the wrapping effect (complex-shaped solution sets are replaced conservatively by more simple outer bounds which are subsequently propagated further) [11, 12, 13, 14]. From this description, it becomes obvious that it is desirable to express uncertainty on the bounds of the domains of reachable states in the frame of simulation and state estimation techniques. Recently, corresponding set representations were developed which allow to express either a notion of variables certainly belonging to a solution set or certainly not belonging to the set (denoted as a _relative distance measure interval arithmetic_[15]); alternatively, the notion of thick intervals, thick boxes, thick functions, and thick ellipsoids can be seen as an approach to handle this kind of uncertainty on set boundaries [16]. Moreover, combinations of fuzzy and interval methods as those presented in [17] represent other currently investigated techniques that aim at expressing uncertainty on state boundaries. Especially the use of ellipsoids as enclosures for reachable sets [18; 19] is a promising approach because such domains naturally arise as descriptions of provable stability domains for nonlinear dynamic systems if the corresponding analysis is performed with the help of quadratic Lyapunov functions. Moreover, propagating ellipsoids over a dynamic system model with sufficient smoothness properties typically leads again to a solution set with boundaries that can effectively be described by enclosing ellipsoids if the domains of interest are sufficiently small. For a proof of this property, see [20], where it has been shown that the Hausdorff distance between the true solution and the ellipsoidal enclosure reduces at least linearly for decreasing domain sizes. Computations of ellipsoidal enclosures inherently have the property of a constant complexity if they are applied recursively. In addition, their enclosure quality enhances automatically for reducing domain sizes which is in contrast to the propagation of polytopic and zonotopic domains [21; 22].