# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 1 Introduction  
Model uncertainty and external disturbances are omnipresent when tasks such as trajectory tracking control or the localization of vessels in marine applications are considered [1, 2]. This equally holds for surface vessels operated on rivers or the open sea in the areas of passenger and goods transportation, as well as for autonomous (underwater) robots performing tasks such as the inspection and maintenance of pipelines or other offshore infrastructure, such as wind turbines. In this paper, we restrict ourselves to the case of surface vessels. For those, uncertain GPS-based position measurements are used to update predicted vessel locations that can be obtained from the evaluation of a ship's state equation. To reflect real-life working conditions, these state equations need to include uncertainty in the system model; for example, imperfectly known actuator characteristics on the one hand and external disturbances due to wind and water currents on the other hand. In combination, the measurement-based localization, an observer-based disturbance reconstruction, and the model-based state prediction allow for forecasting those domains that are reachable by a vessel in a certain time span. In such a way, it becomes possible to verify whether specific maneuvers are guaranteed to be safe, i.e., whether collisions with other vessels or obstacles can be ruled out with certainty. Such guaranteed statements can be made with the help of a set-valued calculus [3, 4, 5, 6, 7, 8, 9]. A suitable representation of uncertainty for a set-valued evaluation of a dynamic system model is the use of ellipsoidal state enclosures. In comparison with axis-aligned interval boxes, ellipsoidal domains have the advantage of being mapped exactly onto ellipsoids during the state prediction if thesystem dynamics are fully linear. For nonlinear models, as well as for the intersection of predicted state domains with measurements of selected state vector components, which are subject to bounded uncertainty, the ellipsoidal estimation framework provides guaranteed outer bounds of those domains that are compatible with the system model and all available data. In addition to providing guaranteed outer bounds, inner state enclosures (describing guaranteed reachable domains) can also be computed by a so-called _thick ellipsoid_ approach to quantify the effects of uncertainties and nonlinearities, which unavoidably lead to overestimation when computing guaranteed outer state enclosures [4; 5]. Moreover, ellipsoidal state enclosures have the advantage over classical interval-based representations that correlations between different components of the state vector are representable by using a single set without the necessity for subdivision approaches or advanced changes in coordinates, leading to affine arithmetic-like set representations. Note, furthermore, that other set-valued representations, such as Taylor models [10; 11], polytopes [12], or zonotopes [13], are commonly more demanding from a computational point of view due to the necessity for strategies that limit the involved polynomial degrees and the number of zonotope vertices. For that reason, such approaches are not further considered in this paper. However, the interested reader is referred to [12], where it was shown that the ellipsoidal technique has a similar performance as the use of polygons (which are optimal for linear systems represented by differential inclusions) with respect to the tightness of the state enclosures if uncertainties in the system dynamics and the available measurements are sufficiently small and measured data are available with a sufficiently high frequency. As a counterpart to set-valued estimation approaches, stochastic filtering techniques [14; 15] are widely used in the frame of localization and disturbance estimation [16]. These approaches rely on the specification of probability density functions for the state vectors, as well as for all uncertain quantities and external disturbances.