## 5 Application Scenario: State and Disturbance Estimation for an Underactuated Hovercraft
### Simulation Results  
To illustrate the performance of the thick ellipsoid state estimation algorithm of Section 4, two different levels of uncertainty are compared for the sensor models (90). These are  
\[\delta_{i}^{2}=0.005 \tag{91}\]  
and  
\[\delta_{i}^{2}=0.0001. \tag{92}\]  
with the help of these parameters, bounded measurement noise (in the sense of scaled uniformly distributed random numbers) has been generated for the simulations presented in this section. Moreover, we assume true initial system states \(\mathbf{x}_{0}\) randomly chosen from the interior of an ellipsoid centered at  
\[\boldsymbol{\mu}_{0}^{*}= \begin{bmatrix}2&0.1&0.01&0.1&0&0\end{bmatrix} ^{T} \tag{93}\]  
with the diagonal shape matrix  
\[\mathbf{Q}_{0}^{*}=\mathrm{diag}\left\{ \begin{bmatrix}1&1&0.1&1&1\end{bmatrix}  \right\}. \tag{94}\]  
The hovercraft's external disturbances originate from setting \(d_{1}=-0.25\) and \(d_{3}=0.05\). The state estimation procedure is initialized with an ellipsoid (containing the true initial state) with the same shape matrix (94) and the shifted center \(\boldsymbol{\mu}_{0}=\boldsymbol{\mu}_{0}^{*}- \begin{bmatrix}0&0&0&0.1&0&0 \end{bmatrix} ^{T}\). Figures 10 and 11 display the evolution of the outer state enclosures for \(k=1000\) steps of the estimation algorithm from Figure 6 in the left columns. In the right columns of these figures, an enlarged view is given which compares outer and inner bounds of the thick ellipsoid after the first prediction step, the propagation of the prediction results in terms of outer bounds and the respective corrected ellipsoids after accounting for the measured data. In addition, the measured and true state values are depicted. Throughout this section, the intersection of the predicted ellipsoids with the degenerate information according to (90) is performed by the application of formula (53). The gap between the outer and inner enclosures for \(k=1\) indicates the non-negligible influence of nonlinearities for sufficiently uncertain values of \(x_{3,k}\). For the sake of forecasting the domains of reachable states, only the outer bounds are investigated further in Figures 12 and 13. It can be seen that the smaller measurement uncertainty (92) leads to reduced uncertainty in the state reconstruction. Most noticeable, however, is the fact that the observer becomes also applicable to reconstruct bounds for the disturbance \(d_{3,k}\) in this case which are tighter than their initialization and consistent with the system model as well as the uncertain state observations, cf. Figure 13d. The disturbance estimation was performed as described at the end of the previous subsection by intersecting the current estimate for the disturbance (after the correction step) with the previously obtained best disturbance bounds. For the case of temporally varying disturbances, this procedure has to be modified. Instead of intersecting the bounds from previous time steps, it would be necessary to account for worst-case variations of the disturbance during the prediction step by inflating the corresponding domains similarly to the influence of process noise in a Kalman filter implementation. For the time-invariant case, future work can also aim at disturbance reconstruction by accounting for the estimates over a window of multiple time steps \(k\) and to find a common solution for the resulting set of algebraic constraints which are obtained when evaluating the state Equations (89) in which both \(x_{i,k}\) and \(x_{i,k+1}\) are replaced by the outcomes of the respective correction steps. Figure 11: State estimation for the case of small uncertainties (92). (**a**) Outer ellipsoidal enclosures for the velocities \(x_{1,k}\) and \(x_{2,k}\). (**b**) Enlarged view of the ellipsoidal enclosures of \(x_{1,k}\) and \(x_{2,k}\). (**c**) Outer ellipsoidal enclosures for the velocities \(x_{1,k}\) and \(x_{3,k}\). (**d**) Enlarged view of the ellipsoidal enclosures of \(x_{1,k}\) and \(x_{3,k}\). Figure 10: State estimation for the case of large uncertainties (91).