# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 4 Modeling of Uncertainty
### Uncertainty in the Dynamic System Model: Bounded Process Noise  
In Equation (10), the control inputs of the DDboats under consideration have been described by \(u_{v}^{(i)}(t)+z_{1}^{(i)}(t)\) and \(u_{\phi}^{(i)}(t)+z_{2}^{(i)}(t)\) as a superposition of an identified or flatness-based input term and a corresponding point-valued disturbance estimate. With this knowledge, the ellipsoidal state estimation can be applied to a discrete-time system model (discretization step size \(T_{\mathrm{s}}=1\,\mathrm{s}\), corresponding to the sampling time of the available GPS sensor) with the state vector  
\[\mathbf{x}_{k}^{(i)}:=\mathbf{x}^{(i)}(t_{k})=\left[x_{1}^{(i)}(t_{k})\quad x _{2}^{(i)}(t_{k})\quad v^{(i)}(t_{k})\quad\phi^{(i)}(t_{k})\right]^{T},\;\;i\in \left\{\mathrm{A},\mathrm{B}\right\}\;. \tag{54}\]  
For the sake of a compact notation of the scheme for uncertainty modeling, the superscript \((\cdot)^{(i)}\) is omitted in the following equations as long as this does not lead to ambiguities. Using an explicit Euler discretization for the continuous-time state equations, which is sufficiently accurate due to the small sampling time \(T_{\mathrm{s}}\) and volume-preserving according to [5] for a [PERSON] car model, the system model  
\[\mathbf{x}(t_{k+1})=\underbrace{ \begin{bmatrix}1&0&T_{\mathrm{s}}\cos(\phi(t_ {k}))&0\\ 0&1&T_{\mathrm{s}}\sin(\phi(t_{k}))&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} }_{\mathbf{\Phi}(\mathbf{p})}\cdot\mathbf{x}(t_{k})+ \underbrace{T_{\mathrm{s}}\cdot \begin{bmatrix}0\\ 0\\ u_{\phi}(t_{k})+\hat{z}_{1}(t_{k})\\ u_{\phi}(t_{k})+\hat{z}_{2}(t_{k})\end{bmatrix} }_{\mathbf{\Psi}(\mathbf{p})}+ \mathbf{w}(t_{k}) \tag{55}\]  
is obtained. This model has the same form as the system model (21) that was used for the derivation of the ellipsoidal state estimation procedure in the previous section. To account for uncertainty in the process dynamics, the term \(\mathbf{\Psi}(\mathbf{p})\) needs to be bounded by an ellipsoid \(\mathcal{K}_{\mathbf{\Psi},k+1}(\boldsymbol{\mu}_{\mathbf{\Psi},k+1},\mathbf{ \Gamma}_{\mathbf{\Psi},k+1})\) according to the definition (40). For the application at hand, this term contains the influence of the control inputs \(u_{v}(t_{k})\) and \(u_{\phi}(t_{k})\) and the actuator uncertainties \(\hat{z}_{1}(t_{k})\) and \(\hat{z}_{2}(t_{k})\), as well as the disturbance estimates  
\[\mathbf{w}(t_{k})=T_{\mathrm{s}}\cdot\mathbf{C}\cdot\mathbf{H}(t_{k})\cdot( \mathbf{y}_{\mathrm{m}}(t_{k})-\mathbf{C}\hat{\mathbf{x}}(t_{k})) \tag{56}\]  
determined by the observer-based a posteriori uncertainty analysis according to Section 2.3.3. In this section, these quantities have been determined in a point-valued form with the estimated state vector \(\hat{\mathbf{x}}(t)\) of the observer (16), where the associated output matrix \(\mathbf{C}\) is defined in (17). To make the ellipsoidal estimation scheme applicable, the point-valued estimates are employed as the midpoint vector  
\[\boldsymbol{\mu}_{\mathbf{\Psi},k+1}=T_{\mathrm{s}}\cdot \begin{bmatrix}0\\ 0\\ u_{v}(t_{k})+\hat{z}_{1}(t_{k})\\ u_{\phi}(t_{k})+\hat{z}_{2}(t_{k})\end{bmatrix} +\mathbf{w}(t_{k}) \tag{57}\]  
of the ellipsoidal disturbance enclosure. The bounds for axis-aligned disturbance intervals are derived from the standard deviations listed in Table 1. If the parameter \(r>0\) in  
\[\gamma_{l}=r\cdot\sigma_{x_{l}}\;,\;l\in\left\{1,\ldots,4\right\}\;, \tag{58}\]is employed as a user-defined degree of freedom to specify a desired confidence level of the ellipsoidal disturbance representation, the factorized shape matrix of the process noise term \(\boldsymbol{\
abla}(\mathbf{p})\) results in the (in this example, time-invariant and purely diagonal) expression  
\[\boldsymbol{\Gamma}_{\boldsymbol{\
abla},k+1}=\sqrt{2}\cdot T_{\mathrm{s}}\cdot \mathrm{diag}\{\left[ \begin{array}{ccc}\gamma_{1}&\ldots&\gamma_{4}\end{array}  \right]\}\enspace, \tag{59}\]  
being equivalent to  
\[\mathbf{Q}_{\boldsymbol{\
abla},k+1}=2\cdot r^{2}\cdot T_{\mathrm{s}}^{2} \cdot\mathrm{diag}\{\left[