# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 2 Autonomous DDboats
### Modeling of the DDboats
#### 2.3.3 A Posteriori Reconstruction of Input Disturbances  
To reconstruct the disturbance signals \(\hat{z}_{1}^{(i)}(t)\) and \(\hat{z}_{2}^{(i)}(t)\) for both alternative input representations described in the preceding two subsections, the state observer  
\[\hat{\mathbf{x}}^{(i)}(t) = \begin{bmatrix}\cos\left(\hat{\phi}^{(i)}(t)\right)\cdot\hat{ \sigma}^{(i)}(t)\\ \sin\left(\hat{\phi}^{(i)}(t)\right)\cdot\hat{\sigma}^{(i)}(t)\\ u_{v}^{(i)}(t)+\hat{z}_{1}^{(i)}(t)\\ u_{\phi}^{(i)}(t)+\hat{z}_{2}^{(i)}(t)\\ 0\end{bmatrix} +\mathbf{H}(t)\cdot\left(\mathbf{y}_{\text{m}}^{(i)}(t)-\mathbf{ C}\hat{\mathbf{x}}^{(i)}(t)\right) \tag{16}\] \[= \begin{bmatrix}0&0&\cos\left(\hat{\phi}^{(i)}(t)\right)&0&0&0\\ 0&0&\sin\left(\hat{\phi}^{(i)}(t)\right)&0&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\end{bmatrix} \cdot\hat{\mathbf{x}}(t)+ \begin{bmatrix}0\\ u_{v}^{(i)}(t)\\ u_{\phi}^{(i)}(t)\\ 0\\ 0\end{bmatrix} +\mathbf{H}(t)\cdot\left(\mathbf{y}_{\text{m}}^{(i)}(t)-\mathbf{ C}\hat{\mathbf{x}}^{(i)}(t)\right)\]  
with the output matrix \[\mathbf{C}= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\end{bmatrix}  \tag{17}\]  
and the measurement vector \(\mathbf{y}_{\mathbf{m}}^{(i)}(t)\) containing the boats' positions, their velocities, and their headings are employed. The corresponding output estimates are, hence, denoted by \(\mathbf{C}\hat{\mathbf{x}}^{(i)}(t)\). The computation of the gain matrix \(\mathbf{H}\) exploits the duality principle between control and observer design (which is motivated by the quasi-static extended Kalman filter design according to [5]) in terms of solving the algebraic Riccati equation  
\[\mathbf{P}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)}\cdot\mathbf{C}^{T}\cdot \mathbf{R}^{-1}\cdot\mathbf{C}\cdot\mathbf{P}\Big{(}\hat{\mathbf{x}}^{(i)} \Big{)}-\mathbf{A}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)}\cdot\mathbf{P}\Big{(} \hat{\mathbf{x}}^{(i)}\Big{)}-\mathbf{P}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)} \cdot\mathbf{A}^{T}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)}-\mathbf{Q}=\mathbf{0} \tag{18}\]  
in each sampling instant \(t_{k}\) at which measured data \(\mathbf{y}_{\mathbf{m}}(t)\) are available. As a scheduling parameter, the most recent state estimate \(\hat{\mathbf{x}}\) is used in (18) as the argument of the state-dependent system matrix \(\mathbf{A}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)}\). Note that this continuous-time observer design is applicable due to the short temporal distance (\(T_{\mathrm{s}}=1\,\mathrm{s}\)) between two subsequent measurement instants and the comparably small variation in the state variables over a single discretization interval. The temporally varying observer gain  
\[\mathbf{H}(t)=\mathbf{P}\Big{(}\hat{\mathbf{x}}^{(i)}\Big{)}\cdot\mathbf{C}^{ T}\cdot\mathbf{R}^{-1} \tag{19}\]  
is recomputed in each sampling instant (due to the a posteriori application for the purpose of identifying process uncertainty, this recomputation does not impose any constraints concerning real-time applicability) for the heuristically chosen weighting matrices  
\[\mathbf{Q}=\text{diag}\{\left[200\quad 200\quad 100\quad 100\quad 10\right] \}\text{ and }\mathbf{R}=\text{diag}\{\left[0.1\quad 0.1\quad 0.1\quad 0.25\right]\}\,. \tag{20}\]  
For a stochastic interpretation of these matrices in terms of the covariances of process and measurement noise, see [14; 15]. With the help of this observer and the effective input estimates according to the preceding two subsections, the results depicted in Figure 4 are obtained. Table 1 summarizes the standard deviations for each of the components of the vector \(\mathbf{H}(t)\cdot\left(\mathbf{y}_{\mathbf{m}}^{(i)}(t)-\mathbf{C}\hat{ \mathbf{x}}^{(i)}(t)\right)\) over the complete horizon of the trajectory tracking experiment. These standard deviations are used in Section 4.1 to define interval bounds for an additive process noise acting independently on each component of the state equations.