# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 3 Ellipsoidal State Estimation Procedure  
For the description of the ellipsoidal state estimation procedure, assume that a discrete-time system model in the form  
\[\mathbf{x}_{k+1}=\boldsymbol{\Phi}(\mathbf{p})\cdot\mathbf{x}_{k}+\boldsymbol{ \psi}(\mathbf{p}) \tag{21}\]  
is given, where \(\boldsymbol{\Phi}(\mathbf{p})\) is the system matrix and \(\boldsymbol{\psi}(\mathbf{p})\) is a bounded disturbance or control input. Although discrete-time system models are considered in this paper, the term \(\boldsymbol{\psi}(\mathbf{p})\) can also be used to account for time discretization errors of continuous-time models. A corresponding approach has recently been investigated in [22]. Due to the fact that time discretization errors are much smaller for the application at hand in comparison with the process noise discussed in the previous section, the further derivation of the ellipsoidal state estimation approach is restricted solely to the discrete-time setting. Both \(\boldsymbol{\Phi}(\mathbf{p})\) and \(\boldsymbol{\psi}(\mathbf{p})\) may depend on uncertain parameters \(\mathbf{p}\in\mathbb{R}^{n_{\mathrm{P}}}\). These parameters may also reflect state dependencies in the system matrix and the input vector if a nonlinear set of state equations is reformulated according to [4] into a quasi-linear form, cf. (16). In both cases (either pure parameter dependencies and/or state dependencies), the vector \(\mathbf{p}= \begin{bmatrix}p_{1}&\dots&p_{n_{\mathrm{P}}}\end{bmatrix} ^{T}\) is assumed to be piecewise constant between two subsequent sampling instants and bounded in terms of a set-valued uncertainty model by the axis-aligned multi-dimensional interval box \(\mathbf{p}\in[\mathbf{p}]= \begin{bmatrix}\mathbf{p}\;;\overline{\mathbf{p}} \end{bmatrix} \), where the element-wise relation \(\underline{p}_{i}\leq p_{i}\leq\overline{p}_{i}\) holds for each component \(i\in\{1,\dots,n_{\mathrm{P}}\}\) of the parameter vector. Figure 4: Visualization of the effective input signals \(u_{v}^{\langle\mathrm{A}\rangle}(t)\) and \(u_{\phi}^{\langle\mathrm{A}\rangle}(t)\) for the DDboat A. (**a**) Reconstruction of the effective input signal \(u_{v}^{\langle\mathrm{A}\rangle}(t)\) without disturbance estimate. (**b**) Reconstruction of the effective input signal \(u_{v}^{\langle\mathrm{A}\rangle}(t)\) with additive disturbance estimate. (**c**) Reconstruction of the effective input signal \(u_{\phi}^{\langle\mathrm{A}\rangle}(t)\) without disturbance estimate. (**d**) Reconstruction of the effective input signal \(u_{\phi}^{\langle\mathrm{A}\rangle}(t)\) with additive disturbance estimate. The ellipsoidal state estimation procedure applied in this paper is obtained as a special case of the thick ellipsoid state estimation presented in [4]. To implement this procedure, we assume that the additive term \(\mathbf{\psi}(\mathbf{p})\) is included in a suitable ellipsoid that--in the case of an initial box-valued enclosure--is obtained by enclosing the additive bounded uncertainty in its associated Lowner-John ellipsoid [23]. In general, ellipsoidal state bounds at the time instant \(k\) are denoted as  
\[\mathcal{E}_{k}(\mathbf{\mu}_{k},\mathbf{\Gamma}_{k}):=\left\{\mathbf{x}_{k}\in\mathbb{R}^ {n}\ \big{|}\ (\mathbf{x}_{k}-\mathbf{\mu}_{k})^{T}\mathbf{\Gamma}_{k}^{-T}\mathbf{\Gamma}_{k}^{-1}(\mathbf{x}_{ k}-\mathbf{\mu}_{k})\leq 1\right\} \tag{22}\]  
for which the shape matrix is given in the factorized form \(\mathbf{\mathrm{Q}}_{k}=\mathbf{\Gamma}_{k}\mathbf{\Gamma}_{k}^{T}\succ 0\) with the positive definite matrix \(\mathbf{\mathrm{Q}}_{k}\). This matrix specifies--depending on the following estimation steps--state domains before the evaluation of a prediction step, or the corrected state estimates after the evaluation of the associated measurement update step. Moreover, denote the ellipsoid midpoint as \(\mathbf{\mu}_{k}\in\mathbb{R}^{n}\).