## 5 Application Scenario: State and Disturbance Estimation for an Underactuated Hovercraft
### Modeling  
According to Figure 9, denote the surge and sway velocities by \(x_{1}\) and \(x_{2}\), respectively, and the hovercraft's yaw rate by \(x_{3}\) in a body-fixed coordinate frame. For the following derivation of the equations of motion, assume that the translational velocities are summarized in the vector \(\mathbf{V}= \begin{bmatrix}x_{1}&x_{2}&0\end{bmatrix} ^{T}\), while the angular velocity vector \(\mathbf{W}\) with respect to the ship's center of gravity (COG) is given by \(\mathbf{W}= \begin{bmatrix}0&0&x_{3}\end{bmatrix} ^{T}\). In this formulation, the dynamics associated with the motion in heavy, roll, and pitch are neglected by setting the corresponding vector entries to zero. As also shown in [59; 60], a force and torque balance in all degrees of freedom yields the equations of motion  
\[ \begin{split}&\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{ \partial E}{\partial\mathbf{V}}\right)+\mathbf{W}\times\left(\frac{\partial E }{\partial\mathbf{V}}\right)=\mathbf{F}\quad\text{and}\\ &\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial E}{\partial \mathbf{W}}\right)+\mathbf{V}\times\left(\frac{\partial E}{\partial\mathbf{V}} \right)+\mathbf{W}\times\left(\frac{\partial E}{\partial\mathbf{W}}\right)= \mathbf{T}\enspace,\end{split}  \tag{84}\]  
where  
\[E=\frac{1}{2}\cdot\left(\mathbf{V}^{T}\cdot \begin{bmatrix}m_{11}&0&0\\ 0&m_{22}&0\\ 0&0&m_{33}\end{bmatrix} \cdot\mathbf{V}+\mathbf{W}^{T}\cdot \begin{bmatrix}J_{11 }&0&0\\ 0&J_{22}&0\\ 0&0&J_{33}\end{bmatrix} \cdot\mathbf{W}\right) \tag{85}\]  
is the total kinetic energy of the ship (under the assumption of purely diagonal mass and inertia matrices [60]) with the strictly non-negative entries \(m_{ii}\geq 0\) and \(J_{ii}\geq 0\), \(i\in\{1,2,3\}\). For a hovercraft, it can be assumed that \(m_{11}=m_{22}=m\) corresponds to the mass of the ship, without additional hydrodynamic effects, while \(J_{33}=J_{\mathrm{r}}\) is the mass moment of inertia of the vessel in the direction of \(x_{3}\). Defining further the vectors  
\[\mathbf{F}= \begin{bmatrix}u_{1}-cx_{1}-d_{1}\\ -cx_{2}-d_{2}\\ 0\end{bmatrix} \quad\text{and}\quad\mathbf{T}= \begin{bmatrix}0\\ 0\\ u_{2}-c_{\mathrm{r}}x_{3}-d_{3}\end{bmatrix}  \tag{86}\]  
Figure 9: Definition of the velocity components in a body-fixed coordinate systems, velocity-proportional damping as well as disturbance forces and torques. of non-conservative, external generalized forces and torques with the control inputs \(u_{1}\) and \(u_{2}\) and independent, velocity-proportional damping effects in each coordinate with the coefficient \(c>0\) for both translational degrees of freedom as well as \(c_{\rm r}>0\) for the rotary motion, the equations of motion (84) simplify to  
\[ \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\\ d_{1}\\ d_{2}\\ d_{3}\end{bmatrix} = \begin{bmatrix}-\frac{c}{m}&x_{3}&0&\frac{1}{m}&0&0\\ -x_{3}&-\frac{c}{m}&0&0&\frac{1}{m}&0\\ 0&0&-\frac{c}{m}&0&0&\frac{1}{m}\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix} \cdot \begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ d_{1}\\ d_{2}\\ d_{3}\end{bmatrix} + \begin{bmatrix}\frac{1}{m}&0\\ 0&0\\ 0&\frac{1}{7}\\ 0&0\\ 0&0\\ 0&0\end{bmatrix} \cdot \begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} \enspace, \tag{87}\]  
where \(d_{i}\), \(i\in\{1,2,3\}\), represent external disturbances according to Figure 9 due to wind and currents. These disturbances are included in the state equations (87) by means of integrator disturbance models. For the following simulation, a linear state feedback controller  
\[ \begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} =- \begin{bmatrix}k_{11}&k_{12}&k_{13}\\ k_{21}&k_{22}&k_{23}\end{bmatrix} \cdot \begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix} \enspace, \tag{88}\]  
corresponding to a feedback of the actual velocity, is parameterized by means of pole assignment to decelerate the hovercraft up to standstill at the desired operating point \(x_{1}=x_{2}=x_{3}=0\).