# Stabilization of an Unconventional Large Airship When Hovering
## 3 Stabilization Strategy
### Scheme of the Control Vector
#### 3.2.4 The Linearizing Control Law  
According to Equation (22), we can deduce the linearizing control \(\alpha\). This control vector will ensure the decoupling and can be written as:  
\[\alpha=D^{-1}(\aleph-\aleph-F_{p_{1}},\Omega) \tag{23}\]Note that the linearization would only be possible if the decoupling matrix \(D\) is invertible. However, the decoupling matrix \(D\) given by:  
\[D=\left( \begin{array}{cc}\Pi&0\\ 0&\Gamma\end{array} \right) \tag{24}\]  
is invertible, and we have:  
\[D^{-1}=\left( \begin{array}{cc}\Pi^{-1}&0\\ 0&\Gamma^{-1}\end{array} \right) \tag{25}\]  
(For the expressions of \(\Pi\), \(\Gamma\), \(\Pi^{-1}\), and \(\Gamma^{-1}\) see Appendix B). Replacing Equation (23) in Equation (22), the equivalent system becomes linear and completely decoupled. It can be written as follows:  
\[\left( \begin{array}{c}\ddot{\xi}_{1}\\ \ddot{\xi}_{2}\\ \ddot{\xi}_{2}\\ \ddot{\xi}_{3}\\ \ddot{\xi}_{4}\\ \ddot{\xi}_{5}\\ \ddot{\xi}_{6}\end{array} \right)=\left( \begin{array}{c}\dot{z}_{2}^{1}\\ \dot{z}_{2}^{2}\\ \dot{z}_{2}^{3}\\ \dot{z}_{4}^{4}\\ \ddot{z}_{5}\\ \dot{z}_{2}^{2}\\ \dot{z}_{6}\end{array} \right)=\left( \begin{array}{c}\aleph_{1}\\ \aleph_{2}\\ \aleph_{3}\\ \aleph_{4}\\ \aleph_{5}\\ \aleph_{6}\end{array} \right) \tag{26}\]  
Finally, we obtain the following six decoupled linear sub-systems with their outputs:  
\[\left\{ \begin{array}{c}\dot{z}_{1}^{i}=z_{2}^{i}\\ \dot{z}_{2}^{i}=\aleph_{j}\\ \ddot{\xi}_{i}=z_{1}^{i}\end{array} \right. \tag{27}\]  
The vector \(\aleph\) is designed according to the control objectives. In general, for a tracking trajectory problem, the expressions of \(\aleph_{i}\) are:  
\[\aleph_{i}=\omega_{i}^{(r_{i})}+\sigma_{r_{i}-1}\Big{(}y_{i}^{(r_{i}-1)}- \omega_{i}^{(r_{i}-1)}\Big{)}+\ldots+\sigma_{1}(y_{i}-\omega_{i}) \tag{28}\]  
\(\omega_{i}^{\ (r_{i})}\) are reference trajectories. In our case these are functions in steps \(\Big{(}\dot{\omega}_{i}^{(r_{i})}=\dot{\omega}_{i}^{(r_{i})}=0\Big{)}\). \(\sigma_{r_{i}-1}\) are the gains. If the \(\sigma_{i}\) are chosen so that the polynomial \(s^{r_{i}}+\sigma_{r_{i}-1}s^{r_{i}-1}+\cdots\sigma_{2}.s+\sigma_{1}=0\), the polynomial is said to be a Hurwitz polynomial (possesses roots with negative real parts), then it can be shown that the error \(\dot{\xi}_{i}=\mathrm{y}_{i}-\Psi_{i}\) tends towards zero. We have chosen the following poles P = \(\{-10,\ -4\}\) for the system in Equation (28) to be stable, then the gains of the control \(\aleph_{i}\), which are solutions of the equation \(s^{2}+\sigma_{2}.s+\sigma_{1}=0\), are given by: \(\sigma_{1}\) = 40 and \(\sigma_{2}\) = 14. This values are needed in the numerical simulations to assess the performance of the proposed controller. Previous developments show that it is possible to anticipate the effect of a gust of wind on the airship. The feedforward control that we have just established is a key part of this objective. Indeed, this control ensures the accelerated attenuation of the effect of the wind disturbance when this is known in advance, and it gives the control vector a certain robustness with respect to any modeling errors. We have also demonstrated the following result:  
**Proposal. ** Consider the Multiple-Input Multiple-Output (MIMO) system of the airship MC500 described by Equation (12). We define \(\big{(}r_{i}\big{)}_{1\leq j\leq 6}\) as the relative degree of the outputs and \(\rho_{11}\) as the relative degree of the disturbance \(\mathrm{F}_{\mathrm{v1}}\) associated with the outputs \(\mathrm{y}_{1}=\mathrm{x}_{0}\). It is assumed that the decoupling matrix D is invertible, then the control vector U given in Equation (23):* anticipates the effect of a gust of wind. * ensures the stability of the airship in the vicinity of a target state. Note:  
1. We have restricted ourselves to a single excitation force along the X axis to limit the size of the equations. However, to take into account the effect of other force and torque components, the same steps described above would be followed. 2.