# Stabilization of an Unconventional Large Airship When Hovering
## 3 Stabilization Strategy  
### Mathematical Tools  
In some applications, exogenous perturbation is known in advance. This is often the case for the airship. The MC500 is equipped with LIDAR sensors that can measure the force of a wind gust. From the control point of view, the purpose of using this information is to reduce or minimize the influence of the disturbance that affects the airship. To deal with the problem of minimizing the impact of a wind gust, we propose the use of feedforward controls. This control vector is built using techniques of differential geometry. The diagram of this method is described in Figure 3.  
The feedforward/feedback method controls a set of nonlinear systems which are as follows:  
\[\left\{\begin{array}{l}\dot{x}=f(x)+g(x)u+\sum\limits_{i=1}^{m}p_{i}d_{i}\\ y=h(x)\end{array}\right. \tag{7}\]  
Here, \(x\in R^{n}\) represents the state system, \(y\) is the output system, \(u\in R^{m}\) the input system, \(g(x)=\left[g_{1}(x),\cdots,g_{m}(x)\right]^{T}\), \(g_{i}(i=1,\cdots,m)\) is an m-dimensional vector field, \(f(x)\in R^{n}\) is a vector field, and \(h(x)=\left[h_{1}(x),\cdots,h_{m}(x)\right]^{T}\), \(h_{i}(i=1,\cdots,m)\) is a sufficiently smooth scalar function, \(d_{i}\) the disturbance inputs, and \(p_{i}\) are vectors detailed in Appendix A.  
**Definition 1**.: _If we assume that the vector fields \(h\) and \(f\) are sufficiently smooth, the Lie derivative of \(h\) with respect to \(f\) can be defined as:_  
\[L_{f}h=\sum\limits_{i=1}^{n}\frac{\partial h_{i}}{\partial x_{i}}f_{i}(x) \tag{8}\]  
**Definition 2**.: _In the system of Equations (7), if the two ensuing conditions are satisfied for all\(x\in R^{n}\) in the vicinity of an equilibrium point \(x_{e}\)[17]:_  
\[L_{g_{i}}\ L_{f}^{k}\ h_{i}(x)\ =\ 0 \tag{9}\]  
Figure 3: Block diagram of the control vector.  
The (m \(\times\) m) matrix of decoupling \(D\) is not singular at \(x=\mathrm{x_{e}}\), with:  
\[D(x)=\left(\begin{array}{cccc}L_{g_{1}}L_{f}^{r_{1}-1}h_{1}(x)&\dots&L_{g_{m}} L_{f}^{r_{1}-1}h_{1}(x)\\ L_{g_{1}}L_{f}^{r_{2}-1}h_{2}(x)&\dots&L_{g_{m}}L_{f}^{r_{2}-1}h_{2}(x)\\ \dots&\dots&\dots\\ L_{g_{1}}L_{f}^{r_{m}-1}h_{m}(x)&\dots&L_{g_{m}}L_{f}^{r_{m}-1}h_{m}(x)\\ \end{array}\right) \tag{10}\]  
Then, the system (7) is said to have a relative degree:  
\[r=\sum_{i=1}^{m}\ r_{i} \tag{11}\]  
The indices \(I,j,\) and \(k\) verify: \(1\leq j,\ i\leq m,\ 1\leq k\leq r_{i}-1.\ r_{i}\) are the relative degree of the outputs. Examples of the derivatives \(L_{gj}\ L_{f}^{k}\ h_{i}(x)\) are presented later in Section 3.2.1.  
**Definition 3**.: _The relative order \(\rho_{ji}\) of the output \(y_{i}\) of \(Y\) with respect to the disturbance input \(d_{i}\) is defined [19; 20] as the smallest integer for which: \(L_{p_{i}}\ L_{f}^{p_{i}-1}\ h_{j}\
eq 0\)_  
**Proposal 1**.: _The system of Equation (7) is feedback linearizable if it exists as a function \(h\in\mathbb{R}^{m}\) sufficiently smooth such that the system has the relative degree \(r=n=dim(x)\), \(x\) is the state._  
We will apply this result to demonstrate that the model of the airship MC500 subjected to a wind gust is linearizable. We can thus apply the input-output linearization method to define a control vector that anticipates the effect of a wind gust pre-detected by LIDAR.