# Aerial Grasping of an Object in the Strong Wind: Robust Control of an Aerial Manipulator
## 3 System Control
### Hex-rotor Control
#### 3.1.2 Hoo Controller  
The new system in Equation (14) becomes a linear system including the interconnection term and disturbance. For a system without disturbance, a linear controller can guarantee the asymptotic stability [29]. In this section, the design of a robust \(H\infty\) control to ensure the BIBO stability of the system with the interconnection term and disturbance is described. The robust \(H\infty\) controller was designed under the assumption that the external disturbance is bounded. Although several works show such an assumption is restrictive [31; 32], such an assumption can be taken in practice for an aerial manipulator system, because the manipulator always works as a fixed base manipulator in real aerial manipulation applications. So, we consider it as bound disturbance in this paper. There exist works in the literature [31; 33] that tackle uncertainty in a better way, which can be considered in the future. **Remark II**: In practice, the unmodeled part disturbance of the manipulator, \(\Delta_{F}\) and \(\Delta_{\tau}\), can be seen as bound disturbance in the working states. For the hex-rotor, the position and yaw angle are usually chosen as system outputs. We use \(p_{b,d}\) to denote the desired position. The system state error is denoted by \(e_{p_{b}}=p_{b}-p_{b,d}\), \(e_{\Phi_{b}}=v_{b}-v_{b,d}\), \(e_{\Phi_{b}}=\Phi_{b}-\Phi_{b,d}\) and \(e_{\ddot{\Phi}_{b}}=\dot{\Phi}_{b}-\dot{\Phi}_{b,d}\). Combining Equation (3) with Equation (14), the error dynamic of the system can be written as the following equation:  
\[\left\{ \begin{array}{l}\dot{x}=Ax+Bu+E\delta(e_{\Phi_{b}})+D\Delta\\ y=Cx\end{array} \right. \tag{15}\]  
where \(x=\left[ \begin{array}{c}e_{p_{b}}\\ e_{\Phi_{b}}\\ e_{\Phi_{b}}\\ e_{\dot{\Phi}_{b}}\end{array} \right]\) \(y=\left[ \begin{array}{c}e_{p_{b}}\\ e_{\dot{\psi}}\end{array} \right]\), \(A=\left[ \begin{array}{cccc}O_{3\times 3}&I_{3\times 3}&O_{3\times 3}&O_{3 \times 3}\\ O_{3\times 3}&O_{3\times 3}&O_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&O_{3\times 3}&O_{3\times 3}&I_{3\times 3}\\ O_{3\times 3}&O_{3\times 3}&O_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&I_{3\times 3}\end{array} \right]\) \(B=\left[ \begin{array}{cc}O_{3\times 3}&O_{3\times 3}\\ I_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&I_{3\times 3}\end{array} \right]\)  
\(C=\left[ \begin{array}{cccc}I_{3\times 3}&O_{3\times 3}&O_{3\times 2}&O_{3 \times 1}&O_{3\times 3}\\ O_{1\times 3}&O_{1\times 3}&O_{1\times 2}&I_{1\times 1}&O_{3\times 3} \end{array} \right]\), \(D=\left[ \begin{array}{cc}O_{3\times 3}&O_{3\times 3}\\ I_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&I_{3\times 3}\end{array} \right]\), \(E=\left[ \begin{array}{cc}O_{3\times 3}&O_{3\times 3}\\ I_{3\times 3}&O_{3\times 3}\\ O_{3\times 3}&I_{3\times 3}\end{array} \right]\), \(\Delta=\left[ \begin{array}{c}\frac{1}{m_{s}}\Delta_{F}\\ T(\Phi_{b})I_{b}^{-1}\Delta_{\tau}\end{array} \right]\), \(u=\left[ \begin{array}{c}v_{1}-\tilde{p}_{b,d}\\ v_{2}-\tilde{\Phi}_{b,d}\end{array} \right]\). For the interconnection term \(\delta(e_{\Phi_{b}})\), we have following theorem. **Theorem 1**. : _For \(\delta(e_{\Phi_{b}})\) in the system (15), there exists a constant \(\sigma\), such that_  
\[\left\|\delta(e_{\Phi_{b}})\right\|\leq\left\|Fx\right\| \tag{16}\]  
_where \(F=\left[ \begin{array}{cccc}O_{3\times 3}&O_{3\times 3}&\sigma I_{3\times 3}&O_{3 \times 3}\\ \end{array} \right]\)._The proof of Theorem 1 is given in the Appendix A. **Theorem 2. **_If a feedback controller \(u=Kx\) and a positive definite symmetric matrix \(P\) can be found satisfying the following inequality:_  
\[P(A+BK)+(A+BK)^{T}P+\frac{1}{\gamma^{2}}PDD^{T}P+\frac{1}{\lambda}PEE^{T}P+C^{T}C +\lambda F^{T}F\leq 0 \tag{17}\]  
_where, \(\lambda\) is a positive constant. Then, system in Equation (15) will be finite gain \(L_{2}\)-stable from disturbances \(\Delta\) to outputs \(y\) and the \(L_{2}\) gain is less than or equal to \(\gamma\)._  
**Proof.