## 4 Controller Design  
### Design of ESO  
Before the succeeding steps, model simplification and transformation should be performed to facilitate the model-based controller design. Due to the high rotation rate of the main rotor, the main rotor flapping dynamics are much faster than the fuselage attitude dynamics. Thus, the quasi-static assumption of flapping dynamics is made here [39], the Equation (4) could be simplified to (9).  
\[B_{1w}=-K_{B}^{-1}(K_{\omega}\omega_{12w}+K_{\theta}\theta_{1w}+\zeta\theta_{0}) \tag{9}\]  
Then the helicopter model could be transformed to the simplified reference model in (10).  
\[\left\{\begin{array}{l}\dot{\Theta}=S^{-1}(\Theta)\omega\\ \dot{\omega}=-J^{-1}(\omega\times I\omega)+\overline{Q}\\ \overline{Q}=\left[\overline{Q}_{x},\overline{Q}_{y},\overline{Q}_{z}\right] ^{T}=f+R\theta\end{array}\right. \tag{10}\]  
Additionally, the variables mentioned above can be written specifically in (11).  
\[\left\{\begin{array}{l}f=\left[f_{1},f_{2},f_{3}\right]^{T}=\left[f_{12}^{T},f_{3}\right]^{T}\\ f_{12}=\left[f_{1},f_{2}\right]^{T}=\int_{0}^{-1}(-Q_{A12}H^{-1}(\beta)K_{B}^{ -1}(K_{\omega}H(\beta)\omega_{12}+\zeta\theta_{0})+\left[T_{i}h_{t},T_{m}I_{m} \right]^{T})\\ f_{3}=(-\rho\Omega_{t}^{2}R_{A}^{2}A_{t\infty}\sigma\lambda_{t1}/8-T_{m}l_{m }\beta_{18}-Q_{m})/l_{zz}\\ \theta=\left[\theta_{1c},\theta_{1s},\theta_{1}\right]^{T}=\left[\theta_{1}^{ T},\theta_{t}\right]^{T}\\ R=\left[\begin{array}{cc}R_{12}&0\\ 0&0&R_{3}\end{array}\right]\\ R_{12}=-\int_{0}^{-1}Q_{A12}K_{B}^{-1}K_{\theta},R_{3}=\rho\Omega_{t}^{2}R_{A }^{2}A_{t\infty}\sigma l_{t}/12l_{zz}\\ Q_{A12}=\left[\begin{array}{cc}Q_{m}&T_{m}h_{m}\\ T_{m}h_{m}&-Q_{m}\end{array}\right],J_{0}=\left[\begin{array}{cc}I_{xx}&0 \\ 0&I_{yy}\end{array}\right]\end{array}\right. \tag{11}\]  
As nonlinearities and disturbances are assembled in one term, they can be compensated for by the controller with no difference. To acquire the disturbance information, which could not be measured directly by the sensor, an ESO is designed, where the structure is shown in (12).  
\[\left\{\begin{array}{l}\dot{z}_{1}=-J^{-1}(\omega\times J\omega)+z_{2}+ \left[\left(R_{12}\theta_{1}\right)^{T},R_{3}\theta_{t}\right]^{T}-B_{01}e\\ \dot{z}_{2}=-B_{02}fal(e,0.5,\delta)\end{array}\right. \tag{12}\]  
where \(z_{1}=\left[z_{p},z_{q},z_{q}\right]^{T}\), \(z_{2}=\left[z_{f1},z_{f2},z_{f3}\right]^{T}\), \(y=\left[p,q,r\right]^{T}\) and \(e=\left[e_{1},e_{2},e_{3}\right]^{T}=z_{1}-y\). \(B_{0x}\) is the parameter matrix of the observer. \(fal=\left[fal_{1},fal_{2},fal_{3}\right]^{T}\) and \(fal_{x}\) are defined by (13).  
\[fal_{x}(e,a,\delta)=\left\{\begin{array}{ll}e_{x}/\delta^{a-1}&|e_{x}|\leq \delta\\ |e_{x}|^{a}sign(e_{x})&|e_{x}|>\delta\end{array}\right. \tag{13}\]  
Due to the characteristics of nonlinear feedback and extended state, ESO could achieve accurate and agile observation of states and disturbances with little reliance on the system model [40]. Thus, it could be seen that the ESO presented above could track the total disturbances accurately as shown in (14).  
\[z_{1}\to y,z_{2}\to f \tag{14}\]  
Then, the observed total disturbances \(f\) could be treated as part of the model in the control system.