# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 2 Dynamic Modeling of Electric Tail Reduction System
The dynamic equation of the tail rotor is obtained as  
\[T_{r}=J_{t}\frac{1}{\lambda}\ddot{\theta}+T_{t} \tag{4}\]  
By the reduction ratio between the tail rotor and the motor, we have  
\[\dot{\theta}_{m}=\frac{1}{\lambda}\dot{\theta} \tag{5}\]  
\[\ddot{\theta}_{m}=\frac{1}{\lambda}\ddot{\theta} \tag{6}\]  
where \(\dot{\theta}_{m}\) and \(\ddot{\theta}_{m}\) are the angular velocity and acceleration of electric tail rotor of unmanned helicopter; \(T_{t}\) is the tail rotor load; and \(J_{t}\) is the rotational inertia of tail rotor. The tail rotor load depends on the torque generated by the main rotor in the helicopter coordinate system and the yaw demand of the whole machine. Therefore, we considered the torque of the main rotor. The relationship between the tail rotor load and main rotor power is [3]  
\[T_{t}=c_{ymp}^{x}(\frac{P_{M}}{\omega_{M}}+M_{f}) \tag{7}\]  
where \(M_{f}\) is the yaw moment for helicopter; constant \(c\) is the proportional factor of tail rotor thrust to tail rotor load; constant \(\xi_{ymp}\) is the maneuver reserve factor; and \(\omega_{M}\) is the main rotor speed. The main rotor power \(P_{M}\) of the helicopter consists of the following three parts by [37]  
\[P_{M}=P_{PM}+P_{iM}+P_{paraM} \tag{8}\]  
where \(P_{PM}\) is the profile power; \(P_{iM}\) is the induced power; and \(P_{paraM}\) is the parasite power. The specific expressions of the above three types of power are as follows:  
\[ \begin{array}{c}P_{PM}=\frac{1}{2}\rho_{M}\pi R_{M}^{2}(R_{M}\omega_{M})^{3} M_{p}\\ P_{iM}=F_{M}V_{iM}\\ P_{paraM}=DV\end{array}  \tag{9}\]  
where \(\rho_{M}\) is the air density; \(R_{M}\) is the main rotor radius; \(V\) is the helicopter horizontal forward flight speed. The specific expressions of profile power coefficient \(M_{p}\), rotor pull force \(F_{M}\), and parasite drag \(D\) are shown as follows:  
\[ \begin{array}{c}M_{p}=\frac{1}{4}K_{p}C_{xp}\sigma[1+0.75(1+5\mu_{m}^{2})a_ {1E}+4.7\mu_{m}^{2}]\\ F_{M}=\sqrt{G^{2}+D^{2}}\\ D=\frac{1}{2}\rho_{M}V^{2}C_{x}/g_{f}K_{df}\end{array}  \tag{10}\]  
where \(K_{p}\) is the blade shape factor; \(C_{xp}\) is the profile coefficient; \(\sigma\) is the rotor solidity; \(C_{xf}\) is the angle of attack resistance coefficient; \(S_{f}\) is the fuselge top view area; \(K_{df}\) is the parasite correction coefficient; \(C_{xp}=0.009+0.0067(\mu_{m}-0.3)\); and \(a_{1E}=0.35\mu_{m}\). The helicopter forward ratio is  
\[\mu_{m}=\frac{V}{R_{M}\omega_{M}} \tag{11}\]The helicopter gravity is  
\[G=mg \tag{12}\]  
The induced power is the power consumed by the helicopter to exhaust the air downward and generate the pulling force. It is closely related to the forward flying speed of the helicopter, and is the main part of the power required by the main rotor when the helicopter hovers or flies forward at a small speed. The induced speed of helicopter hovering can be obtained as  
\[V_{h}=\sqrt{\frac{G}{2\rho_{M}\pi R_{M}^{2}}} \tag{13}\]  
The induced speed of a helicopter at high-speed can be obtained as  
\[V_{g}=\frac{F_{M}}{2\rho_{M}\pi R_{M}^{2}V} \tag{14}\]  
In general, the induced velocity of a helicopter is expressed as a piecewise function of hover induced velocity and high-speed induced velocity. However, when hovering mode and high-speed mode are switched, the induced speed estimated by the above method is easy to produce mutation. Therefore, the following weighted estimation method is applied to calculate the induced velocity of high-speed flight:  
\[V_{iM}=V_{h}e^{\left(-\frac{FV}{V_{h}}\right)}+V_{g}(1-e^{\left(-\frac{FV}{V_{h }}\right)}) \tag{15}\]  
where \(K\) is correction factor. Based on Equations (7)-(15), the tail rotor load can be solved. In this paper, the surface mounted rotor structure based on symmetrical magnetic circuit was used (\(L_{q}=L_{d}\)).