# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 2 Dynamic Modeling of Electric Tail Reduction System  
The electric tail reduction system of unmanned helicopter mainly includes the following parts: permanent magnet synchronous motor (PMSM), reducer, tail rotor, and speed sensor for measuring the tail rotor speed. The schematic diagram is shown in Figure 1. The input shaft of the reducer was directly connected with the outer rotor of the motor, and the output shaft was equipped with a tail rotor. The coupling was omitted and the number of components was reduced to achieve a compact structure, lightweight, and high power/weight ratio. The dynamic model of the electric tail reduction system was established according to the following methods. A PMSM uses permanent magnets embedded in the steel rotor to create a constant magnetic field. PMSMs are mainly controlled using direct torque control and field oriented control. [PERSON] expounded the electromechanical system for the first time and established the dynamic equation using the Lagrange method [36]. The Lagrange-Maxwell Equation in three-phase stator reference frame is expressed as  
\[ \begin{array}{c}\frac{d}{dt}(\psi_{abc})+R_{S}i_{abc}=U_{abc}\\ \int\ddot{\theta}+B\dot{\theta}+T_{L}=i^{T}_{abc}\frac{\partial L_{s}}{ \partial\dot{\theta}}i_{abc}+i^{T}_{abc}\frac{\partial\dot{\theta}_{f}}{ \partial\dot{\theta}}\end{array} \]  
where \(\psi_{abc}\) is the flux vector of three-phase stator shafting; \(R_{s}\) is the resistance; \(i_{abc}\) is the three-phase current vector of motor stator; \(U_{abc}\) is the three-phase voltage vector of motor stator; \(j\) is the rotational inertia of motor; \(B\) is the viscous damping coefficient of motor; \(T_{L}\) is the output torque; \(L_{s}\) is the armature inductance; \(\psi_{f}\) is the permanent magnet flux linkage; \(\ddot{\theta}\) is the angular acceleration; and \(\dot{\theta}\) and \(\theta\) are the angular velocity and rotation angle of motor. The current, voltage, and flux linkage in the above formula are transformed from three-phase stator shafting to two-phase rotor shafting at the same time, and the final electro-mechanical coupling dynamic equation of PMSM is  
\[ \begin{array}{c}\ddot{\theta}+B\dot{\theta}+T_{L}=\frac{3}{2}n_{p}[i_{q} \psi_{f}+(L_{d}-L_{q})i_{d}i_{q}]\end{array}  \tag{1}\]  
\[ \begin{array}{c}T_{e}=\frac{3}{2}n_{p}[i_{q}\psi_{f}+(L_{d}-L_{q})i_{d}i_{q}] \end{array}  \tag{2}\]  
where \(T_{e}\) is the electromagnetic torque; \(T_{L}\) is the motor output torque; \(L_{d}\), \(L_{q}\) respectively is the motor AC-DC axis inductance; \(n_{p}\) is the number of stator poles; \(i_{d}\), \(i_{q}\) respectively is the motor AC-DC axis current. The above shows that the electromagnetic torque \(T_{e}\) has three parts: (1) the torque \(\dot{J}\ddot{\theta}\) generated by rotor rotation, (2) the loss \(B\dot{\theta}\) caused by the viscous damping of the motor, and (3) the output torque \(T_{L}\) of the motor. According to the selected reducer, the dynamic equation of the reducer is  
\[ \begin{array}{c}T_{L}=\frac{1}{\lambda\eta}T_{r}\end{array}  \tag{3}\]  
with  
\[ \begin{array}{c}\eta=\eta_{1}\eta_{2}\end{array} \]  
Figure 1: Electric tail reduction system. where \(T_{L}\) is the motor output torque; \(T_{r}\) is the reducer output torque; \(\lambda\) is the reduction ratio; \(\eta\) is the reducer transmission total efficiency; \(\eta_{1}\) is the efficiency of the bearing; and \(\eta_{2}\) is the efficiency of the gear transmission. The tail rotor balances the counter torque of the main rotor by producing the yaw moment and carries out the heading operation. Then, the dynamic equation of the tail rotor is proposed by considering the time-varying load during the helicopter flight process.