# Robust Control of Electric Tail Reduction System: Uncertainty and Performance Index
## 5 Numerical Simulation
### Step Response under Variable Load Condition  
Figure 2 shows the power variation of the main rotor from hovering state to high-speed horizontal forward flight of the helicopter at a height of 100 m. Figure 2 shows  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(n_{p}\) & 6 & / \\ \(\psi_{f}\) & 0.09 & Wb \\ \(\int_{tr}\) & 0.0408 & kg\(\cdot\)m\({}^{2}\) \\ \(\overline{B}_{tr}\) & 0.01 & N\(\cdot\)m\(\cdot\)s/rad \\ \(\lambda\) & 1.3 & / \\ \(\eta\) & 0.93 & / \\ \(c\) & 0.05 & / \\ \(\xi_{supp}\) & 1.6 & / \\ \(\omega_{M}\) & 89.6 & rad/s \\ \(\rho_{M}\) & 1.2319 & kg\(\cdot\)m\({}^{3}\) \\ \(R_{M}\) & 1.93 & m \\ \(K_{p}\) & 1 & / \\ \(\sigma\) & 0.04758 & / \\ \(\mathrm{C}_{xf}\) & 0.65 & / \\ \(S_{f}\) & 1.5 & m\({}^{2}\) \\ \(K_{df}\) & 1 & / \\ \(m\) & 250 & kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the unmanned helicopter and the electric tail reduction system.  
that the total power \(P_{M}\) is'saddle shaped' with the horizontal forward flight speed \(V\) (the relation between \(V\) and \(t\) is \(\dot{V}(t)=1\)), which basically conforms to the actual working conditions (when the horizontal forward flight speed \(V\) is less than the induced speed of helicopter hovering \(V_{h}\), the helicopter induced speed is considered as induced speed of helicopter hovering).  
Figure 3 shows the variation of tail rotor load \(T_{t}\) and equivalent load \(T_{tr}\) with the horizontal forward flight speed \(V\).  
We set the tracking speed (system constraint) \(\dot{\theta}_{d}=90\pi(rad/s)\) and initial speed \(\dot{\theta}_{m0}=50\pi(rad/s)\) of the electric tail reduction system. The tracking speed error in the simulation was defined as \(\dot{\theta}_{e}=\dot{\theta}_{m}-\dot{\theta}_{d}\). The load of tail rotor followed the actual working  
Figure 3: The tail rotor load varying with the helicopter horizontal forward speed.  
Figure 2: The main rotor power varying with the helicopter horizontal forward speed.  
conditions in Figures 2 and 3. Here, it was assumed that the helicopter flies forward horizontally without yaw demand (i.e., \(M_{f}=0\)). Based on Assumption 2, we chose \(A=1\),\(b=0\). In (29), we chose \(n=4\), then the robust control gain can be expressed as  
\[\gamma(||\mu||)=\frac{(1+\beta_{\varepsilon})^{-1}||\mu||^{2}}{||\overline{\mu} ||(||\mu||^{3}+||\mu||^{2}\varepsilon+||\mu||\varepsilon^{2}+\varepsilon^{3})}\]  
We chose \(\varepsilon=1\), \(P=1\). By choosing five different \(\hat{\alpha}\)'s, their values and the corresponding \(\kappa_{opt}\)'s and \(J_{min}\)'s are given in Table 2. In (25), \(\Delta J_{tr}=0.02\sin(t)\), \(\Delta B_{tr}=0.001\cos(2t)\) and \(\Delta T_{tr}=\sin(t)\) are used to simulate uncertainty and load disturbance of system. Furthermore, according to (32), we chose the bounding function \(\rho\)\(=55\).  
Figures 4-6 show the simulation results under variable tail rotor load including the effects of five different optimized \(\kappa_{opt}\) on system performance. Compared with the LQR control, Figure 4 shows that the robust constraint following control method proposed in this paper has faster response to the system and can approach the constraint in a shorter time. Figure 5 shows that when the control precision was higher, with the increase of design parameter \(\kappa_{opt}\), the tracking  
Figures 2-6 show that with the increase in helicopter horizontal forward flight speed, the required control cost of electric tail reduction system first increased slightly (high induced power at low speed), then decreased, and then increased when the tail rotor speed did not change.