# \(L_{1}\) Adaptive Control for a Vertical Rotor Orientation System
## 5 Simulation Results for the Adaptive Control System  
### Simulation of the Adaptive Controller  
Consider the nominal mechanical parameters in Table 1.  
According to Table 1, the nominal rotational inertias of the rotor about the two axes are: \(J_{xy0}=0.01864\) kg \(\cdot\) m\({}^{2}\) and \(J_{z0}=0.008439\) kg \(\cdot\) m\({}^{2}\). Then, the system parameters in Equation (8) are:  
\[\mathbf{A}=\left[\begin{array}{cccc}0&0&1&0\\ 0&0&0&1\\ 161.4&0&0&-185.1\\ 0&161.4&185.1&0\end{array}\right]\,,\;\mathbf{B}=\left[\begin{array}{cccc}0&0\\ 0&0\\ 56.65&0\\ 0&56.65\end{array}\right] \tag{38}\]  
The control input is:  
\[\mathbf{u}_{\rm m}=-\mathbf{K}_{\rm m}\mathbf{x}=\left[\begin{array}{cccc}0&100&7&0\\ -100&0&0&7\end{array}\right]\mathbf{x} \tag{39}\]  
Therefore,  
\[\mathbf{A}_{\rm m}=\mathbf{A}-\mathbf{B}\mathbf{K}_{\rm m}=\left[\begin{array}{cccc}0&0&1&0 \\ 0&0&0&1\\ 161.4&-5665&-396.55&-185.1\\ 5665&161.4&185.1&-396.55\end{array}\right] \tag{40}\]  
Let the parameter uncertainties be \(\theta_{1}=0.2J_{xy0}\) and \(\theta_{1}=0.2J_{z0}\), the adaptive gain be \(\Gamma=20,000\) and the low-pass filter be:  
\[C(s)=\left[\begin{array}{cc}\dfrac{3\omega_{c}^{2}s+\omega_{c}^{3}}{(s+ \omega_{c})^{3}}&0\\ 0&\dfrac{3\omega_{c}^{2}s+\omega_{c}^{3}}{(s+\omega_{c})^{3}}\end{array}\right] \tag{41}\]  
The bandwidth of \(C(s)\) will influence the stability and performance of the system. A detailed discussion of the low-pass filter \(C(s)\) is in [12].  
The simulation model using MATLAB/Simulink is illustrated in Figure 3. (a) is the parameter adaptive law in Equation (14); (b) is the nonadaptive control input \(\mathbf{u}_{\rm m}\) in Equation (39); (c) is the low-pass filter \(C(s)\) in Equation (41); (d) is the reference input, which is zero in this paper's situation; (e) is the state predictor in Equation (12); (f) is the plant system in Equation (2); (g) gives the parameter uncertainties; (h) is the external disturbance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Material & Iron & – \\ Outer diameter & 150 & mm \\ Inner diameter & 142 & mm \\ Height & 150 & mm \\ Mass & 1.737 & kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rotor parameters.  
The simulation results for the prediction error in every component are shown in Figure 4. All of the components of the prediction error get reduced as time progresses.  
The variation trends of system parameter prediction, \(\hat{\mathbf{K}}_{\theta}\), and one of the prediction error components, \(e_{1}\), under different adaptive gains are shown in Figure 5. It can be seen that, when \(\Gamma\) is larger, \(\hat{\mathbf{K}}_{\theta}\) and \(e_{1}\) will converge more quickly. Therefore, \(\Gamma\) should be as large as possible, as long as the actuator of the control system can be achieved.  
Figure 4: Prediction error simulation results for: (**a**) the \(x\)-direction component \(e_{1}\); (**b**) the \(y\)-direction component \(e_{2}\); (**c**) the \(x\)-direction angular error \(e_{3}\); and (**d**) \(y\)-direction angular error \(e_{4}\).  
Figure 3: Block diagram in Simulink.