# Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode
## 3 Case Analysis  
### Model Verification and Method Comparison  
According to [18], the drag characteristics of the entire flight stage have been considered in the overall design of the missile, so that the \(x\) directional position and velocity \(v\) can be considered uncontrolled. Therefore, the \(x\)-directional position and velocity \(v\) in this paper assume nominal values, and only the \(y\)-directional position and trajectory inclination \(\theta\) can be controlled. To verify the effectiveness of this method, this paper conducts simulation analysis using different initial deviations.  
Sliding mode controller parameters: \(\alpha=1.01\), \(\beta=0.01\), \(\gamma=11.9\), \(\lambda=0.01\), \(\overline{\beta}=1.5\), \(k_{1}=1.2\), \(\varepsilon_{i}=0.5\).  
Simulation Analysis 1: only the initial position deviation is present: \(\Delta y=-100\) m.  
For the case where only an initial position deviation is present, the simulation results obtained using the three methods designed in this paper are shown in Figures 4-6.  
The Figures 4-6 reveal that in the case of an initial position deviation, compared to LQR and ADRC, the AHSTC controller designed in this paper can achieve good tracking control under the initial deviation, and the control accuracy is high. In addition, Table 2 reveals that the AHSTC method has relatively small terminal state error, compared to the LQR method. It is noted that the terminal state \(y\) and \(\theta\) are measured by all-state sensors.  
Figure 4: Trajectory tracking curve.  
Figure 5: Height \(y\) tracking curve.  
The Figures 7-9 reveal that when an initial trajectory inclination deviation is present, the AHSTC controller designed in this paper can achieve good tracking control under the initial deviation, and the control accuracy is high. The position tracking can achieve fast convergence, the angle tracking can achieve convergence in approximately 7 s, and the model convergence speed is fast. Table 3 reveals that the trajectory inclination error between the AHSTC and LQR methods is small, but the AHSTC method has a small terminal height error.  
Simulation analysis 3: an initial position deviation is present: \(\Delta y=-100\)m, and the initial trajectory inclination deviation: \(\Delta\theta=3^{0}\).  
For the cases with an initial position deviation and initial trajectory inclination deviation, the simulation results using three methods designed in this paper are shown in Figures 10-14.  
In the AHSTC method, the control tracking curve is presented in Figure 11, which shows that the control curve quickly converges to the nominal control curve, the overall tracking control variable is relatively smooth, and the tracking effect is good. The tracking curves of the two state quantities are shown in Figures 12 and 13, indicating that the tracking trajectory is not only smooth overall but also achieves good tracking of the nominal trajectory. Figure 14 shows the sliding mode surface change curve and the sliding mode surface change rate curve, respectively. The sliding mode surface  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Terminal State** & **AHSTC** & **LQR** & **ADRC** \\ \hline \(y\) coordinate/\(m\) & \(-2.8481\) & \(5.1634\) & \(-0.75889\) \\ Trajectory & \(-0.0057\) & \(-0.0046\) & \(0.1101\) \\ inclination \(\theta/^{\circ}\) & \(-0.0057\) & \(-0.0046\) & \(0.1101\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tracking error based on Simulation Analysis 2.  
Figure 10: Trajectory tracking curve.  
Figure 9: Control tracking curve.