# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 4 Simulation and Experimental Verification
### Tracking and Anti-Disturbance Performance  
Moreover, in order to verify the developed controller, the following two controllers, LQR and adaptive sliding mode (ASM) [28], are compared. The LQR controller is expressed as \(v=k_{1}e_{\theta}+k_{2}\alpha+k_{3}\dot{e}_{\theta}+k_{4}\dot{\alpha}\), and ASM [28] is designed as  
Figure 6: Estimated results of an actual system. (The red dashed lines represents the nominal value of the corresponding parameter.)  
\(\left(\hat{\eta}x_{2}+\hat{\eta}\alpha-\hat{\theta}_{d}+k_{1}s-k_{2}\alpha+\hat{k} sign(s)\right)\), where \(s=\dot{e}_{\theta}+\lambda e_{\theta},\dot{\hat{k}}=k_{3}|s|-k_{4}\hat{k}\). These three controller parameter values are shown in Table 1.  
Firstly, a step curve with an amplitude of 20\({}^{\circ}\) and a sine curve with an amplitude of 5\({}^{\circ}\) and a frequency of 0.16 Hz are applied to evaluate the tracking performance of the three controllers. Figures 7 and 8 show the step and sinusoidal responses of the three controllers.  
In order to evaluate the anti-disturbance performance of the three controllers, the system is pointed at 0\({}^{\circ}\), and an interference pulse with an amplitude of 1 V and a width of 0.2 s is added at 1.5 s of the control input, which is used to simulate the impact scene of the system. Figure 9 shows the anti-disturbance of the three controllers.  
\begin{table}
\begin{tabular}{c c c} \hline
**Control Scheme** & **Parameter** & **Value** \\ \hline LQR & \(k_{1},k_{2},k_{3},k_{4}\) & \(-14.0,20.0,\)\(-1.4,0.2\) \\ ASM & \(k_{1},k_{2},k_{3},k_{4},\)\(\lambda\) & \(88.0,\)\(1.16\times 10^{3},\)\(200.0,\)\(1.0,7.95\) \\  & & \(600.0,\)\(1.2\times 10^{5},\)\(8\times 10^{6},\) \\ ANTSM & \(\eta_{1},\eta_{2},\eta_{3},\)\(\lambda,\)\(\gamma,\)\(k_{1},k_{2},k_{3}\) & \(0.135,\)\(1.2,\)\(700.0,\) \\  & & \(1.27\times 10^{5},\)\(1.0\) \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of three different control methods.  
Figure 7: Step response curve of the three controllers.  
Figure 8: Sinusoidal response curve of the three controllers.  
Figure 9: Anti-disturbance response curve of the three controllers.  
Based on the above experimental results, the performance parameters of the system are shown in Table 2. First, it can be seen that the step response time of the three controllers is 0.65 s, 0.55 s, and 0.49 s, respectively. The ANTSM controller based on an ESO has a shorter response time than the other two controllers. Then, the data show that the maximum sinusoidal tracking errors of the three controllers are 2\({}^{\circ}\), 1\({}^{\circ}\), and 0.4\({}^{\circ}\), respectively. Therefore, the finite-time sliding mode controller based on the ESO has the best tracking performance. Finally, we can see that the anti-disturbance errors of the three controllers are 2.4\({}^{\circ}\), 1.8\({}^{\circ}\), and 0.6\({}^{\circ}\), respectively. The above values show that in terms of anti-disturbance, the proposed controller has a stronger anti-disturbance ability than the other two controllers. Furthermore, it can be seen from the voltage curves in Figures 8 and 9 that the control inputs of the three controllers have different degrees of oscillation. However, the oscillation amplitude of the control voltage of the proposed controller is significantly smaller than that of the other controllers. In summary, the proposed controller has a better transient response, tracking performance, and disturbance rejection ability and can effectively attenuate the oscillation of the traditional controller.