# Finite-Time Consensus Tracking Control for Speed Sensorless Multi-Motor Systems
## 5 Simulations
### Performance Evaluations of Observers  
The estimated accuracy of the disturbance directly determines the performance of the following controllers. First, the disturbance observers are designed and evaluated. To verify the performance of the disturbance observers, four disturbance signals \(d_{1}\), \(d_{2}\), \(d_{3}\) and \(d_{4}\) are added to the follower motors 1, 2, 3 and 4, respectively. Figures 2-5 show the comparison of four disturbance signal curves and corresponding observed value curves. From Figure 2, it shows that the disturbance value of motor 1 and its estimated value achieve consensus tracking after 0.5 s and the maximum steady-state error ratio of the observer is 0.575%. Similarly, from Figures 3-5, it can be concluded that the maximum tracking time of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Motor** & \(R(\Omega)\) & \(I_{eq}\) (kg - m\({}^{2}\)) & \(k_{e}\) (mv/rad/s) & \(k_{t}\) & **Initial Value \(\theta\) (0) (Rad)** \\ \hline
0 & 0.2 & 0.025 & 0.12 & 0.006 & 0.5 \\
1 & 0.6 & 0.035 & 0.24 & 0.012 & -0.8 \\
2 & 0.5 & 0.03 & 0.24 & 0.009 & 1.3 \\
3 & 0.7 & 0.06 & 0.25 & 0.018 & -2.1 \\
4 & 0.8 & 0.04 & 0.3 & 0.02 & 2.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter settings of all motors.  
Figure 1: Topology structural diagram of the interaction between motors.  
In addition, the accuracy of state estimation also has an important impact on the performance of the controller. Figures 6-9 show the comparison of the unmeasurable states and their corresponding estimated values of four subsystems. In Figure 6, it shows that the unmeasurable state value \(\omega_{1}\) and its corresponding estimated value achieve consensus tracking after 0.25 s and the maximum steady-state error ratio of the observer is 0.342%. Similarly, from Figures 7-9, it can be concluded that the maximum tracking time of the observers is 0.3 s, 0.28 s, and 0.3 s, respectively. The maximum steady-state error ratios of all the observers are 0.3  
Case 1: The leader's output is a ramp function \(\theta_{0}=t(rad)\). Figures 10 and 11 show the dynamic tracks of the positions and velocities of all motors under the consensus protocol. The output curves of the follower motors 1, 2, 3, and 4 are marked by solid black, solid red, solid blue, and solid pink lines, respectively. As shown in Figures 10 and 11, the output amplitudes of all following motors are showing a decreasing trend before 0.65 s. Figure 10 shows that the four following motors track the position signal of the virtual leader motor. The simulation result indicates that the positions of the motors in the system tend to be consistent after approximately 0.65 s. Figure 11 shows that the speed signals of the followers are gradually consistent with the speed of the virtual leader, which converge to 1 rad/s by theoretical calculation. The maximum steady-state error ratio of case 1 is 0.25%.  
Case 2: The leader's output is a sinusoidal signal \(\theta_{0}=3\sin(2t)\). Figures 12 and 13 show the position and velocity signals of all motors under the consensus protocol. As shown in Figures 12 and 13, the output amplitudes of all following motors show a decreasing trend and are gradually approaching the output amplitude of the leader motor before 0.61 s. Figure 12 shows that the four following motors track the position signal of the virtual leader motor under the reference signal \(\theta_{0}=3\sin(2t)\). The simulation result indicates that the positions of the motors in the system tend to be consistent after approximately 0.61 s.  
Figure 11: Consensus dynamic tracking curve of all motors under the reference signal \(\theta_{0}=t(rad)\): velocity signals of all motors.  
Figure 9: Estimated performance evaluation curves of four motor velocities: velocity \(\omega_{4}\) and its estimated value curve.