# Finite-Time Tracking Control of a Flexible Link Manipulator Based on an Extended State Observer
## 4 Simulation and Experimental Verification
### Estimation of Unknown Parameters of the System  
The unknown parameters of the system are estimated by numerical simulation to verify the convergence of the proposed observer. The control input is set to a square wave signal with an amplitude of 1 V and a frequency of 1 Hz. Three different sets of adaptive gains are selected to illustrate the effect of adaptive gains on parameter estimation. The initial value of the model parameter is set to \(\hat{b}(0)=\hat{m}(0)=\hat{m}(0)=0\). Three groups of different adaptive gains are set to \(\kappa_{1-1}=3.75\times 10^{5},\kappa_{1-2}=7.5\times 10^{5},\kappa_{1-3}=1.5 \times 10^{6}\), \(\kappa_{2-1}=3\times 10^{5},\kappa_{2-2}=6\times 10^{5},\kappa_{2-3}=1.2 \times 10^{6}\), and \(\kappa_{3-1}=5\times 10^{7},\kappa_{3-2}=1\times 10^{8},\kappa_{3-3}=2 \times 10^{8}\) respectively. The bandwidth is selected as \(\omega_{0}=50\), and \(\sigma_{1},\sigma_{2},\sigma_{3}\) are set to a small positive number \(1.0\times 10^{-4}\). Figure 3 shows that the proposed adaptive ESO can achieve an accurate estimation of model parameters under different adaptive gains. The results indicate that the larger the adaptive gain, the faster the parameter convergence. Moreover, pulses with an amplitude of 2 V and width of 1 s are added to the control input at 200 s to simulate the scene of impact disturbance. Figure 4 shows that the larger the adaptive gain, the more susceptible the parameter estimation is to the impact of disturbance. However, when the impact disturbance disappears, the parameter converges faster. Figure 3: Estimation results of unknown parameters of the system under different adaptive law gains. (The red dashed lines represents the nominal value of the corresponding parameter. )Three different bandwidths are selected to illustrate the influence of bandwidth on parameter estimation. The initial value of the model parameters is set to \(\hat{b}(0)=\hat{m}(0)=\hat{n}(0)=0\), and the parameters of the adaptive ESO are selected to be \(\kappa_{1}=1.5\times 10^{6},\kappa_{2}=1.2\times 10^{6},\kappa_{3}=2\times 10^{8}\). The bandwidth is set to \(\omega_{0-1}=50,\omega_{0-2}=60,\omega_{0-3}=70\). Figure 5 shows that the adaptive ESO with large bandwidth can effectively reduce the influence of external disturbance on parameter estimation. Figure 4: Estimation results of unknown parameters of the system with impact disturbance under different adaptive law gains. (The red dashed lines represents the nominal value of the corresponding parameter.) Figure 5: The estimation results of the unknown parameters of the system with impact disturbance under different bandwidths. (The red dashed lines represents the nominal value of the corresponding parameter.) From the curves shown in Figures 4 and 5, we can see that when the system is affected by external disturbances, the parameter estimates of the adaptive ESO are out of equilibrium and fluctuate during the convergence process. This is because when the system is affected by external disturbances, the state estimation error of the adaptive ESO is out of the equilibrium point. At the same time, the control input is a square-wave signal, and the state estimation error changes with the control input, causing the parameter estimation to fluctuate. The fluctuation amplitude is affected by the adaptive gains, and the larger the gains, the larger the fluctuation. As the state estimation error returns to equilibrium, the parameter estimation fluctuation disappears. Finally, the parameter estimation of the system is carried out in the actual system. Figure 6 shows the parameter estimates, \(\hat{b}=51\), \(\hat{m}=-38\), \(\hat{n}=455\), respectively. The above experimental results are close to the nominal values. The difference between the two is caused by the following reasons: the error between the nominal parameters and the parameters of the actual system, the high-order dynamics of the flexible link neglected in the dynamic model, the friction existing in the actual system, and the measurement accuracy of the sensor and other factors.