## 4 Experimental Study  
In order to verify the feasibility of the proposed control scheme in practice, the following experimental study was conducted by an AC servomotor equipped with an encoder (10,000 counts/rev). The dynamics of the motor (i.e., \(\dot{J}\ddot{\theta}_{p}=u-F\)) is similar to the system analyzed in Section 2, where the mass \(\hat{M}\) in (22) is replaced by the moment of inertia \(\hat{J}\), and the tracking error is defined as \(e=\theta_{d}-\theta_{p}\). Apply the method presented in Section 4; the estimation nominal values of \(\hat{J}\) and \(\partial_{2}\) in (22) are \(6.0608\times 10^{-4}\) (\(\mathrm{kg}\cdot\mathrm{m}\cdot\mathrm{A}/\mathrm{N}\)) and \(0.0018\) (\(\mathrm{kg}\cdot\mathrm{m}\cdot\mathrm{A}/\mathrm{N}/\mathrm{s}\)), respectively. An industrial computer equipped with a motion control card is used to implement the control algorithm, where the sampling rate is 500 Hz. According to the analysis in Section 3, a precise estimation of the disturbance variation rate will be appreciated. However, due to the restriction of sensor resolution, it is hard to identify the parameters in the pre-sliding region (i.e., \(\sigma_{0}\), \(\sigma_{1}\)). Nevertheless, a suitable switching gain can reasonably be applied, such that the approaching condition is satisfied. In the experiments, both positioning control (i.e., \(x_{d}\) is constant) and tracking control were performed to demonstrate the effectiveness of the proposed method. For positioning control, the reference position is \(2\pi\) rad (10,000 counts). The periodic reference command for tracking control is \(R[1+\sin(2\pi ft+1.5\pi)]\), where \(R=2\pi\) rad (10,000 counts) and \(f=1\) Hz. The control algorithm (22) was adopted for both positioning and tracking control. The value of \(w\) was altered to illustrate the resulting robustness. The values of control gains \(k_{1}\), \(k_{2}\) and resulting performance indexes are listed in Table 3. Moreover, with the consideration of about 10% modeling errors and the selected gains, the robust stability (54) is feasible for \(w=0\), where:  
\[\mathbf{P}=\left[ \begin{array}{cc}116.7322&0.0308\\ 0.0308&0.2588\end{array} \right],\ \ \ \mathbf{Q}=\left[ \begin{array}{cc}8.0203&-0.0078\\ -0.0078&0.0028\end{array} \right],\ \ \ \varepsilon=11.6058 \tag{40}\]  
As a result, one can gradually increase the value of the robust gain step-by-step to achieve a satisfactory result without inducing system instability. Figure 7 shows the positioning response, where the steady state error (\(e_{ss}\)) caused by friction was eliminated via using gain \(w\). In addition, a larger \(w\) also reduces overshoots and settling time \(t_{ss}\). Unlike the general PID controller, experiments showed that zero positioning error can be achieved without obvious overshoot. The control effort is shown in Figure 8, where the chattering phenomenon induced by the conventional sliding mode control was improved. System output responses, error response and control effort of tracking control with respect to \(w\) are shown in Figures 9-11. The experimental results are consistent with (20) in that the larger the \(w\), the larger the uncertain variation rates can be suppressed and, thereby, a better tracking performance can be obtained in the pre-sliding region (i.e., \(\dot{x}_{p}\approx 0\)). All of the experiments show that the proposed method is capable of achieving good positioning/tracking performance, as well as avoiding significant control chattering.