## 3 Simulation Results  
For general controller design and system physical behavior validation purposes, system parameters need to be identified in advance [30]. In Section 2, the proposed control law (22) involves part of the system nominal parameters, including system mass and viscous coefficient. A systematic estimation method is going to present in the following. For the servo system with apparent velocity, the main contribution of the friction effect comes from Coulomb friction and the viscous coefficient; that is, \(F\approx\sigma_{2}v+f_{\rm C}sign(v)\), where \(v=\dot{x}_{p}\). Reconsider (1) in the absence of external disturbance; one has:  
\[M\frac{dv}{dt}=u-(\sigma_{2}v+f_{\rm C}sign(v)) \tag{25}\]  
From the practical realization point of view, the corresponding discrete model can be directly derived by taking the backward difference:  
\[\frac{dv}{dt}\approx\frac{v(k)-v(k-1)}{T_{s}} \tag{26}\]  
where \(T_{s}\) denotes as the sampling period. Hence, the dynamic Equation (25) can be approximated by:  
\[v(k)=\frac{M}{M+\sigma_{2}T_{s}}v(k-1)+\frac{T_{s}}{M+\sigma_{2}T_{s}}u(k)+ \frac{-f_{\rm C}\cdot T_{s}}{M+\sigma_{2}T_{s}}sign(v(k)) \tag{27}\]  
Based on several measurements, (27) can be represented by:  
\[\left[ \begin{array}{c}v(k)\\ v(k+1)\\ \vdots\\ \vdots\\ v(k+m)\end{array} \right]=\left[ \begin{array}{ccc}v(k-1)&F(k)&sign(v(k))\\ v(k)&F(k+1)&sign(v(k+1))\\ \vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots\\ v(k+m-1)&F(k+m)&sign(v(k+m))\end{array} \right]\left[ \begin{array}{c}X_{1}\\ X_{2}\\ X_{3}\end{array} \right] \tag{28}\]  
where:  
\[X_{1}=\frac{M}{M+\sigma_{2}T_{s}},\;X_{2}=\frac{T_{s}}{M+\sigma_{2}T_{s}},\;X _{3}=\frac{-f_{\rm C}\cdot T_{s}}{M+\sigma_{2}T_{s}} \tag{29}\]  
Define:  
\[\mathbf{Y}=\left[ \begin{array}{c}v(k)\\ v(k+1)\\ \vdots\\ \vdots\\ v(k+m)\end{array} \right],\;\;\;\mathbf{A}=\left[ \begin{array}{ccc}v(k-1)& F(k)&sign(v(k))\\ v(k)&F(k+1)&sign(v(k+1))\\ \vdots&\vdots&\vdots\\ \vdots&\vdots&\vdots\\ v(k+m-1)&F(k+m)&sign(v(k+m))\end{array} \right],\;\;\;\mathbf{X}=\left[ \begin{array}{c}X_{1}\\ X_{2}\\ X_{3}\end{array} \right] \tag{30}\]  
The optimal solution is:  
\[\mathbf{X}_{opt}\equiv\mathbf{X}=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1} \mathbf{A}^{T}\mathbf{Y} \tag{31}\]  
Therefore, the servo motor parameters can be calculated by:  
\[M=\frac{X_{1}}{X_{2}}T_{s},\;\sigma_{2}=\frac{1-X_{1}}{X_{2}},\;f_{\rm C}=- \frac{X_{3}}{X_{2}} \tag{32}\]  
In the following simulation study, periodical reference commands of the form \(x_{d}=x_{d1}=R\sin(2\pi ft)\) shown in Figure 1 and \(x_{d}=x_{d2}=R[1+\sin(2\pi ft+1.5\pi)]\) shown in Figure 2 were adopted. The simulation results were performed by MATLAB code ode45 (Runge-Kutta method) with maximum step size 0.000025 (s). The same control gains were used for both cases and are listed in Table 1. Figure 3 is the comparison of tracking performance. It can be found that the tracking error with \(x_{d1}\) is comparatively large at the beginning of control process, which is caused by initial velocity error, but this phenomenon can be avoided with \(x_{d2}\). The tracking error is of the order \(10^{-7}\) in the simulation,  
Next, a comparison study on the PID controller and the proposed EBSMC is presented. It is well known that for tracking applications, larger tracking errors are usually induced when motions change direction. During the velocity reversal, the variation of the nonlinear friction forces is apparently  
Figure 4: Control force comparison. Figure 5: Sliding surface comparison. Figure 3: Tracking error comparison. fast such that the linear controller (PID) is not able to overcome the perturbations properly for this critical situation. For the motion control system, the error dynamics can be represented by:  
\[ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=-\frac{\sigma_{2}}{M}e_{2}-\frac{1}{M}u+\frac{\sigma_{2}}{M}\dot{x} _{d}+\ddot{x}_{d}+\frac{1}{M}F_{B}(\kappa,\dot{\kappa})+\frac{1}{M}d\end{array}