## 3 Simulation Results
 \tag{33}\]  
Consider (22), but now, the control algorithm is modified in the form of a PID controller:  
\[u=\hat{M}K_{P}e_{1}+\hat{M}K_{D}e_{2}-\hat{\sigma}_{2}e_{2}+\partial_{2}\dot{x }_{d}+\hat{M}\ddot{x}_{d}+\hat{M}K_{I}\int_{0}^{t}e_{1}d\tau \tag{34}\]  
Substituting (34) into (33) yields:  
\[ \begin{array}{l}\dot{e}_{1}=e_{2}\\ \dot{e}_{2}=-K_{P}e_{1}-K_{D}e_{2}-K_{I}\int_{0}^{t}e_{1}d\tau+\Gamma\end{array}  \tag{35}\]  
where:  
\[ \begin{array}{l}\Gamma\ \ =\frac{\bar{M}}{M}K_{P}e_{1}+\frac{\bar{M}}{M}K_{D}e_{2 }+\frac{\bar{M}}{M}K_{I}\int_{0}^{t}e_{1}d\tau\\ \ \ -\frac{\bar{\sigma}_{2}}{M}e_{2}+\frac{\bar{\sigma}_{2}}{M}\dot{x}_{d}+ \left(1-\frac{\bar{M}}{M}\right)\ddot{x}_{d}+\frac{1}{M}F_{B}(\kappa,\dot{ \kappa})+\frac{1}{M}d\end{array}  \tag{36}\]  
To address the closed-loop stability and control performance of (35), let \(\dot{e}_{1}=e_{2}\) and \(\dot{e}_{2}=e_{3}\); one can derive the following augmented system:  
\[\dot{\mathbf{e}}_{a}=(\mathbf{A}_{a}+\Delta\mathbf{A}_{a})\mathbf{e}_{a}+ \mathbf{\Gamma}_{a} \tag{37}\]  
where \(\mathbf{e}_{a}=\left[ \begin{array}{cc}e_{1}&e_{2}&e_{3}\end{array} \right]^{T}\) and:  
\[ \begin{array}{l}\mathbf{A}_{a}=\left[\begin{array}{ccc}0&1&0\\ 0&0&1\\ -K_{I}&-K_{P}&-K_{D}\end{array}\right],\ \ \ \Delta\mathbf{A}_{a}=\left[ \begin{array}{ccc}0&1&0\\ 0&0&1\\ \frac{\bar{M}}{M}K_{I}&\frac{\bar{M}}{M}K_{P}&\frac{1}{M}\left(\widetilde{M}K_ {D}-\widetilde{\
u}_{2}\right)\end{array}\right]\\ \mathbf{\Gamma}_{a}=\left[\begin{array}{ccc}0&0\\ \frac{\widetilde{\
u}_{2}}{M}\ddot{x}_{d}+\left(1-\frac{\bar{M}}{M}\right) \dddot{x}_{d}+\frac{1}{M}\dot{F}_{B}(\kappa,\dot{\kappa})+\frac{1}{M}\dot{d} \end{array}\right]\end{array}  \tag{38}\]  
It is clear that the control issue turns into a robust stabilization problem, and the stability issue can also be solved by ways of LMIs. Similar to the steps from (47) to (56), it is easy to show that by the properly selected triple \((K_{P},K_{I},K_{D})\), there must exist a region \(\mathbf{\Phi}\), such that:  
\[\|\mathbf{e}_{a}\|\rightarrow\mathbf{\Phi}:=2\frac{\lambda_{\max}(\mathbf{P}) }{\lambda_{\min}(\mathbf{Q})}\|\mathbf{\Gamma}_{a}\| \tag{39}\]  
Equation (39) implies that the control performance is relevant to the size of \(\mathbf{\Gamma}_{a}\). Even though the variation of the lumped perturbation is finite, the PID control precision is going to be limited by the magnitude of the variation. In other words, better tracking performance needs to be achieved by means of high gain control. However, different from the PID control, asymptotic tracking performance remains available by the proposed EBSMC even without the use of high gain manner, as long as the upper bound of the uncertainty variation is known. Simulations are going to demonstrate this feature. Comparing with the PID controller, to fairly address the main advantage of the proposed EBSMC, an equivalent control gain setting (that is \(K_{P}=1+k_{1}k_{2}\), \(K_{D}=k_{1}+k_{2}\) and \(K_{I}=w\)) is applied. Moreover, in the following simulations, two cases are applied. Cases I and II are with the consideration of time varying external disturbance and nonlinear friction, respectively. The disturbance \(d=\alpha\sin(2\pi ft)\) is applied, where \(\alpha=2\) and \(f=10\) Hz. Figure 6a,b are the simulation results of Cases I and II, respectively. It is clear that the tracking performances of the proposed EBSMC for both cases is better than that by the PID controller. Detailed control performance comparisons are summarized in Table 2. Apparently, once the variation of the lumped perturbations is fast, the PID control performance will be degraded obviously. On the contrary, the EBSMC is able to achieve much better control precision even without the use of high gain control.