## 5 Numerical Simulations
### Comparison Results with CBC and AFCFC Methods  
In order to further show the superior performance of the proposed RESO based command filtered controller (RBCFC), the following two controllers are selected to make a comparison study. (1) **Conventional backstepping controller (CBC) proposed in [16]. ** The structure of the controller is given as:  
\[\left\{ \begin{array}{l}\alpha_{1}=-k_{1}\epsilon_{1}+\dot{y}_{\rm d}\\ \alpha_{2}=\dfrac{1}{K}(-k_{2}\epsilon_{2}-e_{1}+MgL\sin x_{1}+Kx_{1}+I\& \lambda_{1})\\ \alpha_{3}=-k_{3}\sigma_{3}-Ke_{2}+\lambda_{2}\\ u=-k_{4}\epsilon_{4}-e_{3}-K(x_{1}-x_{3})+I\&\lambda_{3}\end{array} \right. \tag{52}\]  
(2) **Adaptive fuzzy command filtered controller (AFCFC) proposed in [30]. ** The structure of the controller is given as:  
\[\left\{ \begin{array}{l}\alpha_{1}=-k_{1}\epsilon_{1}+\dot{y}_{\rm d}\\ \alpha_{2}=\dfrac{1}{M^{-1}K}(-k_{2}\epsilon_{2}-\dfrac{1}{2}v_{2}- \epsilon_{1}+z_{1\rm d}-\dfrac{v_{2}\dot{\theta}\dot{\psi}_{1}^{\top}\dot{ \psi}_{1}}{2l_{1}^{2}})\\ \alpha_{3}=-k_{3}\sigma_{3}-M^{-1}Ke_{2}+z_{2\rm d}\\ u=B(\epsilon_{4}+z_{3})+K(e_{3}+z_{2}-e_{1}-y_{\rm d})+J(-k_{4}\epsilon_{4}-e_ {3}+z_{3\rm d})\\ \dot{\theta}=\dfrac{r\psi_{2}^{\top}v_{2}\psi_{1}^{\top}\dot{ \psi}_{1}}{2l_{1}^{2}}-r\dot{\theta}\end{array} \right. \tag{53}\]  
where \(z_{\rm id}(i=1,2,3)\) are the outputs of SCFs, \(l_{1},r,\bar{r}\) are positive design parameters, \(\dot{\theta}\) is the estimation of the adaptive parameter \(\theta\), \(\psi_{1}\) is the vector of fuzzy basis functions. More details about these control parameters are referred to in Ref. [30]. **Remark 1**. : _It is noticed that the structure of the CBC is the simplest among the three controllers, but it suffers from the drawback of computational complexity, and does not consider the influence of disturbances. Both the proposed RBCFC and AFCFC can address these problems, but the AFCFC employs a fuzzy system with an online adaptive learning law to approximate the unknown disturbances, while the proposed RBCFC utilizes two simple RESOs to estimate the disturbances._  
Since the three controllers are designed using the same backstepping methodology, their gains are chosen as the same for the sake of fair comparison, i.e., \(k_{1}=7\), \(k_{2}=25\), \(k_{3}=8\), \(k_{4}=8\). Meanwhile, the same parameters of the SCFs are chosen for the proposed  
Figure 9: Comparison results between \(\dot{\alpha}_{1}\) and the output of the SCF \(z_{\rm 1d}\). controller and the AFCFC, i.e., \(\omega_{n}=1100\), \(\zeta=0.8\). Other parameters for the AFCFC are chosen as in Ref. [30]. No measurement noises are considered in this circumstance for a clear and fair comparison. The simulation results of the single-link FJM under the three controllers are depicted in Figures 10-13, which record the curves of the tracking performance of the system output \(y\), the tracking error \(e_{1}\), the state of the motor angle \(\theta\), and the control torque \(u\), respectively. Figure 11: Tracking error \(e_{1}\) under the three controllers. Figure 12: System state \(x_{2}\) (the motor angle \(\theta\)) under the three controllers. Figure 10: Tracking performance of the system output \(y\) under the three controllers. From Figure 10, it is seen that both the proposed controller and the AFCFC can attenuate the influence of lumped disturbances and achieve satisfactory control performance. However, the system output with the CBC is seriously affected by the disturbances, and fluctuates around the reference trajectory. This is mainly because the CBC does not contain a compensation term for the disturbances. The same results can be further verified by Figures 11 and 12, which respectively show that the tracking error \(e_{1}\) and the motor angle \(\theta\) under the three controllers converge to a small neighborhood around zero. It is clearly seen that the convergence radius of the proposed controller is the smallest among the three controllers.