## 3 RESO-Based Backstepping Control Design
### Second-Order Command Filtered Backstepping (SCFB) Controller
**Step 3:** To stabilize \(v_{3}\), the third Lyapunov function candidate is chosen as:  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]7  
The time derivative of \(v_{3}\) is computed by  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]8 \[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]9Taking the time derivative of \(V_{3}\) along (29) and using the third equation of (16) yields  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]0 \[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]1 \[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]2  
To make \(V_{3}\) negative, the virtual control law \(\alpha_{3}\) is chosen as  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]3  
where \(k_{3}\) is a positive design parameter. Substituting (31) into (30) yields  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]4  
If \(v_{4}=0\) and \(\tilde{d}_{1}=0\), then \(\dot{V}_{3}\leq 0\). **Step 4:** To stabilize \(v_{4}\), the final Lyapunov function candidate is chosen as:  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]5  
The time derivative of \(v_{4}\) is  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]6  
Taking the time derivative of \(V_{4}\) along (34) and using the fourth equation of (16) yields  
\[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]7 \[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]8 \[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]9 \[\dot{V}_{1} =v_{1}\dot{v}_{1}=v_{1}\big{(}v_{2}+z_{1}-\dot{y}_{\rm d}+\xi_{2}- \dot{\xi}_{1}\big{)} \tag{20}\]0 \[\dot{V}_{1} =v_{1}\dot{v}_{1}=v_{1}\big{(}v_{2}+z_{1}-\dot{y}_{\rm d}+\xi_{2}- \dot{\xi}_{1}\big{)} \tag{20}\]1  
To make \(V_{4}\) negative, the actual control law \(u\) is designed as  
\[\dot{V}_{1} =v_{1}\dot{v}_{1}=v_{1}\big{(}v_{2}+z_{1}-\dot{y}_{\rm d}+\xi_{2}- \dot{\xi}_{1}\big{)} \tag{20}\]2  
where \(k_{4}\) is a positive design parameter, and \(\hat{d}_{2}\) is the estimate of \(d_{2}\) from the RESO (9). Substituting (36) into (35) yields  
\[\dot{V}_{1} =v_{1}\dot{v}_{1}=v_{1}\big{(}v_{2}+z_{1}-\dot{y}_{\rm d}+\xi_{2}- \dot{\xi}_{1}\big{)} \tag{20}\]3  
If \(\tilde{d}_{1}=0\) and \(\tilde{d}_{2}=0\), then \(\dot{V}_{4}\leq 0\). Summarizing the above design steps, the proposed SCFB controller is composed of the following control laws. \[\left\{ \begin{array}{l}\alpha_{1}=-k_{1}e_{1}+\dot{y}_{\text{d}}\\ \alpha_{2}=\frac{1}{K}\Big{(}-k_{2}e_{2}+Kx_{1}+MgL\sin x_{1}+Iz_{1\text{d}}-v_ {1}-I\hat{d}_{1}\Big{)}\\ \alpha_{3}=-k_{3}e_{3}+z_{24}-Kv_{2}\\ u=-k_{4}e_{4}-K(x_{1}-x_{3})+Jz_{3\text{d}}-v_{3}-J\hat{d}_{2}\end{array} \right. \tag{38}\]  
where \(z_{\text{id}}(i=1,2,3)\) come from the SCFs (4), and \(\hat{d}_{1},\hat{d}_{2}\) are from the RESOs (8) and (9). The block diagram of the proposed RESO based SCFB composite control system for the single-link FJM is shown in Figure 3.