## 3 RESO-Based Backstepping Control Design
### Second-Order Command Filtered Backstepping (SCFB) Controller  
Based on the RESOs, a feedback tracking controller is designed by using the SCFB technique, and an error compensation system is proposed to reduce the filtering errors. Firstly, we define the tracking errors as  
\[\left\{ \begin{array}{l}e_{1}=y-y_{\mathrm{d}}\\ e_{2}=x_{2}-z_{1}\\ e_{3}=x_{3}-z_{2}\\ e_{4}=x_{4}-z_{3}\end{array} \right. \tag{15}\]  
where \(z_{i}(i=1,2,3)\) are the outputs of the SCFs defined in (4). In order to reduce the filtering errors produced by the SCFs, an error compensation dynamic system is designed as:  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=-k_{1}\xi_{1}+\xi_{2}+(z_{1}-\alpha_{1 })\\ \dot{\xi}_{2}=-\frac{1}{l}k_{2}\xi_{2}+\frac{K}{l}[\xi_{3}+(z_{2}-\alpha_{2})] \\ \dot{\xi}_{3}=-k_{3}\xi_{3}+\xi_{4}+(z_{3}-\alpha_{3})\\ \dot{\xi}_{4}=-k_{4}\xi_{4}\end{array} \right. \tag{16}\]  
where \(k_{i}>0\ (i=1,2,3,4)\) are design parameters, and the initial values of \(\xi_{i}\ (i=1,2,3,4)\) are \(\dot{\xi}_{i}(0)=0\). Then the compensated tracking errors are defined as  
\[\left\{ \begin{array}{l}v_{1}=e_{1}-\xi_{1}\\ v_{2}=e_{2}-\xi_{2}\\ v_{3}=e_{3}-\xi_{3}\\ v_{4}=e_{4}-\xi_{4}\end{array} \right. \tag{17}\]  
Next, we derive the control laws to stabilize the tracking errors \(v_{i}\) by using the recursive backstepping methodology. The whole design procedure is divided into the following four steps. **Step 1:** To stabilize \(v_{1}\), the first Lyapunov function candidate is chosen as:  
\[V_{1}=\frac{1}{2}v_{1}^{2} \tag{18}\]  
Based on (2), (15) and (17), the time derivative of \(v_{1}\) is computed by  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\] \[=x_{2}-\dot{y}_{\mathrm{d}}-\dot{\xi}_{1}=v_{2}+z_{1}-\dot{y}_{ \mathrm{d}}+\xi_{2}-\dot{\xi}_{1}\]  
Taking the time derivative of \(V_{1}\) along (19) and using the first equation of (16) yields\[\dot{V}_{1} =v_{1}\dot{v}_{1}=v_{1}\big{(}v_{2}+z_{1}-\dot{y}_{\rm d}+\xi_{2}- \dot{\xi}_{1}\big{)} \tag{20}\] \[=v_{1}(v_{2}+\alpha_{1}-\dot{y}_{\rm d}+k_{1}\xi_{1})\] \[=v_{1}(v_{2}+\alpha_{1}-\dot{y}_{\rm d}+k_{1}e_{1}-k_{1}v_{1})\]  
To make \(V_{1}\) negative, the virtual control law \(\alpha_{1}\) is chosen as  
\[\alpha_{1}=-k_{1}e_{1}+\dot{y}_{\rm d} \tag{21}\]  
where \(k_{1}\) is a positive design parameter. Substituting (21) into (20) yields  
\[\dot{V}_{1}=-k_{1}v_{1}^{2}+v_{1}v_{2} \tag{22}\]  
Obviously, if \(v_{2}=0\), then \(V_{1}\leq 0\). **Step 2:** Similarly, to stabilize \(v_{2}\), the second Lyapunov function candidate is chosen as:  
\[V_{2}=V_{1}+\frac{1}{2}lv_{2}^{2} \tag{23}\]  
The time derivative of \(v_{2}\) is obtained as  
\[\dot{v}_{2} =\dot{e}_{2}-\dot{\xi}_{2}=\dot{x}_{2}-\dot{z}_{1}-\dot{\xi}_{2} \tag{24}\] \[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]0 \[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]1  
Taking the time derivative of \(V_{2}\) along (24) and using the second equation of (16) yields  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]2 \[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]3 \[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]4  
To make \(V_{2}\) negative, the virtual control law \(\alpha_{2}\) is chosen as  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]5  
where \(k_{2}\) is a positive design parameter; \(\dot{d}_{1}\) is the estimate of \(d_{1}\) from the RESO (8). Subsisting (26) into (25) yields  
\[\dot{v}_{1} =\dot{e}_{1}-\dot{\xi}_{1}=\dot{x}_{1}-\dot{y}_{\mathrm{d}}-\dot{ \xi}_{1} \tag{19}\]6  
If \(v_{3}=0\) and \(\ddot{d}_{1}=0\), then \(\dot{V}_{2}\leq 0\).