## 3 Singular Perturbation Decomposition and Control Law Design
### The Slow Subsystem
 \tag{44}\]  
where, \(\hat{D}_{0\text{s}}(\mathbf{q})\) is \(D_{0\text{s}}(\mathbf{q})\)'s estimate matrix, \(\hat{H}_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\) is \(H_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\)'s estimate matrix. \(\Delta D_{0\text{s}}(\mathbf{q})\) and \(\Delta H_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\) are the error. It is assumed that the system's uncertain parameters are bounded. So, from Properties 1 and 2, we know that \(D_{0\text{s}}(\mathbf{q})\) and \(H_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\) are bounded. Therefore, \(\hat{D}_{0\text{s}}(\mathbf{q}),\hat{H}_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\), \(\Delta D_{0\text{s}}(\mathbf{q})\) and \(\Delta H_{0\text{s}}(\mathbf{q},\dot{\mathbf{q}}_{0})\) are also bounded. We design a robust control law:  
\[ \begin{array}{c}\tau_{\text{s}}=\hat{D}_{0\text{s}}(x_{0},x_{1})\dot{x}_{2 \text{d}}+\hat{H}_{0\text{s}}(x_{0},x_{1},\dot{x}_{2})(x_{2\text{d}}+A\ddot{x} _{1})+A\B\ddot{x}_{1}+[\hat{D}_{0\text{s}}(x_{0},x_{1})\mathbf{A}+B]\ddot{x}_{2}+ \mathbf{u}\end{array}  \tag{45}\]  
where, \(\hat{H}_{0\text{s}}(x_{0},x_{1},\dot{x}_{2})\) is a new matrix by letting \(x_{2}\)=\(\dot{x}_{2}\) in \(\hat{H}_{0\text{s}}(x_{0},x_{1},x_{2})\). \(B\) is a positive definite diagonal matrix, \(\mathbf{u}\) is a robust compensation controller for compensating the uncertainty of the system. As can be seen from Equation (45), the control method does not include the speed variable \(x_{2}\) of the system, so there is no need to measure the system's speed in real time during the control process. By substituting Equation (45) into Equation (44) and combining with Equations (32), (33) and (39), we have:  
\[\dot{V}_{2}=-z_{2}^{\mathsf{T}}B\mathbf{z}_{2}+z_{2}^{\mathsf{T}}(\mathbf{\eta}-\mathbf{u}) \tag{46}\]  
where,  
\[ \begin{array}{c}\mathbf{\eta}=[D_{0\text{s}}(x_{0},x_{1})-\hat{D}_{0\text{s}}(x _{0},x_{1})](\dot{x}_{2\text{d}}+A\ddot{x}_{2})-[D_{0\text{s}}(x_{0},x_{1}) \mathbf{A}+B]\ddot{x}_{2}+\\ \hskip 14.226378 pt[H_{0\text{s}}(x_{0},x_{1},x_{2})-\hat{H}_{0\text{s}}(x_{0},x _{1},x_{2})](x_{2\text{d}}+A\ddot{x}_{1})+\tau_{\text{d}}\end{array}  \tag{47}\]  
Defining variable:  
\[\dot{\mathbf{z}}_{2}=\ddot{x}_{2}+A\ddot{x}_{1} \tag{48}\]  
Substituting Equations (17), (32), (33) and (39) into Equation (48), we have:  
\[\dot{\mathbf{z}}_{2}=\mathbf{z}_{2}+\ddot{x}_{2} \tag{49}\]  
Then, by substituting Equation (49) into Equation (46):  
\[\dot{V}_{2}=-z_{2}^{\mathsf{T}}B\mathbf{z}_{2}+\ddot{\mathbf{z}}_{2}^{\mathsf{T}}(\mathbf{ \eta}-\mathbf{u})-\ddot{\mathbf{x}}_{2}^{\mathsf{T}}(\mathbf{\eta}-\mathbf{u}) \tag{50}\]  
Combining with Equations (14), (17), (33) and (48), we can rewrite Equation (47) as:  
\[ \begin{array}{c}\mathbf{\eta}=\Delta D_{0\text{s}}(x_{0},x_{1})(\ddot{x}_{2 \text{d}}+A\ddot{x}_{2})-[D_{0\text{s}}(x_{0},x_{1})\mathbf{A}+B]\ddot{x}_{2}+[ \Delta H_{0\text{s}}(x_{0},x_{1},\mathbf{x}_{2})+\hat{H}_{0\text{s}}(x_{0},x_{1}, \overline{x}_{2})](\dot{2}_{2}+\dot{x}_{2})+\tau_{\text{d}}\end{array}  \tag{51}\]  
As we know,  
\[ \begin{array}{c}\|\mathbf{\eta}\|\leq\|\Delta D_{0\text{s}}(x_{0},x_{1})\|(\| \dot{x}_{2\text{d}}\|+A\Big{\|}\ddot{x}_{2}\Big{\|})+\|D_{0\text{s}}(x_{0},x_{1}) \mathbf{A}+B\|\cdot\|\ddot{x}_{2}\|+\\ (\|\Delta H_{0\text{s}}(x_{0},x_{1},x_{2})\|+\big{\|}\hat{H}_{0\text{s}}(x_{0},x _{1},\overline{x}_{2})\big{\|})\cdot(\|\dot{x}_{2}\|+\|\dot{x}_{2}\|)+\|\tau_{ \text{d}}\|\end{array}  \tag{52}\]  
According to property 2, property 3 and Equation (14), \(\hat{H}_{0\text{s}}(x_{0},x_{1},\overline{x}_{2})\) is bounded. \(A\) and \(B\) are positive definite matrices, according to Property 1, \(D_{0\text{s}}(x_{0},x_{1})\mathbf{A}+B\) is bounded. From the above analysis we know \(\ddot{x}_{2}\) is bounded. According to Property 4, \(\tau_{\text{d}}\) is bounded. So \(\eta\) is bounded, \(\|\mathbf{\eta}\|\leq p,p\) is a positive constant. The robust controller \(u\) is designed as following:  
\[\mathbf{u}=\left\{