## 3 Adaptive Robust Control Design
### Robust Control Law Design  
To facilitate the controller design, it is assumed that the system dynamic functions \(f_{1}(\mathbf{x})\), \(f_{2}(\mathbf{x})\), and \(g_{2}(\mathbf{x},u)\), which are approximated later, are completely known. In addition, the mismatched/matched lumped disturbances and angular velocity of the inertial load are measurable. The control objective is to design a robust control law that ensures high-accuracy tracking performance. The tracking errors are defined as follows:  
\[\begin{split} e_{1}&=x_{1}-x_{1d}\\ e_{2}&=x_{2}-\hat{x}_{1d}\end{split} \tag{13}\]  
where \(x_{1d}\) and \(\hat{x}_{1d}\) are the desired angular position and velocity.  
The sliding surface for the actual tracking error is constructed by  
\[s=e_{2}+\lambda e_{1} \tag{14}\]  
where \(\lambda\) is a positive constant. To guarantee the convergence for the sliding variable (14), an auxiliary sliding surface is designed as  
\[\sigma=\hat{s}+ks \tag{15}\]  
where \(k\) is a positive constant.  
The time derivative of the auxiliary sliding surface \(\sigma\) defined in (15) can be derived as  
\[\sigma=\hat{s}+ks \tag{16}\]  
Figure 2: The control scheme of the proposed control strategy.  
For the sake of simplicity, we use \(f_{1}\), \(\dot{f}_{1}\), \(f_{2}\), and \(g_{2}\) instead of \(f_{1}(\mathbf{x})\), \(\dot{f}_{1}(\mathbf{x})\), \(f_{2}(\mathbf{x})\), and \(g_{2}(\mathbf{x},u)\). Taking the first and second-order derivatives of the sliding variable (14) and combining them with the system dynamics (11), one obtains  
\[\begin{split}\dot{s}&=x_{3}+f_{1}+d_{1}-\bar{x}_{1d }+\lambda e_{2}\\ \ddot{s}&=f_{2}+g_{2}u+d_{2}+\dot{f}_{1}+\dot{d}_{1}- x_{1d}^{(3)}\\ &+\lambda(x_{3}+f_{1}+d_{1}-\bar{x}_{1d})\end{split} \tag{17}\]  
where \(x^{(3)}\) represents the third-order derivative of \(x\).  
Substituting (17) into (16), the derivative of \(\sigma\) is given by  
\[\begin{split}\dot{\sigma}&=f_{2}+d_{2}+g_{2}u+\dot{ f}_{1}+d_{1}-x_{1d}^{(3)}+k\lambda(x_{2}-\hat{x}_{1d})\\ &+(\lambda+k)(x_{3}+f_{1}+d_{1}-\bar{x}_{1d})\end{split} \tag{18}\]  
Based on this, the equivalent control is synthesized as  
\[u_{eq}=\frac{1}{g_{2}}\begin{pmatrix}-f_{2}-d_{2}-\dot{f}_{1}-\dot{d}_{1}+x_ {1}^{(3d)}\\ -(\lambda+k)(x_{3}+f_{1}+d_{1}-\bar{x}_{1d})\\ -k\lambda(x_{2}-\hat{x}_{1d})\end{pmatrix} \tag{19}\]  
The switching control term is designed as  
\[u_{sw}=-\frac{1}{g_{2}}(\eta_{1}\text{sgn}(\sigma)+\eta_{2}\sigma) \tag{20}\]  
where \(\eta_{1}\) and \(\eta_{2}\) are positive constants that are determined later.  
Based on (19) and (20), the complete control law is derived as  
\[u=u_{eq}+u_{sw} \tag{21}\]  
**Theorem 1**.: _The control laws (19)-(21) guarantee the finite-time convergence of the auxiliary sliding variable without a singularity. Based on that, the tracking error asymptotically converges to zero in the case that the functions \(\dot{f}_{1}(\mathbf{x})\), \(f_{2}(\mathbf{x})\), and \(g_{2}(\mathbf{x},u)\) are known and the angular velocity along with the lumped disturbances \(d_{1}\) and \(d_{2}\) are measurable._  
Proof of Theorem 1.: Consider the candidate Lyapunov function as  
\[V_{c}=\frac{1}{2}\sigma^{2} \tag{22}\]  
Taking the time derivative of (22), one obtains  
\[\dot{V}_{c}=\sigma\dot{\sigma} \tag{23}\]  
Substituting (19)-(21) into (23), we derive  
\[\dot{V}_{c}\leq-\eta_{1}|\sigma| \tag{24}\]  
Based on (23) and (24), one can obtain  
\[\dot{V}_{c}\leq-\eta\sqrt{2}V_{c}^{1/2} \tag{25}\]  
It can be observed from (25) that \(V_{c}\) converges to zero in finite time with convergence time is computed as  
\[t_{r}\leq\frac{\sqrt{2}}{\eta}V_{c}^{1/2}(0) \tag{26}\]Based on this, it can be concluded that by using the control laws (19), (20), and (21), the auxiliary sliding variable converges to zero in finite time; hence, the control system is asymptotically stable.  
This completes the proof of Theorem 1.