## 3 Control Design and Stability Analysis
### Boundedness and Convergence Analysis  
**Theorem 1** (Boundedness of the closed loop signals). : _Consider the model (1) to (3), subject to input constraint (4) and Assumptions 1 to 5. If the control law (58), update laws (65) to (71) and auxiliary system (47a), (47b) are applied, then (Ti) the signals \(z_{1},z_{2},\,\hat{\theta}_{1},\,\hat{\theta}_{3},\,\hat{\theta}_{4},\,\hat{ \theta}_{rx1},\,\hat{\theta}_{2},\,\hat{\theta}_{ilb},\,\hat{\theta}_{ub}\) are bounded; (Tii) the signals \(b_{m}u,\,b_{m}\Delta u\) are bounded._  
Proof. : Arranging and integrating Equation (72), yields  
\[V+k_{1}\int_{to}^{t}f_{z1}dt+k_{2}\int_{to}^{t}f_{z2}dt\leq V(\bar{x}_{to}) \tag{73}\]  
Therefore, \(V(\bar{x})\leq V(\bar{x}_{to})\), so that \(V\in L_{\infty}\). In view of (60), one further obtains \(V_{z}\in L_{\infty}\), \(V_{\theta}\in L_{\infty}\). Further, considering definitions of \(V_{z}\) (29), \(V_{z1}\) (9), \(V_{z2}\) (30) one obtains \(z_{1}\in L_{\infty}\), \(z_{2}\in L_{\infty}\). Further, considering \(V_{\theta}\in L_{\infty}\) and definition (61), it follows that \(\hat{\theta}_{1},\,\hat{\theta}_{3}\), \(\hat{\theta}_{4},\,\hat{\theta}_{rx1},\,\hat{\theta}_{2},\,\hat{\theta}_{ilb}, \,\hat{\theta}_{ub}\) are bounded. This completes the proof of Ti. Considering Equation (58), and the boundedness of all the closed loop signals involved therein, one obtains that \(b_{m}v\) is bounded. Further, considering definition (37), the bounded nature of \(v\) and \(b_{m}\) stated in assumption 2.5, it follows that \(b_{m}\Delta u\) is bounded. This completes the proof of Tii. **Theorem 2** (Convergence of signals \(z_{1}\), \(z_{2}\) and \(e\)). : _Consider the model (1) to (3), subject to input constraint (4) and Assumptions 1-5. If the control law (58), update laws (65) to (71) and auxiliary system (47a), (47b) are applied, then (Ti) the signal \(z_{1}\) converges asymptotically to \(\Omega_{z1}\), \(\Omega_{z1}=\{z_{1}:|z_{1}|\leq C_{b}\}\); (Tii) the signal \(z_{2}\) converges asymptotically to \(\Omega_{z2}\), \(\Omega_{z2}=\{z_{2}:|z_{2}|\leq C_{b}\}\); (Tiii) if \(\Delta u\) vanishes, then \(e=x_{1}-y_{d}\) converges asymptotically to \(\Omega_{e0}\), \(\Omega_{e0}=\{e:|e|\leq C_{b}\}\); (Tiv) \(e=x_{1}-y_{d}\) converges asymptotically to \(\Omega_{e}\),_  
\[\Omega_{e}=\left\{e:|e|\leq C_{b}+\frac{1}{\sqrt{2k_{o}\beta_{2}(K _{\varphi 2}-1/2)}}\sqrt{\sup_{0\leq s\leq t}(W_{\psi}^{2})}\right\} \tag{74}\] \[W_{\psi}=\hat{\theta}_{ub}b_{m}\Delta u\] \[k_{o}=2\min\{K_{\varphi 1}-1/2,\ \beta_{1}(K_{\varphi 2}-1/2)\} \tag{75}\]  
_where \(\beta_{1}\), \(\beta_{2}\) are constants that satisfy \(1=\beta_{1}+\beta_{2},\ \ \beta_{1}\in(0,\ 1),\ \ \beta_{2}\in(0,\ 1)\)._  
Proof. : From Equation (73) it follows that  
\[k_{1}\int_{to}^{t}f_{z1}dt\leq V(\bar{x}_{to}),\ k_{2}\int_{to}^{t}f_{z2}dt\leq V (\bar{x}_{to}) \tag{76}\]  
so that \(f_{z1}\in L_{1}\). Applying the [PERSON]'s lemma [27], yields \(\lim_{t\to+\infty}f_{z1}^{2}=0\). Furthermore, considering the definition of \(f_{z1}\) (11), it follows that \(z_{1}\) converges asymptotically to \(\Omega_{z1}\). This completes the proof of Ti. From Equation (76) if follows that \(f_{z2}\in L_{1}\). Applying the [PERSON]'s lemma [27], yields \(\lim_{t\rightarrow+\infty}f_{z2}^{2}=0\). Furthermore, considering the definition of \(f_{z2}\) (32), it follows that \(z_{2}\) converges asymptotically to \(\Omega_{z2}\). This completes the proof of Tii. From the definition of \(z_{1}\) (6), it follows that \(e=x_{1}-y_{d}\) can be expressed as  
\[e=z_{1}+\psi_{1} \tag{77}\]  
From Equations (47a) and (47b), it follows that if \(\Delta u\) vanishes, then \(\psi_{1}\) and \(\psi_{2}\) converge to zero. From (77), accounting for the convergence of \(z_{1}\) to \(\Omega_{z1}\), it follows that \(e\) converges asymptotically to \(\Omega_{e0}\). This completes the proof of Tiii. We choose the quadratic form  
\[V_{\psi}=(1/2)\psi_{1}^{2}+(1/2)\psi_{2}^{2} \tag{78}\]  
Differentiating with respect to time, yields \(\dot{V}_{\psi}=\psi_{1}\psi_{1}+\psi_{2}\psi_{2}\).