## 5 RBFNN Design and Analysis of Position Output-Constrained Robust Adaptive Controller
### System Stability Analysis
\begin{array}{l}\dot{V}\leq-\alpha\log\frac{k^{2}}{k^{2}-c_{1}^{2}}-(\beta- \frac{1}{2})e_{2}^{2}-\frac{1}{2}\widetilde{\mathbf{W}}^{\top}\widetilde{\mathbf{W}}- \frac{1}{2}\sigma_{1}\widetilde{a}^{2}-\frac{1}{2}\sigma_{2}\widetilde{b}^{2}- \frac{1}{2}\sigma_{3}\widetilde{c}^{2}\\ \quad\quad+\frac{1}{2}\epsilon^{*2}+\frac{1}{2}\widetilde{\mathbf{W}}^{*2}+\frac{1} {2}\sigma_{1}a^{*2}+\frac{1}{2}\sigma_{2}b^{*2}+\frac{1}{2}\sigma_{3}c^{*2} \end{array}  \tag{48}\]  
Accordingly, the following can be defined:  
\[\mu=\min[2\alpha,2\beta-1,\chi,\lambda_{1}\sigma_{1},\lambda_{2}\sigma_{2}, \lambda_{3}\sigma_{3}], \tag{49}\]  
\[l=\frac{1}{2}\epsilon^{*2}+\frac{1}{2}\mathbf{W}^{*2}+\frac{1}{2}\sigma_{1}a^{*2} +\frac{1}{2}\sigma_{2}b^{*2}+\frac{1}{2}\sigma_{3}c^{*2}. \tag{50}\]  
Thus,  
\[ \begin{array}{l}\dot{V} \leq-\mu\left[\frac{1}{2}\log\frac{k^{2}}{k^{2}-c_{1}^{2}}+\frac{ 1}{2}e_{2}^{2}+\frac{1}{2\chi}\widetilde{\mathbf{W}}^{\top}\widetilde{\mathbf{W}}+ \frac{1}{2\lambda_{1}}\widetilde{a}^{2}+\frac{1}{2\lambda_{2}}\widetilde{b}^{ 2}+\frac{1}{2\lambda_{3}}\widetilde{c}^{2}\right]+l\\ \leq-\mu\dot{V}+l\end{array}  \tag{51}\]  
By selecting the appropriate control parameters \(\alpha\), \(\beta\), \(\chi\), \(\lambda_{i}\) and \(\sigma_{i}\), \(\mu>\frac{l}{\xi}\) can be satisfied. If \(V>\zeta\), then \(\dot{V}<0\); that is, \(V\leq\zeta\) is an invariant set. If \(V(0)\leq\zeta\) holds, then \(\forall t\geq 0\); that is, \(V(t)\leq\zeta\). According to Lemma 1, we can know that \(V\) is bounded, and \(|e_{1}|<k\). Accordingly, all error signals \(e_{1}\), \(e_{2}\), \(\widetilde{\mathbf{W}}\), \(\widetilde{a}\), \(\widetilde{b}\), and \(\widetilde{c}\) are bounded; thus, \(\widetilde{\mathbf{W}}\), \(\hat{a}\), \(\hat{b}\), and \(\hat{c}\) are bounded. According to Equation (51), it can be concluded that  
\[0\leq V(t)\leq\frac{l}{\mu}+\left[V(0)-\frac{l}{\mu}\right]\exp^{-\mu t}. \tag{52}\]  
The inequality in Equation (52) means that there is a moment \(T\), and, for \(\forall t>T\), \(0\leq V(t)\leq\frac{l}{\mu}\) is always established. According to Equation (38), we can get that \(e_{2}\), \(\widetilde{\mathbf{W}}\), \(\widetilde{a}\), \(\widetilde{b}\), and \(\widetilde{c}\) converge to the following compact sets, respectively:  
\[\Omega_{e_{2}}=\left\{e_{2}\middle|\middle|e_{2}\middle|\leq\sqrt{\frac{2l}{ \mu}}\right\}, \tag{53}\]  
\[\Omega_{\widetilde{\mathbf{W}}}=\left\{\widetilde{\mathbf{W}}\middle|\middle|\widetilde{ \mathbf{W}}\middle|\leq\sqrt{\frac{2\chi l}{\mu}}\right\}, \tag{54}\]  
\[\Omega_{\widetilde{a}}=\left\{\widetilde{a}\middle|\middle|\widetilde{a} \middle|\leq\sqrt{\frac{2\lambda_{1}l}{\mu}}\right\}, \tag{55}\]  
\[\Omega_{\widetilde{b}}=\left\{\widetilde{b}\middle|\middle|\widetilde{b} \middle|\leq\sqrt{\frac{2\lambda_{2}l}{\mu}}\right\}, \tag{56}\]  
\[\Omega_{\widetilde{c}}=\left\{\widetilde{c}\middle|\middle|\widetilde{c} \middle|\leq\sqrt{\frac{2\lambda_{3}l}{\mu}}\right\}. \tag{57}\]  
From the above content, it can be seen that the tracking error of the FAO can be adjusted to any small value by selecting the appropriate control parameters. In addition, the initial estimation errors \(\widetilde{\mathbf{W}}(0)\), \(\widetilde{a}(0)\), \(\widetilde{b}(0)\), and \(\widetilde{c}(0)\), and their influence on the system performance can also be adjusted to any small value by selecting appropriate control parameters. \(\square\)