## 3 ARNNC of Single-Rod Pneumatic Cylinders
Noting (14) \(q^{\bigstar}+\lambda_{3s}s_{3}\) is considered to compensate for the unmodeled dynamics and random disturbances \(\widetilde{f_{a}}\), \(\widetilde{f_{b}}\), \(\dot{p}_{u}\), and the modeling errors of adaptive estimation \(\psi_{3}T\widetilde{\theta}\), and \(-\lambda_{3s}s_{3}\) is the robust term explained in the next subsection. At last, control input can be calculated with Equation (3) and the equation of effective orifice area against \(u\)[29]. **Theorem 1**. : _Under the assumptions A1, A2, and with control law (10), (16), and adaptation law \(\sigma=\sigma_{2}+\sigma_{3}\) for \(\theta_{1},\theta_{2},\theta_{3},\theta_{4}\), the following properties can be obtained:_  
_A. All signals of the whole closed system are bounded. Furthermore, the positive definite function is defined by_  
\[V_{s}=\frac{1}{2}w_{2}s_{2}{}^{2}+\frac{1}{2}w_{3}s_{3}{}^{2} \tag{21}\]  
_is bounded by_  
\[V_{s}\leq\varepsilon^{-\lambda t}V_{s}(0)+\frac{\varepsilon}{\lambda}(1-e^{- \lambda t}), \tag{22}\]  
_where \(\lambda=\min\{2\theta_{1}\lambda_{2},2\lambda_{3}\}\); \(\varepsilon=w_{2}\varepsilon_{22}+w_{3}\varepsilon_{32}\)._  
_B. In addition, provided \(\forall i=1,a,b\) that \(f_{i}=0\) and the \(\theta\) exist in the fictitious bound \(\delta\), the asymptotic stable state can be achieved._  
**Proof of Theorem 1. ** The derivative of \(V_{s}\) is given by  
\[ \begin{array}{l}\dot{V}_{s}=w_{2}s_{2}\dot{s}_{2}+w_{3}s_{3}\dot{s}_{3}\\ \quad=w_{2}s_{2}(\theta_{1}(\overline{S}_{a}(p+s_{3})-F)+\theta_{2}+\overline{f }_{1}-\dot{a}_{r})+w_{3}s_{3}(q+q_{r}+\psi_{3}{}^{T}\theta-\frac{w_{2}}{w_{3}} \overline{S}_{a}s_{2}s_{3}\theta_{1}+\overline{f}_{a}-k_{s_{4}/S}\overline{f} _{b}-\dot{p}_{u})\end{array}  \tag{23}\]  
Taking \(p\) and \(q\) into (22) and considering 2 of (14) and (20),  
\[ \begin{array}{l}\dot{V}_{s}=-\theta_{1}\lambda_{2}w_{2}s_{2}{}^{2}+w_{2}s_{2 }(\theta_{1}p^{\bigstar}\overline{S}_{a}-\psi_{2}{}^{T}\overline{\theta}+ \overline{f}_{1})-\lambda_{3}w_{3}s_{2}{}^{2}+w_{3}s_{3}(q^{\bigstar}-\psi_{3} {}^{T}\overline{\theta}+\overline{f}_{a}-k_{s_{4}/S}\overline{f}_{b}-\dot{p} _{u})\\ \quad\leq-\theta_{1}\lambda_{2}w_{2}s_{2}{}^{2}-\lambda_{3}w_{3}s_{3}{}^{2}+w_ {2}\epsilon_{22}+w_{3}s_{32}\leq-\lambda V_{s}+\epsilon\end{array}  \tag{24}\]  
which leads to (22) and part A of Theorem 1 is proved. For B of Theorem 1, taking the Lyapunov function,  
\[V_{a}=\frac{1}{2}w_{2}s_{2}{}^{2}+\frac{1}{2}w_{3}s_{3}{}^{2}+\frac{1}{2} \overline{\theta}{}^{T}\Sigma^{-1}\overline{\theta}. \tag{25}\]  
Differentiating (25), consider the P2 of (7), (14), (20),  
\[ \begin{array}{l}\dot{V}_{a}=-\theta_{1}\lambda_{2}w_{2}s_{2}{}^{2}+w_{2}s_{2 }(\theta_{1}p^{\bigstar}\overline{S}_{a}+\overline{f}_{1})-\lambda_{3}w_{3}s_{ 3}{}^{2}+w_{3}s_{3}(q^{\bigstar}+\overline{f}_{a}-k_{s_{4}/S_{3}}\overline{f} _{b}-\dot{p}_{u})+\overline{\theta}{}^{T}\Sigma^{-1}(\dot{\theta}-\sigma)\\ \leq-\theta_{1}\lambda_{2}w_{2}s_{2}{}^{2}-\lambda_{3}w_{3}s_{3}{}^{2}+w_{3}s_ {3}\dot{p}_{u}\end{array}  \tag{26}\]  
Since \(\theta\) is constant or a slow time-varying signal and considering Assumption 1 and P1 of (7), \(\dot{\theta},\dot{\theta},\overline{\theta}\) is bounded. And \(s_{1},s_{2},s_{3}\) are bounded because \(V_{s}\) is bounded. From the above, \(V_{a}\) is bounded. If the PE condition is satisfied [34], \(\overline{\theta}\to 0\) as \(t\rightarrow\infty\). From (15), we obtain \(\widehat{\dot{x}}_{2}\to 0\) and \(\dot{p}_{u}\to 0\) as \(t\rightarrow\infty\) and \(\dot{V}_{a}{<}\)\(0\) except \(s_{2}=s_{3}=0\). From (9) and (15), \(\dot{s}_{2},\dot{s}_{3}\) is bounded, and it can be checked \(\ddot{V}_{a}\) is bounded. Using [PERSON]'s lemma, \(\dot{V}_{a}\to 0\) as \(t\rightarrow\infty\), so do \(s_{1},s_{2},s_{3}\to 0\). \(\Box\)