## 3 ARNNC of Single-Rod Pneumatic Cylinders  
In this section, an ARNNC control strategy for pneumatic systems will be proposed. As shown in Figure 2, the presented design methodology integrates ARC philosophy and the NN philosophy intelligently. The ARC part obtains an accurate parameters' estimation by adaptation and closed-loop stability of the whole system by robust feedback terms. The NN provides an uncertain compensation for the unmodeled dynamics of complicated pneumatic systems, unpredictable disturbances in working conditions, and estimation errors of the adaptive part. The following section will describe the ARNNC control scheme for a single-rod cylinder pneumatic system in detail. where \(\varepsilon_{21}\), \(\varepsilon_{22}\) are small real positive constants. Noting (14) \(p^{\bigstar}+\frac{1}{\theta_{\rm{lim}}\overline{s}_{a}}\lambda_{2s}s_{2}\) is considered to compensate for the unmodeled dynamics, random disturbances \(\widetilde{f_{1}}\), and the estimated errors of the adaptive part \(\psi_{2}T\widetilde{\theta}\), and \(-\frac{1}{\theta_{\rm{lim}}\overline{s}_{a}}\lambda_{2s}s_{2}\) is the robust term explained in the next subsection. Let \(s_{3}=\left(x_{3}-k_{S_{a}/S_{b}}x_{4}\right)-p\), the dynamics of \(s_{3}\) are given as:  
\[\dot{s}_{3}=q-(\frac{\gamma_{S_{a}}}{V_{a}}x_{2}x_{3}+k_{S_{a}/S_{b}}\frac{ \gamma_{S_{b}}}{V_{b}}x_{2}x_{4})+\frac{\gamma-1}{L_{p}V_{a}}\dot{Q}_{a}-k_{S_ {a}/S_{b}}\frac{\gamma-1}{L_{p}V_{b}}\dot{Q}_{b}+\theta_{3}-k_{S_{a}/S_{b}} \dot{\theta}_{4}+\widetilde{f_{a}}-k_{S_{a}/S_{b}}\widetilde{f_{b}}-\dot{p}_{ c}-\dot{p}_{u} \tag{15}\]  
where \(q=\frac{\gamma R}{L_{p}V_{a}}(\dot{m}_{atin}T_{r}-\dot{m}_{aout}T_{a})-k_{S_{ a}/S_{b}}\frac{\gamma R}{L_{p}V_{b}}(\dot{m}_{bin}T_{r}-\dot{m}_{bout}T_{b})\); \(\dot{p}_{c}=\frac{\partial p}{\partial x_{1}}\dot{x}_{1}+\frac{\partial p}{ \partial x_{2}}\dot{\hat{x}}_{2}+\frac{\partial p}{\partial\hat{\theta}}\dot {\hat{\theta}}+\frac{\partial p}{\partial\hat{\Psi}_{p}}\hat{\hat{W}}_{p}+ \frac{\partial p}{\partial\hat{S}_{p}}\hat{S}_{p}+\frac{\partial p}{\partial \hat{t}}\), and \(\dot{p}_{u}=\frac{\partial p}{\partial x_{2}}\widetilde{\hat{x}}_{2}\) represents the calculated and uncalculated part of \(\dot{p}\), respectively. \(\frac{\partial p}{\partial\hat{W}_{p}}\hat{\Psi}_{p}+\frac{\partial p}{\partial \hat{S}_{p}}\hat{S}_{p}\) is the derivative of \(\hat{W}_{p}g_{p}\) which will be explained later. Design a virtual input for \(q\) and adaptation function for \(\sigma_{3}\) as:  
\[q=q_{a}+q_{s}+q_{mn}, \tag{16}\]  
\[q_{a}=-q_{r}-\psi_{3}T\hat{\theta}, \tag{17}\]  
\[q_{s}=-\lambda_{3}s_{3}, \tag{18}\]  
\[\sigma_{3}=w_{3}\psi_{3}s_{3}, \tag{19}\]  
where \(q_{r}=-(\frac{\gamma_{S_{a}}^{S_{a}}}{V_{a}}x_{2}x_{3}+k_{S_{a}/S_{b}}\frac{ \gamma_{S_{b}}}{V_{b}}x_{2}x_{4})+\frac{\gamma-1}{S_{p}V_{a}}\dot{Q}_{a}-k_{S_ {a}/S_{b}}\frac{\gamma-1}{S_{p}V_{b}}\dot{Q}_{b}-\dot{p}_{c}+\frac{\partial p }{\partial\widetilde{x}_{2}}\dot{\hat{x}}_{2}\); \(\psi_{3}\!=[\frac{w_{2}}{w_{3}}\overline{S}_{a}s_{2}s_{3}-\frac{\partial p}{ \partial x_{2}}(\overline{S}_{a}(x_{3}-k_{S_{a}/S_{b}}x_{4})-F),-\frac{\partial p }{\partial x_{2}},1,-k_{S_{a}/S_{b}}]^{T}\); \(w_{3}\) is the weight constant; \(\lambda_{3}\) is a positive constant for feedback term \(q_{s}\); \(q_{mn}\) is the NN compensator to be demonstrated in the next subsection. The optimal value \(q^{\bigstar}\) for \(q_{mn}\) can be designed similar to (14) with the following properties [28]:  
\[ \begin{array}{ll}1.&(q^{\bigstar}+\lambda_{3s}s_{3})-\psi_{3}T\widetilde{ \theta}+\widetilde{f_{a}}-k_{S_{a}/S_{b}}\widetilde{f_{b}}-\dot{p}_{u}< \varepsilon_{31}\\ 2.&s_{3}(q^{\bigstar}-\psi_{3}T\widetilde{\theta}+\widetilde{f_{a}}-k_{S_{a}/ S_{b}}\widetilde{f_{b}}-\dot{p}_{u})<\varepsilon_{32}\\ 3.&s_{3}q^{\bigstar}<0\end{array} , \tag{20}\]  
where \(\varepsilon_{31}\), \(\varepsilon_{32}\) are small real positive constants.