## 2 Dynamics and Problem Formulation  
A single-rod pneumatic cylinder depicted in Figure 1 is considered as the object in this paper. According to [19], the dynamics of piston movement of the system can be described as:  
\[M\ddot{x}=(p_{a}-p_{0})S_{a}-(p_{b}-p_{0})S_{b}-F_{g}-F_{f}+f_{0}+\widetilde{f_{0}}, \tag{1}\]  
where \(M\) represents the inertia of piston and lumped payload; \(x\) is the displacement of the piston, \(p_{0}\), \(p_{a}\), and \(p_{b}\) represent the absolute pressure of the atmosphere, cylinder chamber \(A\) and \(B\), respectively. \(S_{a}\) and \(S_{b}\) are the effective areas of the forward and backward chamber; \(F_{g}\) is the component of gravitational force along the direction of the piston motion; \(F_{f}\) is the friction force which is a combination of viscous friction \(b\dot{x}\) in which \(b\) represents the viscous friction coefficient and the Coulomb friction \(F_{c}\), i.e., \(F_{c}=A_{f}S_{f}(\dot{x})\); and \((f_{0}+\widetilde{f_{0}})\) represents the lumped modeling errors including uncertain nonlinearities, such as unmodeled friction forces, and external disturbances in which \(f_{0}\) represents the nominal value. The complete pneumatic cylinder dynamics are described in [29], which is too complex for controller design. For simplicity, suppose the gas temperature follows the polytropic law [30] and can be written as:  
\[\left\{ \begin{array}{l}T_{l}=T_{r}(\frac{p_{l}}{p_{ab}})^{\frac{n-1}{n}},p_ {eb}=0.8077p_{r}\\ \frac{dp_{l}}{dt}=\frac{\gamma\kappa}{V_{l}}(\dot{m}_{lm}T_{r}-\dot{m}_{lout}T_ {l})-\frac{\gamma p_{l}}{V_{l}}\frac{d\
u_{l}}{dt}+\frac{\gamma-1}{V_{l}}\dot {Q}_{l}+f_{lm}+\widetilde{f_{lm}}\end{array} \right., \tag{2}\]  
where \(l=a,b\) which represent each chamber of cylinder; \(T_{l}\) is the temperature of gas inside the chamber; \(T_{r}\) is the temperature of air supply; \(p_{eb}\) is the equilibrium pressure when the spool of the control valve is at the central position; \(n=1.35\) is the polytropic coefficient; \(p_{r}\) is the pressure of the air supply; \(\gamma\) is the specific heat ratio; \(R\) is the gas constant; \(\dot{m}_{lim}\), \(\dot{m}_{lout}\) represent supply flow and return flow, respectively; \(V_{l}=V_{0\,l}+S_{l}(0.5L\pm x)\) is the volume of the chamber which \(V_{0l}\) is the dead volume and \(L\) is the stroke; \(\dot{Q}_{l}\) is the heat transfer of air and the inside barrel of cylinder; \((f_{lm}+\widetilde{f_{lm}})\) represents the lumped modeling error including neglected temperature dynamics, uncertainties, and random disturbances in which \(f_{lm}\) is the nominal value. The dynamic features of MPYE valve spool will not be taken into consideration since it has a much high bandwidth than the frequency of the pneumatic system; the relationship between mass flow and the control input voltage can be established as [19]:  
\[ \begin{array}{l}q=S(u)K_{q}(p_{u},p_{d},T_{u})=\\ \left\{\begin{array}{cc}S(u)C_{d}C_{1}\frac{p_{u}}{\sqrt{T_{u}}};&\frac{p_{d }}{p_{u}}\leq p_{cr}\\ S(u)C_{d}C_{1}\frac{p_{u}}{\sqrt{T_{u}}}\sqrt{1-\left(\frac{\frac{p_{d}}{p_{u }}-p_{cr}}{(1-p_{cr})}\right)^{2}}&p_{cr}<\frac{p_{d}}{p_{u}}<1\end{array}\right., \\ \end{array}  \tag{3}\]  
Figure 1: Schematic of the single-rod pneumatic cylinder servo system. where \(q\) is the mass flow; \(u\) is the voltage and \(S(u)\) is the effective area of the orifice; \(p_{u}\) is the pressure of the upstream flow and \(p_{d}\) is the pressure of the downstream flow; and \(p_{cr}\) is the critical pressure ratio. The state space model of the whole pneumatic system is described by [19]  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\theta_{1}\Big{(}\overline{S}_{a}(x_{3}-k_{S_{4}/S_{5}}x_{4})-F \Big{)}+\theta_{2}+\overline{f_{1}}\\ \dot{x}_{3}=\frac{\gamma R}{L_{P}V_{a}}\dot{m}_{a}-\frac{\gamma S_{a}}{V_{a}}x_ {2}x_{3}+\frac{\gamma-1}{L_{P}V_{a}}\dot{Q}_{a}+\theta_{3}+\overline{f_{a}}\\ \dot{x}_{4}=\frac{\gamma R}{L_{P}V_{b}}\dot{m}_{b}+\frac{\gamma S_{b}}{V_{b}}x _{2}x_{4}+\frac{\gamma-1}{L_{P}V_{b}}\dot{Q}_{b}+\theta_{4}+\overline{f_{b}} \end{array}