# Pneumatic Rotary Actuator Position Servo System Based on ADE-PD Control
## Nomenclature  
\(A\)  
effective area of the piston of actuator [m\({}^{2}\)]  
\(b_{j}^{l_{1}};b_{j}^{L}\)  
two \(D\)-dimensional initial vectors  
\(B\)  
viscosity damping coefficient [N-s/m]  
\(c_{0}\)  
critical pressure ratio = 0.21  
\(C\)  
flow coefficient of orifice of proportional valve = 0.68  
\(CR;CR_{\text{max}};CR_{\text{min}}\)  
crossover probability  
\(d_{f}\)  
pitch diameter of the pinion [m]  
\(D\)  
the number of dimensions of vectors  
\(e(n);e(n)\)  
input and output of the filter at the \(n\)-th sampling  
\(f_{a,b};f_{a,b}t\)  
function of mass flow passing through the port of proportional valve  
\(f_{c}\)  
target function  
\(F\)  
scaling factor  
\(g\)  
the number of evolutionary iterations  
G  
the maximum generation  
\(I_{i};I_{i}\)  
optimal index; cost function value  
\(J\)  
moment of inertia of pinion [kg-m\({}^{2}\)]  
\begin{tabular}{l l} \(k\) & adiabatic index of ideal gas = 1.4 \\ \(k_{d}\) & differential coefficient \\ \(k_{p}\) & proportional coefficient \\ \(K_{a}\) & gain of servo amplifier [V/rad] \\ \(K_{ca,cb}\) & flow-pressure coefficients \\ \(K_{m}\) & sum of flow-pressure coefficient and flow gain \\ \(K_{ma,mb}\) & flow gains flowing through **a** port and **b** port of proportional valve \\ \(K_{T}\) & gain of pneumatic rotary actuator [rad/m] \\ \(K_{v}\) & gain of proportional directional control valve [m/V] \\ \(m\) & mass of one piston in the actuator [kg] \\ \(M_{a,b}\) & mass of gas in chamber **a** or **b** [kg] \\ \(M_{f}\) & friction moment [N-m] \\ \(M_{L}\) & load moment [N-m] \\ \(\dot{M}_{a,b}\) & mass flow flowing into chamber **a** or flowing out from chamber **b** [kg/s] \\ \(n\) & sampling times \\ \(N_{p}\) & the number of \(D\)-dimensional vectors \\ \(p_{a,b}\) & pressure of chamber **a** or **b** [Pa] \\ \(p_{ai,bi}\) & pressure of chamber **a** or **b** in working position \(i\)** [Pa] \\ \(p_{e}\) & atmospheric pressure [Pa] \\ \(p_{L}\) & difference between \(p_{a}\) and \(p_{b}\) [Pa] \\ \(p_{s}\) & supply pressure [Pa] \\ \(rand_{j}(0,1)\) & random number generated in the interval [0,1] \\ \(R\) & gas constant = 287 [J/(kg\cdot K)] \\ \(t\) & time [s] \\ \(T_{0}\) & initial temperature [K] \\ \(T_{a,b}\) & gas temperature in chamber **a** or **b** [K] \\ \(T_{ai,bi}\) & temperature of chamber **a** or **b** in working position \(i\)** [K] \\ \(T_{d}\) & differential time constant \\ \(T_{p}\) & sampling period [ms] \\ \(T_{s}\) & ambient temperature = 293 [K] \\ \(u_{i,g^{2}}\), \(u_{j,j,g}\) & test vector; \(j\)-th parameter of \(i\)-th test vector \\ \(U(s)\) & Laplace transform of the output voltage of servo amplifier \\ \(v_{i,g^{2}}\), \(v_{j,i,g}\) & mutant vector; \(j\)-th parameter of \(i\)-th mutant vector \\ \(V_{0}\) & value of the volume of gas [m\({}^{3}\)] \\ \(V_{a,b}\) & volume of gas in chamber **a** or **b** [m\({}^{3}\)] \\ \(V_{ai,bi}\) & volume of gas of chamber **a** or **b** in working position \(i\)** [m\({}^{3}\)] \\ \(w_{1,2,3}\) & weights \\ \(W\) & geometric cross-sectional area gradient of orifice in proportional valve [m] \\ \(x\) & displacement of proportional valve spool [m] \\ \(x_{0}\) & the gap between valve spool and inwall of the valve [m] \\ \(x_{i}\) & displacement of proportional valve spool in working position \(i\) [m] \\ \(x_{i,g^{2}}\), \(x_{j,i,g}\) & the \(i\)-th vector of the \(g\)-th generation\(j\)-th parameter of \(i\)-th vector \\ \(y\) & displacement of the actuator piston [m] \\ \(z(n)\) & output of the controller \\ \(Z\) & total points of sampling \\ \(\alpha\) & filter coefficient \\ \(\zeta\) & damping ratio \\ \(\theta\) & rotation angle of the actuator [rad] \\ \(\theta_{x,M_{p}}(s)\) & Laplace transform of rotation angle of the actuator when input is \(x\), \(M_{L}\) + \(M_{f}\) and \(p_{b}\) \\ \(\theta_{r}(s)\) & Laplace transform of the system input angle \\ \(\rho_{a,b}\) & density of gas in chamber **a** or **b** [kg/m\({}^{3}\)] \\ \(\tau\) & time constant \\ \(\omega_{0}\) & natural frequency [Hz] \\ \end{tabular}