## 4 Nonlinear Dynamic Model of Polishing Pressure Control System  
The polishing force control system is shown in Figure 5, which is mainly composed of an air compressor, an electromagnetic reversing valve, an industrial computer (IPC), a pressure sensor, a cylinder, a proportional valve, a D/A conversion device, a power amplifier, and a servo driver. The electric spindle drives the polishing flap wheel to rotate at high speed, and the cylinder is used to achieve micro-displacement of the electric spindle. The sensor is used to detect the polishing force value and feed it back to the IPC. The detected values are processed by the industrial computer, which is used to control the gas flow of the proportional valve to control the output force of the cylinder. The dynamic model of the proportional valve is regarded as a first-order system, and the neutral dead zone of the proportional valve is compensated to improve the control system performance and reduce the complexity of the system model, as shown in Equation (12),  
\[\left\{ \begin{array}{l}\hat{S}_{\rm v}=-k_{1}S_{\rm v}+k_{f}u_{\rm i}\\ u_{\rm i}\Bigg{\{}\begin{array}{l}U-U_{\rm d}(\rm intake)\\ U_{\rm h}-U(\rm exhaust)\end{array}\end{array} \right. \tag{12}\]  
where \(k_{1}\) and \(k_{f}\) are the undetermined coefficients, \(S_{\rm v}\) is the effective sectional area of the proportional valve spool, \(U\) is the actual output voltage, and \(U_{\rm h}\) and \(U_{\rm d}\) are the upper and lower limits of the dead voltage. The gas passing through the proportional valve port can be regarded as an isentropic flow of frictionless ideal gas. The flow equation of the proportional valve port can be written as follows:  
\[\dot{M}_{\rm q}\sqrt{T_{\rm w}}=\delta_{\rm f}S_{\rm v}P_{\rm z}\tau_{\rm s}(P _{\rm z},P_{\rm b}) \tag{13}\]where \(\delta_{\rm f}\) is the flow coefficient, \(\tau_{\rm s}(P_{\rm a},P_{\rm b})\) is the flow function, \(T_{\rm w}\) is the cylinder operating temperature, \(M_{\rm q}\) is the total gas mass, and \(P_{\rm a}\) and \(P_{\rm b}\) are the left and right chamber pressures of the cylinder, respectively. The flow function can be derived as follows:  
\[\tau_{\rm s}(P_{\rm a},P_{\rm b})=\left\{ \begin{array}{l}\xi_{1},\frac{P_{\rm b }}{P_{\rm a}}\leq\varepsilon_{\rm y}\\ \xi_{2}\sqrt{\left(\frac{P_{\rm b}}{P_{\rm a}}\right)^{2/k}-\left(\frac{P_{\rm b }}{P_{\rm a}}\right)^{(k+1)/k}},\ \frac{P_{\rm b}}{P_{\rm a}}>\varepsilon_{\rm y}\\ \xi_{1}=\sqrt{\frac{k}{R_{\rm i}}\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}}\\ \xi_{2}=\sqrt{\frac{2k}{R_{\rm i}(k-1)}},\ \varepsilon_{\rm y}\ =\ (\frac{2}{k+1})^{k/(k-1)} \end{array} \right. \tag{14}\]  
where \(k\) is the gas specific heat ratio, \(\varepsilon_{\rm y}\) is the critical pressure ratio, and \(R_{\rm i}\) is the ideal gas constant. When the gas leakage flow and cylinder pressure reach a steady state in the cylinder, the model equation can be written as follows:  
\[\left\{ \begin{array}{l}\dot{M}_{\rm o}=\rho_{1}P_{3}+\rho_{2}\\ \dot{P}_{3}=\frac{R_{\rm i}T_{\rm w}}{V_{\rm t}}(\dot{M}_{\rm i}-\dot{M}_{\rm o })-\frac{\dot{V}_{\rm t}P_{3}}{V_{\rm t}}\\ V_{\rm t}=V_{\rm w}+S_{\rm q}l_{\rm x},\ 0\leq l_{\rm x}\leq L_{\rm T} \end{array} \right. \tag{15}\]  
where \(\rho_{1}\) and \(\rho_{2}\) are the undetermined coefficients, \(V\) is the total volume of the cylinder chamber, \(l_{\rm x}\) and \(L_{\rm T}\) are the piston displacement and the total stroke, \(S_{\rm q}\) is the effective cross-sectional area of cylinder, \(V_{\rm w}\) is the cylinder invalid volume, \(P_{3}\) is the cylinder internal pressure, and \(M_{\rm i}\) and \(M_{\rm o}\) are the inflow and outflow gas masses of the cylinder, respectively. Figure 5: Polishing force control system. According to the above derivation, the polishing force control system is a second-order system and can be written as follows:  
\[\left\{