## 2 Process and System Description
### System Description  
The basic structure of a more common closed-loop control system for particleboard mixing and dosing is shown in Figure 2. The obtained actual flow by the flow sensor is compared with the input value, and the error is sent to the controller. To minimize the error, the controller constantly adjusts the frequency of the converter to change the speed of the squirrel-cage motor. In this way, the controlled supply by the glue pump results in steady supply of glue. In this paper, the dynamics of the electrical components of the linear motor XY table have been ignored, and its velocity model can be expressed as a first-order differential equation for each axis [30]. \[v_{i}+\frac{b_{i}}{m_{i}}v_{i}+\frac{F_{Coul,i}}{m_{i}}sign(v_{i})=\frac{k_{i }}{m_{i}}u_{i} \tag{1}\]  
where \(v_{i}\) is velocity; \(m_{i}\) is mass; \(b_{i}\) is viscous damping; \(k_{i}\) is input gain, \(F_{Coul,i}\) is coulomb friction and \(i=x\), \(y\), respectively. Figure 1: Particleboard glue mixing and dosing system. Figure 2: The basic structure of flow control system. The mathematical model of a simple and practical variable frequency pressure regulating system that supplies glue is derived for engineering, and when the voltage and frequency ratio of the power supply of squirrel-cage motors remain unchanged, its small deviation linear model is as follows [31]:  
\[J\dot{\omega}=-(D+pK_{0})\omega+pK_{0}\omega_{1}-pm \tag{2}\]  
\[K_{0}=\frac{p}{r_{2}}{\left(\frac{V_{10}}{\omega_{10}}\right)^{2}} \tag{3}\]  
where \(\omega_{1}\), \(\omega\) are the deviations of stator power frequency and rotor electric angular velocity, respectively; \(V_{10}\), \(\omega_{10}\) are the value on static work point of voltage and frequency of the stator power supply, respectively; \(p\) is role pair number; \(r_{2}\) is the rotor resistance converted to stator side; \(D\) is friction coefficient; \(m\) is the deviation of load torque. For the pump load, the torque is proportional to the square of the rotational speed, and the relationship of the deviations is as shown in Equation (4) when linearized by the small deviation. \[m=K_{m}\omega \tag{4}\]  
where \(K_{m}\) is a constant. Invoking Equations (3) and (2), we get the following:  
\[J\dot{\omega}=-(D+pK_{0}+K_{m})\omega+pK_{0}\omega_{1} \tag{5}\]  
Converted to transfer function, it can be represented as follows:  
\[\frac{\omega(s)}{\omega_{1}(s)}=\frac{K_{d}}{1+T_{d}s} \tag{6}\]  
where \(T_{d}=(D+p(K_{0}+K_{m}))/J\), \(K_{d}=pK_{0}/J\). The mathematical model of the \"pump channel\" part is a set of partial differential equations. We can get the glue flow response curve of this link by a test method. The curve is fitted with an appropriate linear link, which can be described with the following transfer function:  
\[\frac{\omega(s)}{\omega_{1}(s)}=\frac{K_{d}}{1+T_{d}s} \tag{7}\]  
where \(h\) is flow deviation; \(T_{b}\) is time constant on the glue inertia; \(K_{b}\) is a constant; \(K_{g}\) is the differential coefficient of the liquid \"anti-regulation\" role. According to the load characteristics, if we set appropriate parameters, such as acceleration and deceleration time, the curve can be set up to make its speed gradient less than flywheel torque of the motor driving system. For the pump load, the linear mode or half \(S\) mode can be used for the acceleration and deceleration curve. As the frequency response of the converter itself is very fast, a large integrator must be designed to weaken the influence of water hammer on the control signal when it is changed. This is done to enable the output frequency to change slowly into the new value. Therefore, the frequency converter is described with the following inertia:  
\[v_{i}+\frac{b_{i}}{m_{i}}v_{i}+\frac{F_{Coul,i}}{m_{i}}sign(v_{i})=\frac{k_{i }}{m_{i}}u_{i} \tag{1}\]0  
where \(u\) is control signal deviation; \(K_{v}\) and \(T_{v}\) are constants; \(T_{v}\) can be set by the user over a considerable range.