# Constant Force Control of Centrifugal Pump Housing Robot Grinding Based on Pneumatic Servo System
## 5 Stability Analysis of the Pneumatic Control System of the Robot Grinding and Polishing Device
### Stability Analysis of the Control System of the Robot Grinding and Polishing Device  
For the pneumatic control system of the centrifugal pump shell robot grinding and polishing device, firstly, the Hurwitz method was used to analyze whether the system was stable, so as to clarify the stability performance of the control system.  
The characteristic equation of the system is  
\[a_{n}s^{n}+a_{n-1}s^{n-1}+a_{n-2}s^{n-2}+\cdots+a_{0}=0(a_{n}>0,n=0,1,2,\cdots n) \tag{28}\]  
According to the Hurwitz criterion, we arrange the characteristic equation coefficients of Equation (28) into the Hurwitz matrix, so that  
\[H=\begin{bmatrix}a_{n-1}&a_{n-3}&a_{n-5}&\cdots&0&0\\ a_{n}&a_{n-2}&a_{n-4}&\cdots&0&0\\ 0&a_{n-1}&a_{n-3}&\cdots&0&0\\ 0&a_{n}&a_{n-2}&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&\cdots&0&a_{0}.\end{bmatrix} \tag{29}\]  
If the principal sub-formulas of each order of the Hurwitz matrix are greater than zero, that is, \(\det(H_{n})>0(n=1,2,3,\cdots n)\), then the system is stable.  
According to the structural parameters of different cylinders, the characteristic equation of the system can be obtained based on the system transfer function, so that  
\[\det(H_{n})>0(n=1,2,3) \tag{30}\]  
According to the above analysis, the principal sub-formulas of each order of the Hurwitz matrix, which are composed of the coefficients of each order of the characteristic equation of the system, are greater than zero, so the system is stable.  
For the cylinders with different diameters and strokes, the step function was used as the input to obtain the position output response of the system. Figure 7 shows the output response curve of the cylinder position with different strokes and the same cylinder  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Rod Diameter** & **Stroke** & **100 mm** & **80 mm** & **50 mm** & **25 mm** \\ \hline
80 mm & 25 mm & 25 mm & 25 mm & 25 mm \\ \hline
50 mm & 20 mm & 20 mm & 20 mm & 20 mm \\ \hline
32 mm & 12 mm & 12 mm & 12 mm & 12 mm \\ \hline
25 mm & 10 mm & 10 mm & 10 mm & 10 mm \\ \hline \end{tabular}
\end{table}
Table 2: Structural parameters of the different cylinders.  
The force output response curves are shown in Figures 9 and 10. Figure 9 shows the output force response curves of different strokes and Figure 10 shows the output force response curves of different cylinder diameters. According to Figure 9a,b, when the cylinder diameter D = 80 mm and the piston rod diameter d = 25 mm, the greater the stroke, the greater the oscillation amplitude, but the time to reach stability is basically the same, being approximately 0.3 s. It can be seen in Figure 9c,d that the time to reach stability is approximately 0.4 s when the cylinder diameter D = 50 mm and the piston rod diameter d = 20 mm, and the oscillation amplitude is the smallest when the stroke L = 25 mm. It can be seen in Figure 9e,f that the oscillation amplitude is the largest when the cylinder diameter D = 32 mm, the piston rod diameter d = 12 mm, and the stroke L = 80 mm. According to Figure 9g,h, when the cylinder diameter D = 25 mm and the piston rod diameter d = 10 mm, the greater the stroke, the greater the oscillation amplitude.  
Figure 8: Output response curves of cylinder positions with different cylinder diameters. (**a**) L = 100 mm; (**b**) L = 80 mm; (**c**) L = 50 mm; (**d**) L = 25 mm.  
According to Figure 10a-c, when the cylinder diameter D = 50 mm and the piston rod diameter d = 20 mm, the oscillation amplitude is the largest. Figure 10d shows that the oscillation is obvious when the cylinder diameter D = 80 mm and the piston rod diameter d = 25 mm, and others have almost no oscillation. As can be seen in Figures 9 and 10, the force output response has impact and oscillation in the initial stage, which needs to be corrected in order to obtain constant force control.  
Figure 9: _Cont._