# 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
### Steady-State Error Analysis of the Pneumatic Control System of the Robot Grinding Device  
The steady-state error of the pneumatic control system of the centrifugal pump housing in the robot grinding device is an important technical index, which reflects the accuracy of the system. The steady-state error of the pneumatic control system is related not only to the form of external action but also to the structural parameters of the control system itself. Since the pneumatic control system of the centrifugal pump housing robot grinding device studied in this work is stable, its accuracy needs to be further studied. The stability error of the system is analyzed below.  
Figure 9: Response curves of the output force at different strokes. (**a**) D = 80 mm, d = 25 mm; (**b**) D = 50 mm, d = 25 mm (partial enlarged view at the initial stage); (**c**) D = 50 mm, d = 20 mm; (**d**) D = 50 mm, d = 20 mm (partial enlarged view at the initial stage); (**e**) D = 32 mm, d = 12 mm; (**f**) D = 32 mm, d = 12 mm (partial enlarged view at the initial stage); (**g**) D = 25 mm, d = 10 mm; (**h**) D = 25 mm, d = 10 mm (partial enlarged view at the initial stage).  
The feedback control model of the system is shown in Figure 11.  
The deviation signal of the system is defined as \(\varepsilon(s)\), which is shown in Figure 11.  
\[\epsilon(s)=U(s)-B(s)=U(s)-H(s)Y(s) \tag{31}\]  
where \(U(s)\) is the input signal of the control system; \(B(s)\) is the main feedback signal of the control system; \(Y(s)\) is the actual output signal of the control system; and \(H(s)\) is the transfer function of the main feedback channel.  
Figure 10: Output force response curves of different cylinder diameters. (**a**) L = 100 mm; (**b**) L = 80 mm; (**c**) L = 50 mm; (**d**) L = 25 mm.  
If the error signal of the system is defined as \(E(s)\), then  
\[E(s)=Y_{r}(s)-Y(s) \tag{32}\]  
where \(Y_{r}(s)\) is the desired output signal of the control system.  
If \(\varepsilon(s)=0\), then \(Y_{r}(s)=Y(s)\), and if we substitute this into Equation (29), we get  
\[Y_{r}(s)=\frac{U(s)}{H(s)} \tag{33}\]  
From Equations (31)-(33), we get  
\[E(s)=Y_{r}(s)-Y(s)=\frac{U(s)}{H(s)}-Y(s)=\frac{U(s)-H(s)Y(s)}{H(s)}=\frac{ \varepsilon(s)}{H(s)} \tag{34}\]  
namely  
\[E(s)=\frac{\varepsilon(s)}{H(s)} \tag{35}\]  
The steady-state error \(e_{ss}\) of the control system is the steady-state component of the system error signal \(e(t)\), namely  
\[e_{ss}=\lim_{t\to\infty}e(t) \tag{36}\]  
According to the terminal value theorem of the Laplace transformation, from Equation (36), we get  
\[e_{ss}=\lim_{t\to\infty}e(t)=\lim_{s\to 0}E(s) \tag{37}\]  
The transfer function of the error is  
\[G_{e}(s)=\frac{E(s)}{U(s)} \tag{38}\]  
From Equations (31), (35), and (38), we get  
\[G_{e}(s)=\frac{1}{H(s)}\frac{1}{1+G(s)H(s)} \tag{39}\]  
From Equations (37)-(39), we get  
\[e_{ss}=\lim_{s\to 0}G_{e}(s)U(s)=\lim_{s\to 0}s\frac{U(s)}{H(s)(1+G(s)H(s))} \tag{40}\]  
It can be seen from Equation (40) that the steady-state error \(e_{ss}\) of the control system depends on the structural parameters \(G(s)\) and \(H(s)\) and the input signals of the system, \(U(s)\).  
It is assumed that the pneumatic servo control system of the robot polishing device ignores the disturbance, and the input signal of its force control system is a step signal;  
Figure 11: Block diagram of the feedback control system.  
hence, there is \(U(s)=\frac{A}{s}\), where \(A\) is the amplitude of the input signal. Then, the steady-state error of the pneumatic control system of the robot grinding device is  
\[e_{ss}=\underset{s\to 0}{\lim}\frac{U(s)}{H(s)(1+G(s)H(s))}=\underset{s\to 0}{\lim}\frac{A}{H(s)(1+G(s)H(s))} \tag{41}\]  
In the unit feedback control system, \(H(s)=1\), and from Equations (27) and (41), we get  
\[e_{ss}=\underset{s\to 0}{\lim}\frac{A}{1+G(s)}=0 \tag{42}\]