## 3 Design of Hybrid Control Scheme
### Constant Feed-Forward Compensator  
As seen from Figures 3 and 5\(F_{\text{d}}\) is a measureable and bounded disturbance with large magnitude. From the perspective of control, using the conventional feed-forward control can suppress this disturbance, because the feed-forward control can compensate for the effect of this disturbance on system output before adverse effects. If the feed-forward regulator compensates the disturbance just right, the controlled variable will not produce deviation.  
A control strategy based on the structure invariance principle is the most traditional one to compensate for this disturbance force because it does not change the structure of the closed loop, and  
Figure 5: Frequency characteristic set of the extraneous force with uncertainty.  
it can be realized by software easily. According to the structure invariance principle, if we choose the velocity \(\dot{x}_{\rm M}\) as the input (the observed motion disturbance) of the compensator and make  
\[U_{c}(s)=G_{\rm com}(s)\dot{X}_{\rm M}(s)=\frac{N_{\rm D}(s)}{N_{\rm U}(0)G_{\rm v }(s)}\dot{X}_{\rm M}(s) \tag{10}\]  
the disturbance force \(F_{\rm d}\) caused by \(\dot{x}_{\rm M}\) will be canceled, and that means the extraneous force is suppressed completely. However, the robustness of this compensation is bad for system nonlinearity and time-varying parameters. In general, the velocity of the booster cannot be measured directly, and the rapidity and accuracy of the velocity obtained by the differential method or velocity estimation seriously affect the restraining ability. Moreover, \(G_{\rm com}(s)\) is a non-proper transfer function, and it is difficult in physical implementation.  
In this case, the frequency of \(\dot{x}_{\rm M}\) is less than 3 Hz, and then \(G_{\rm v}(s)\) can be approximately equal to \(G_{\rm v}(0)\). It is well known that most of the disturbance force is related to the velocity of the motion disturbance in the low-frequency range. Then, a compensation term can be determined when ignoring the high-order factors in \(N_{\rm D}(s)\) to eliminate the main part of the disturbance force as follows  
\[G_{\rm com}(s)=\frac{N_{\rm D}(0)}{N_{\rm U}(0)G_{\rm v}(0)} \tag{11}\]  
There are time-varying parameters in \(N_{\rm D}(0)\) and \(N_{\rm U}(0)\), and we take their middle values to obtain the constant compensation term \(G_{\rm com}(0)=0.0486\).  
The characteristics of the disturbance force of models in \(P_{0}(s)\), which are compensated by the constant compensator, are depicted in Figure 6. The velocity \(\dot{x}_{\rm M}\) is acquired by the direct differential method. In Figure 6, \(F_{\rm d}^{\prime({\rm i})}\) represents the remaining disturbance force after compensation, and \(F_{\rm d}^{({\rm i})}\) represents the original disturbance force.  
Although the restraining effect on the disturbance force of models in \(P_{0}(s)\) differs within the concerned frequency range, the constant compensator at least attenuates the disturbance force 16 dB in magnitude; thus, at least 84% of the disturbance force is eliminated. However, the feed-forward compensator is sensitive to phase and the accuracy of the velocity signal. The restraining effect will degrade because it is difficult to get an ideal velocity signal in practical implementation. Therefore, the remaining extraneous force, after being compensated, will not be small enough, it will still affect the force-tracking performance, and it is necessary to restrain the remaining extraneous force further.