## 3 Design of Hybrid Control Scheme
### DOB-based Compensator  
The remaining extraneous force, after been compensated by the constant velocity compensation, becomes an immeasurable disturbance with relatively small magnitude. From the perspective of control, the immeasurable disturbance can be compensated by an observer. Figure 6: Disturbance force after constant compensation. If controllers of the loading system are designed based on a double-loop cascade composition control strategy, two problems of disturbance rejection (including the remaining extraneous force) and force tracking can be solved separately. The inner loop controller is utilized to eliminate the remaining extraneous force and other disturbances, enhance system robustness, and make the actual plant behave as a nominal model approximately within certain operating bandwidth. The nominal model-based outer loop controller is used to realize the desired force-tracking performance. It is more reasonable to estimate the disturbance on the control channel than to estimate the disturbance itself because we can use the control input to improve the disturbance rejection performance. Then the disturbance observer (DOB) method is introduced in Figure 7, where symbol \(d_{0}\) represents the equivalent disturbance caused by un-modeled dynamics and unknown disturbances including friction. The uncertainty of the system can be treated as a multiplicative perturbation of the nominal system, and it is given as  
\[P(s)=P_{\rm n}(s)(1+\Delta_{\rm L}(s)) \tag{12}\]  
where \(\Delta_{\rm L}\) is an allowable multiplicative uncertainty. The DOB estimates a lumped disturbance, which includes the effects due to model uncertainties as well as the external disturbances, from the difference between the input-output relation of an actual plant \(P(s)\) and its nominal model \(P_{n}(s)\). The amount of the equivalent disturbance \(d_{0}\) is estimated through the inverse nominal model \(P_{n}^{-1}(s)\), and then the equivalent compensation is introduced into the control through a low-pass filter \(Q(s)\), _i.e._, the cancellation input \(u_{\rm do}\) is fed back into the control input, and thus a complete restraining of the equivalent disturbance is achieved. Defining \(d=(u_{\rm vc}+d_{0})P-F_{\rm d}\), the transfer functions from each relevant input to output \(F_{\rm L}\) can be written as  
\[G_{F_{\rm L}u_{\rm c}}=\frac{F_{\rm L}}{u_{\rm c}}=\frac{PP_{\rm n}}{Q(P-P_{ \rm n})+P_{\rm n}} \tag{13}\]  
\[G_{F_{\rm L}d}=\frac{F_{\rm L}}{d}=\frac{P_{\rm n}(1-Q)}{Q(P-P_{\rm n})+P_{\rm n}} \tag{14}\]  
\[G_{F_{\rm L}\dot{\rm u}M}=\frac{F_{\rm L}}{\dot{x}_{\rm M}}=\frac{(G_{\rm com}( 0)P-G_{\rm d})P_{\rm n}(1-Q)}{Q(P-P_{\rm n})+P_{\rm n}} \tag{15}\]  
\[F_{\rm L}=G_{F_{\rm L}u_{\rm c}}u_{\rm c}+G_{F_{\rm L}d}d \tag{16}\]  
Figure 7: Disturbance observer (DOB) structure for the compensated system. In Equation (15), if \(Q(s)\approx\ 1\) in the low-frequency range \([0,\ 2]\) (Hz), the output from \(\dot{x}_{\rm M}\) to \(F_{\rm L}\) approximately equals zero, which means the disturbance force is well suppressed. It is noticed that there is the same multiplicative factor \(1-Q(s)\) from \(d\) and \(\dot{x}_{\rm M}\) to the output force \(F_{\rm L}\), which means that the disturbance force can be regarded as the output resulting from the external disturbances. So the disturbance force can be regarded as a low-frequency disturbance. The sensitivity function \(S_{\rm DOB}\) and the complementary sensitivity function \(T_{\rm DOB}\) of the DOB loop are derived as  
\[\left\{ \begin{array}{l} S_{\rm DOB}=\frac{1-Q(s)}{1+Q(s)\Delta_{ \rm L}(s)}\\ T_{\rm DOB}=\frac{(1+\Delta_{\rm L}(s))Q(s)}{1+Q(s)\Delta_{ \rm L}(s)}\end{array} \right. \tag{17}\]  
From the above analysis, the design of the DOB in our hybrid suppression scheme coincides with the design of the general DOB. When \(Q(s)\) is designed, its cut-off frequency must be higher than the frequency of the disturbance force.