## 3 Results
### Model-Based Robust Feedback Control Design  
A robust controller needs to be derived for the whole set of admissible plant models formed in the variable mass position scenario. For this sake, the \(H_{\infty}\) method outlined in the previous sections was used. A method of gridding was applied to compute five representative linearised and reduced-order control-relevant models, with the load-mass position being sampled at points  
\[l=\left\{180,185,190,195,200\right\}mm,P_{k}(s)=\frac{\omega_{m}(s)}{T_{m}(s)}= P(s)|_{l=l_{k}};\;k=1..5, \tag{46}\]  
with the five transfer functions describing the dynamics between the torque applied by the actuator \(T_{m}\) and angular velocity observed at the actuator side \(\omega_{m}\). Linear models of order 25 were obtained containing 11 dominant flexible modes, one rigid mode of the system, and second-order actuator dynamics appended at the plant input due to the internal current control loop. Figure 19: Assumed parametric-uncertainty: variable load mass position and its effect on the first two resonance modes location, showing the FRFs for motor and load side dynamics; the higher modes above 250 Hz frequencies correspond to actuator and manipulator base dynamics and remain unaffected. A standard fixed-structure PI controller (33), which is implemented in most of industrial servo-amplifiers, was assumed to close the velocity control loop. Only the first flexible mode of the system can be stabilised actively due to the low order of the compensator. The reader is referred to our previous work with a more thorough discussion of achievable performance in flexible motion systems equipped with PID controllers [14; 36]. Therefore, the controller is complemented by a notch-filter and low-pass filter to improve its high-frequency roll-off to improve achievable closed-loop bandwidth and avoid unwanted destabilisation due to higher bending modes, resulting in controller transfer function in the form of  
\[C(s)=\frac{T_{m}(s)}{e_{\omega}(s)}=\left(k_{p}+\frac{k_{i}}{s}\right)\left( \frac{s^{2}+2\xi\omega_{n}+\omega_{n}^{2}}{s^{2}+2\omega_{n}+\omega_{n}^{2}} \right)\left(\frac{\omega_{f}^{2}}{s^{2}+\sqrt{2}\omega_{f}s+\omega_{f}^{2}} \right), \tag{47}\]  
where \(T_{m}\) denotes the torque demand and \(e_{\omega}\) is the velocity tracking error. The notch filter was tuned for the resonance frequency of \(w_{n}=2640\) rad/s corresponding to actuator-arm bending mode to avoid its excitation. The bandwidth of the 2 nd order Butterworth low-pass filter was set to \(w_{f}=4750\) rad/s to cover the rest of the high-frequency flexible dynamics. The design problem for the derivation of the PI gains in the controller (47) is formulated as a loop-shaping inequality  
\[M_{S}=||S_{k}||_{\infty}=\underset{\forall\omega}{sup}|S_{k}(j\omega)|<1.4\; \forall k=1..5;\;S_{k}(s)=\frac{1}{1+C(s)P_{k}(s)}, \tag{48}\]  
with \(S_{k}\) denoting the closed-loop sensitivity function obtained from the fixed-structure controller (47) and all the plant models in (46). The design requirement (48) enforces a specified closed-loop stability margin \(s_{m}\), because the \(M_{S}\) value is indirectly proportional to a closest distance at which the open-loop Nyquist plot approaches the critical point \([-1,j0]\) in the complex plane  
\[\frac{1}{M_{S}}=s_{m}=\underset{\forall\omega}{inf}|1-L(j\omega)|,\;L(j\omega )=C(j\omega)P(j\omega), \tag{49}\]  
thus directly affecting the amount of uncertainty the loop can handle before getting unstable in the closed-loop setting. This is a special case of the generic requirement (34) that can be handled with the aforementioned \(H_{\infty}\) regions method. The set of admissible controllers is computed and visualised in the parametric \(k_{p}-k_{i}\) plane in Figure 20 using the software tool from [34]. Different regions are computed for each member of the plant models set defined by (46). The intersection of all the regions defines the admissible set of controllers fulfilling the design requirement (48) for all the plant models simultaneously.