# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 7 Design by Robust Control Toolbox
The resulting eight-order controllers are described by thesimulation block diagram in Figure 14, in which the coefficients of the controllers are as follows:  
\[K_{11}(s)=\frac{\sum\limits_{i=1}^{9}n_{11i}s^{9-i}}{\sum\limits_{i=1}^{9}d_{i}s ^{9-i}}\;\;K_{12}(s)=\frac{\sum\limits_{i=1}^{9}n_{12i}s^{9-i}}{\sum\limits_{i=1 }^{9}d_{i}s^{9-i}} \tag{54}\]  
\[K_{21}(s)=\frac{\sum\limits_{i=1}^{9}n_{21i}s^{9-i}}{\sum\limits_{i=1}^{9}d_{i} s^{9-i}}\;\;K_{22}(s)=\frac{\sum\limits_{i=1}^{9}n_{22i}s^{9-i}}{\sum\limits_{i=1 }^{9}d_{i}s^{9-i}} \tag{55}\]  
\[d_{1}=1,d_{2}=340080,d_{3}=3304\times 10^{5},d_{4}=88649\times 10^{6},d_{5}=10 726\times 10^{9}\] \[d_{6}=10669\times 10^{11},d_{7}=71224\times 10^{12},d_{8}=13929 \times 10^{13},d_{9}=69122\times 10^{12}\] \[n_{111}=93440,n_{112}=15902\times 10^{6},n_{113}=69416\times 10^{8},n_{114}=10391\times 10^{11},n_{115}=10348\times 10^{13}\] \[n_{116}=10057\times 10^{15},n_{117}=15931\times 10^{16},n_{118}=74 338\times 10^{16},n_{119}=59402\times 10^{16}\]  
\[n_{211}=-31360,n_{212}=-22071\times 10^{6},n_{213}=-79095\times 10^{8},n_{ 214}=-10297\times 10^{11},n_{215}=-10989\times 10^{13}\] \[n_{216}=-88284\times 10^{14},n_{217}=-10268\times 10^{16},n_{218}= -26206\times 10^{16},n_{219}=-16809\times 10^{16}\]  
\[n_{221}=65920,n_{222}=18683\times 10^{6},n_{223}=68687\times 10^{8},n_{ 224}=93408\times 10^{10},n_{225}=10108\times 10^{13}\] \[n_{226}=83294\times 10^{14},n_{227}=15567\times 10^{16},n_{228}=89 339\times 10^{16},n_{229}=74595\times 10^{16}\]  
\[K_{21}(s)=\frac{\sum\limits_{i=1}^{9}n_{21i}s^{9-i}}{\sum\limits_{i=1}^{9}d_{i} s^{9-i}}\;\;K_{22}(s)=\frac{\sum\limits_{i=1}^{9}n_{22i}s^{9-i}}{\sum\limits_{i=1 }^{9}d_{i}s^{9-i}} \tag{55}\]0  
\[K_{21}(s)=\frac{\sum\limits_{i=1}^{9}n_{21i}s^{9-i}}{\sum\limits_{i=1}^{9}d_{i} s^{9-i}}\;\;K_{22}(s)=\frac{\sum\limits_{i=1}^{9}n_{22i}s^{9-i}}{\sum\limits_{i=1 }^{9}d_{i}s^{9-i}} \tag{55}\]1  
The response of the system to unity double-step disturbances is shown in Figure 15, indicating a very low rejection compared with all other type of controllers, i.e., after a 0.3 sec full rejection has been obtained. The initial request for the sensitivity function imposed by \(\mathbf{W}_{1}(s)\) can in fact be satisfied by a 75% rejection, but full rejection was obtained. This controller is not practically realisable because of its higher order and because of the high gains required. There is also a maximum direct controller input-output transmission of a gain of 93,440, which would amplify the input signal noise, inhibiting effective system control. It should be noted that in Figures 9 and 15, the response is per unit step, i.e., and it depends on the unit of the amplitude of the step disturbance. As explained by Figure 8, the PI controller can reduce the singular values to 0 db, but there is an unacceptable price to be paid for this in terms of the transient responses. This penalty cannot be avoided by the \(H_{\infty}\) optimal controller. The poor performance of the \(H_{\infty}\) optimal PI controller can be explained via Figure 13, in which the maximum singular value of the control effort matrix is greater than the other two and is above the zero db line at all frequencies. The outcome is evident from the undesirable poor response in Figure 9 with excessive oscillations. Therefore, the shape of the sensitivity matrix's singular values should not be misinterpreted, even if several \(H_{\infty}\) norms have been minimised. A more accurate evaluation can be obtained by interpreting the singular value curves of the combined sensitivity and control effort matrix via a Bode plot, as in the case of the minimum-effort controller. In the final analysis, the controller must provide appropriate response characteristics in order to fulfil the dynamic and steady-state performance specifications, in addition to satisfying the \(H_{\infty}\) optimality condition. It can be suggested that \(H_{\infty}\) controllers are computational, and their performance is justified by simulation only (i.e., mathematically).