# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 5 Computation of the \(H_{\infty}\) Optimal PI Controller
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Initial Values of** & **Final Values of** & **Number of** & **Minimum Value of** \\ \(\mathbf{k_{11}^{p}}\mathbf{k_{22}^{p}}\mathbf{k_{11}^{p}}\mathbf{k_{22}^{p}}\) & \(\mathbf{k_{11}^{p}}\mathbf{k_{22}^{p}}\mathbf{k_{11}^{p}}\mathbf{k_{22}^{p}}\) & **Iterations** & **\(\mathbf{H_{\infty}}\) Norm** \\ \hline \(k_{11}^{p}=1,k_{22}^{p}=1\) & \(k_{11}^{p}=2.1,k_{22}^{p}=2.1\) & & \\ \(k_{11}^{p}=4\) & \(k_{11}^{p}=22.1\) & & N = 85 & \(\left\|\begin{bmatrix}\mathbf{W_{1}}\mathbf{S}\\ \mathbf{W_{2}}\mathbf{K}\mathbf{S}\end{bmatrix}\right\|_{\infty}=0.995\) \\ \(k_{22}^{p}=4\) & \(k_{11}^{p}=22.1\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Searching for a PI controller which results in the minimum \(H_{\infty}\) norm. Figure 5: Flow chart for the Hooke and Jeeves method for computing the \(H_{\infty}\) optimal PI controller. Figure 6: Flow chart of an exploration step to search for lower \(H_{\infty}\) norm. The resulting pre-compensator, which can achieve stability, together with the lowest possible norm, can be expressed as follows:  
\[\mathbf{K}(s)\ = \begin{bmatrix}\left(2.1+\frac{22.1}{s}\right)&0\\ 0&\left(2.1+\frac{22.1}{s}\right)\end{bmatrix}  \tag{45}\]  
The maximum and minimum singular values of the \((\mathbf{W}_{1}\mathbf{S},\mathbf{W}_{2}\mathbf{K}\mathbf{S})\) matrix are shown in Figure 7, which shows that the upper and lower bounds are fairly close together. Both of the curves are below the 0 db line, as required. In Figure 8, the Bode plot for the sensitivity matrix of the \(H_{\infty}\) optimal PI controller is shown, and this appears to be very satisfactory. The singular values are below 0 db for all frequencies, which is a significant achievement. Moreover, the condition number is very low because the maximum and minimum curves are very close together. This low condition number is not a result of optimisation. In fact, because the PI controller exhibits robustness, a low condition number is always associated with it. Although it seems that the above PI controller satisfies the \(H_{\infty}\) optimality condition, the simulation results show unsatisfactory responses, as indicated in Figure 9. Regardless of minimising the cost function in (41), there are significant oscillations in the responses following double-step disturbances. The rejection time is also high, which is a sign of poor relative stability. This occurs because of the closed-loop pole and zero locations, which cannot be identified via the \(H_{\infty}\) optimality criteria. Figure 7: The Bode plot of maximum and minimum singular values of the \((\mathbf{W}_{1}\mathbf{S},\mathbf{W}_{2}\mathbf{K}\mathbf{S})\) mixed-sensitivity matrix for the \(H_{\infty}\) optimal PI controller. Figure 8: The Bode plot of maximum and minimum singular values of the \(S\) sensitivity matrix for the \(H_{\infty}\) optimal PI controller. Figure 9: Response of the system with the \(H_{\infty}\) optimal PI controller to double-step disturbances, indicating excessive oscillations. It should be remembered that the PI controller in [12], with following form  
\[\mathbf{K}(s)\ =\ \begin{bmatrix}\left(0.1+\frac{14}{s}\right)&0\\ 0&\left(0.1+\frac{14}{s}\right)\end{bmatrix}  \tag{46}\]  
exhibits more appropriate responses as shown in [12] but fails the \(H_{\infty}\) optimality test herein, with the resulting cost function of the following:  
\[\left\| \begin{bmatrix}W_{1}S\\ W_{2}K\mathbf{S}\end{bmatrix} \right\|_{\infty}=1.385 \tag{47}\]  
The Bode plot of the singular values of the cost function is shown in Figure 10, where \(\overline{\sigma}\) reaches 2.7 db and seems to indicate failure but as shown in [12] it works properly.