# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 5 Computation of the \(H_{\infty}\) Optimal PI Controller  
PI controllers always provide full disturbance rejection when they face step disturbances. This cannot be interpreted as robustness. However, we can adjust the PI parameters such that the mixed-sensitivity \(H_{\infty}\) norm can be minimised. We are interested to know if, by this adjustment, the system response can be improved and robustness can be achieved. In this section, we will determine the PI parameters such that the \(H_{\infty}\) norm of the mixed-sensitivity problem is minimised. Herein, we introduce an iterative algorithm, based on the Hooke and Jeeves pattern search method. The \(H_{\infty}\) norm of \(\boldsymbol{W}_{1}\mathbf{S},\boldsymbol{W}_{2}\mathbf{K}\mathbf{S}\) is considered as an objective function in the optimisation procedure, which could be denoted by the following:  
\[\Delta=\left\| \begin{bmatrix}\mathbf{W_{1}}\mathbf{S}\\ \mathbf{W_{2}}\mathbf{K}\mathbf{S}\end{bmatrix} \right\|_{\infty} \tag{41}\]  
The iterations continue until the norm \(\Delta\) in (41) is minimised, enabling the condition \(\Delta<1\) to be achieved. In order to conserve stability, the following constraint is imposed:  
\[|\mathbf{I}+\mathbf{G}(s)\mathbf{K}(s)|=\frac{\prod\limits_{i=1}^{z}(s+\alpha_{i})}{\prod \limits_{i=1}^{p}(s+\beta_{i})}\]  
\[\text{sup}(-\alpha_{i})<0 \tag{42}\]  
The flow chart of the pattern searches for finding the gain and integral rate, which results in a minimum \(H_{\infty}\) norm, is shown in Figure 5. The main subroutine of the search is known as an exploration step about a base point, which, if successful, is followed by pattern moves. The details of the exploration procedures for finding the lower norm are described by the flow chart in Figure 6. The bottom left block in Figure 6 indicates that if the variable \(x_{i+1}=k_{i+1,i+1}\) is explored (bottom right block), then the next variable \(x_{i+2}=k_{i+2,i+2}\) (if any) should be explored. Otherwise (or), if the previous \(x_{i}=k_{i,j}\) is kept, then the variable \(x_{i+1}=k_{i+1,i+1}\) should be explored. The exploration step ends when all the variables are explored. Therefore, any exploration step in the Hooke and Jeeves optimisation method is deterministic. An iteration procedure is also included in the robust control toolbox [20] in MATLAB, based on the parameter \(\gamma\) in (36), which is known as the \(\gamma\) iteration. The main difference between the present approach and that of MATLAB is that in the latter, there is no choice in selecting the order controller structure. However, the required performance of the controller could be imposed by the weighting functions in (36) and (37). If a solution exists, then the program determines the specifications for the \(H_{\infty}\) controller. The final \(\gamma\) value, obtained via the iterations, determines whether or not the desired sensitivity limit has been achieved. For the following weighting matrices  
\[\mathbf{W}_{1}(s) = \begin{bmatrix}0.4\left(\frac{s+10}{s+1}\right)&0\\ 0&0.4\left(\frac{s+10}{s+1}\right)\end{bmatrix}  \tag{43}\] \[\mathbf{W}_{2}(s) = \begin{bmatrix}0.5\left(\frac{s+0.01}{s+10}\right)&0\\ 0&0.5\left(\frac{s+0.01}{s+10}\right)\end{bmatrix}  \tag{44}\]  
a controller matrix consisting of PI diagonal elements was calculated using the Hooke and Jeeves search method. Table 1 shows the search results including the initial PI parameters, the number of iterations, and the final resulting PI parameters, which are used to obtain the minimum rejection time when a step disturbance is imposed on the rotor system.