# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 4 Mixed-Sensitivity \(H_{\infty}\) Control  
This form of a control strategy was employed for disturbance rejection purposes by [PERSON] and [PERSON] [19] for a helicopter control system. The minimum effort control method herein, which can be converted into a mixed-sensitivity \(H_{\infty}\) form, enabling \(H_{\infty}\) optimality, is to be confirmed. The control system configuration is shown in Figure 4, describing how disturbance rejection can be achieved. The output Is a combination of \(\mathbf{z}_{1}(s)\), which is the weighted error of the system, and \(\mathbf{z}_{2}(s)\), which is the weighted control effort, where  
\[\mathbf{z}= \begin{bmatrix}\mathbf{z}_{1}^{T}&\mathbf{z}_{2}^{T}\end{bmatrix} ^{T} \tag{22}\]  
Figure 4: Mixed-sensitivity \(H_{\infty}\) control for disturbance rejection. Because the set point vector is null, the error vector \(\mathbf{v}(s)\) and measured variable vector \(\mathbf{y}(s)\) are equal, with opposite signs. The \(\mathbf{z}\) vector can be computed from the following:  
\[\mathbf{z}_{1}(s)=\mathbf{W}_{1}(s)\ \mathbf{v}(s) \tag{23}\]  
\[\mathbf{z}_{2}(s)=\mathbf{W}_{2}(s)\ \mathbf{u}(s) \tag{24}\]  
By defining the sensitivity matrix of the system as follows:  
\[\mathbf{S}(s)=\left[\mathbf{I}+\mathbf{G}(s)\mathbf{K}(s)\right.\right]^{-1} \tag{25}\]  
It can be shown that (23) and (24) may be rewritten as follows:  
\[\mathbf{z}_{1}(s)=\mathbf{W}_{1}(s)\ \mathbf{S}(s)\ \mathbf{d}(s) \tag{26}\]  
\[\mathbf{z}_{2}(s)=\mathbf{W}_{2}(s)\ \mathbf{K}(s)\mathbf{S}(s)\ \mathbf{d}(s) \tag{27}\]  
Then the sub-matrices of the plant \(\mathbf{P}(s)\) can be computed from the following relations \(\mathbf{O}\) is empty matrix:  
\[\mathbf{P}_{11}(s)= \begin{bmatrix}\mathbf{W}_{1}(s)\\ \mathbf{O}\end{bmatrix} \ \mathbf{P}_{12}(s)= \begin{bmatrix}\mathbf{W}_{1}(s)\ \mathbf{G}(s)\\ -\mathbf{W}_{2}(s)\end{bmatrix} \]  
\[\mathbf{P}_{22}=-\mathbf{G}(s)\ \ \mathbf{P}_{21}=-\mathbf{I} \tag{28}\]  
The partitioned matrix relationship is as follows:  
\[ \begin{bmatrix}\mathbf{z}_{1}\\ \mathbf{z}_{2}\\ \mathbf{y}\end{bmatrix} = \begin{bmatrix}\mathbf{P}_{11}(s)&\ \ \mathbf{P}_{12}(s)\\ \mathbf{P}_{21}(s)&\ \ \mathbf{P}_{22}(s)\end{bmatrix}\begin{bmatrix}\mathbf{d}\\ \mathbf{u}\end{bmatrix}  \tag{29}\]  
Upon the substitution of (28) into (16), if the LFT of the system can be computed as  
\[\mathbf{F}_{l}(\mathbf{P},\mathbf{K})\ = \begin{bmatrix}\mathbf{W}_{1}\mathbf{S}\\ \mathbf{W}_{2}\mathbf{K}\mathbf{S}\end{bmatrix}  \tag{30}\]  
then relation (19) in this case would be as follows:  
\[\max_{d(t)\
eq 0}\left(\frac{\left\|\mathbf{z}(t)\right\|_{2}}{\left\|\mathbf{d}(t) \right\|_{2}}\right)=\left\| \begin{bmatrix}\mathbf{W}_{1}(s)\mathbf{S}(s)\\ \mathbf{W}_{2}(s)\mathbf{K}(s)\ \mathbf{S}(s)\end{bmatrix} \right\|_{\infty} \tag{31}\]  
Therefore, if a stabilising controller \(\mathbf{K}\) is selected for the system which satisfies the relation  
\[\left\| \begin{bmatrix}\mathbf{W}_{1}\mathbf{S}\\ \mathbf{W}_{2}\mathbf{K}\mathbf{S}\end{bmatrix} \right\|_{\infty}=\gamma \tag{32}\]  
then the output of the system \(\mathbf{z}\) is bounded by following inequality:  
\[\left\|\mathbf{z}(t)\right\|_{2}\leq\gamma\left\|\mathbf{d}(t)\right\|_{2} \tag{33}\]  
It is common practice to select the appropriate weighting matrices \(\mathbf{W}_{1}\) and \(\mathbf{W}_{2}\) for the error and control effort vectors. These matrices are diagonal as follows:  
\[\mathbf{W}_{1}(s)=diag(w_{1}(s),\ldots,w_{i}(s),\ldots w_{n}(s)) \tag{34}\]  
\[\mathbf{W}_{2}(s)=diag\big{(}w_{1}^{\prime}(s),\ldots,w_{l}^{\prime}(s),\ldots w_{ n}^{\prime}(s)\big{)} \tag{35}\]  
The shape of \(w_{i}(s)\) in (34)-(35) depends on the frequency range of interest. In [PERSON] et al. [18], \(w_{i}(s)\) functions are recommended for process control applications. The weighting function \(w^{\prime}_{i}(s)\) for the control effort can usually be formulated in order to achieve an appropriate minimum. Moreover, both the low and high frequency singular value characteristics are of interest in machine vibration problems.