# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 2 Open-Loop Transfer Function Matrix Estimation
\begin{bmatrix}\frac{\frac{ \Gamma}{2\Gamma}\Big{(}\cos\sqrt{\Gamma}L-\cosh\sqrt{\Gamma}L\Big{)}}{2\sqrt{ \Gamma}}\begin{bmatrix}\frac{\zeta_{0}}{2\Gamma\sqrt{\Gamma}}\begin{bmatrix} \sin\sqrt{\Gamma}L-\sinh\sqrt{\Gamma}L\end{bmatrix}&0&0\\ \frac{-\zeta_{0}}{2\sqrt{\Gamma}}\Big{(}\sinh\sqrt{\Gamma}L+\sin\sqrt{\Gamma}L \Big{)}&\frac{\zeta_{0}}{2\Gamma\sqrt{\Gamma}}\begin{bmatrix}\sin\sqrt{ \Gamma}L-\sinh\sqrt{\Gamma}L\\ 0&0&\frac{-\zeta_{0}}{2\sqrt{\Gamma}}\Big{(}\sinh\sqrt{\Gamma}L+\sin\sqrt{ \Gamma}L\Big{)}&\frac{\zeta_{0}}{2\Gamma\sqrt{\Gamma}}\begin{bmatrix}\sin\sqrt{ \Gamma}L-\cosh\sqrt{\Gamma}L\end{bmatrix}\end{bmatrix}\\ 0&0&\frac{-\zeta_{0}}{2\sqrt{\Gamma}}\Big{(}\sinh\sqrt{\Gamma}L+\sin\sqrt{ \Gamma}L\Big{)}&\frac{\zeta_{0}}{2\Gamma\sqrt{\Gamma}}\begin{bmatrix}\cos\sqrt{ \Gamma}L-\cosh\sqrt{\Gamma}L\end{bmatrix}\end{bmatrix}\end{bmatrix} \]  
\[ \begin{bmatrix}\mathbf{T_{lr}^{i}}\end{bmatrix} =\begin{bmatrix}\frac{\frac{ \Gamma}{2\Gamma}\Big{(}\cos\sqrt{\Gamma}L-\cosh\sqrt{\Gamma}L\Big{)}}{2\zeta_ {0}} \begin{bmatrix}\sin\sqrt{\Gamma}L-\sinh\sqrt{\Gamma}L\end{bmatrix} &0&0\\ \frac{-\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma \Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma\Gamma \GammaThe matrices \(\mathbf{D}\) in (1) are defined by the following:  
\[\mathbf{D}_{\mathbf{ll}}^{i}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} =\mathbf{D}_{\mathbf{rr}}^{i}\ \mathbf{D}_{\mathbf{lr}}^{i}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} \] \[\mathbf{D}_{\mathbf{rl}}^{i}= \begin{bmatrix}0&J_{t}\,s&0&\Omega\, J_{p}\,s\\ m\,s^{2}&0&0&0\\ 0&-\Omega\,J_{p}\,s&0&J_{t}\,s\\ 0&0&m\,s^{2}&0\end{bmatrix} \]  
\(J_{p}\) and \(J_{t}\) are the polar and transverse moments of inertia of the lumped disc. In the above equations, \(s\) is the Laplace variable, \(m\) is the mass of each lumped element, \(\Omega\) is the rotational speed of the shaft, and \(\rho\) is the density of the shaft material. \(L\) is the length of each distributed element. The matrices \(\mathbf{B}\) are defined by the following:  
\[\mathbf{B}_{\mathbf{ll}}^{i}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} =\mathbf{B}_{\mathbf{rr}}^{i}\ \mathbf{B}_{\mathbf{lr}}^{i}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} \] \[\mathbf{B}_{\mathbf{rl}}^{i}= \begin{bmatrix}0&0&0&0\\ K_{yy}+s\,C_{yy}&0&K_{yz}+s\,C_{yz}&0\\ 0&0&0&0\\ K_{zy}+s\,C_{zy}&0&K_{zz}+s\,C_{zz}&0\end{bmatrix} \]  
The above partitioned flexibility matrix is described in [12] based on the dynamic stiffness matrix method (DSMM), in which each of the sub-matrices is described in [12]. Frequency response data can be obtained from each transfer function, so that a vector of the frequency response data, as a function of \(\omega\), could be written as follows. \[G_{pp}^{\prime}(j\omega)=x_{pp}+j\,y_{pp} \tag{3}\]  
\[G_{pq}^{\prime}(j\omega)=x_{pq}+j\,y_{pq} \tag{4}\]  
\(x_{pq}\) and \(y_{pq}\) in (3) and (4) are the real and imaginary parts of the transfer function \(G^{{}^{\prime}}\). From the multivariable frequency response matrix in Equations (3) and (4), estimates of the transfer functions can be obtained in the following form [12]:  
\[G_{ij}^{\prime}(s)\cong G_{ij}(s)=\frac{\sum\limits_{k=1}^{m}a_{k}s^{k}}{\sum \limits_{k=1}^{n}b_{k}s^{k}}\ \ m\leq n \tag{5}\]  
The schematic diagram of the closed-loop vibration control system for the rotor system is shown in Figure 2:  
In the above open-loop system, the transfer function matrix is as follows:  
\[ \begin{bmatrix}y_{1}(s)\\ y_{2}(s)\end{bmatrix} = \begin{bmatrix}G_{11}(s)&G_{12}(s)\\ G_{21}(s)&G_{22}(s)\end{bmatrix}\begin{bmatrix}u_{1}(s)\\ u_{2}(s)\end{bmatrix}