# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 3 Formulation of the \(H_{\infty}\) Control Problem  
In the early eighties, investigations aimed at reducing closed-loop control system sensitivity were undertaken. Initially, a sensitivity reduction analysis for MIMO systems was formulated by [PERSON] [13] as an optimisation problem. He employed a functional analysis and used transformations, matrix norms, and approximate inverses in his procedures. Thereafter, this area of research was developed by introducing the robust control theory and \(H_{\infty}\) optimal control to design controllers. By defining new transformations, further algorithms were developed, simplifying the design procedures. General control configurations can be investigated by \(H_{\infty}\) methods, described by the block diagram in Figure 3:  
The input disturbance vector is denoted by \(\mathbf{d}(s)\), the output error and control signal vector is \(\mathbf{z}(s)\), the input control effort vector is \(\mathbf{u}(s)\), and the measurement variable vector is \(\mathbf{y}(s)\). The controller model is designated by the matrix \(\mathbf{K}(s)\) and the plant by the matrix \(\mathbf{P}(s)\). This general control configuration can be expressed by the following:  
\[ \begin{bmatrix}\mathbf{z}\\ \mathbf{y}\end{bmatrix} =\mathbf{P}(s) \begin{bmatrix}\mathbf{d}\\ \mathbf{u}\end{bmatrix} = \begin{bmatrix}\mathbf{P}_{11}(s)&\mathbf{P}_{12}(s)\\ \mathbf{P}_{21}(s)&\mathbf{P}_{22}(s)\end{bmatrix}  \tag{11}\]  
Figure 2: Closed-loop control rotor-bearing system with sensors and actuators. The plant \(\mathbf{P}(s)\) consists of four sub-matrices, which satisfy the following relations:  
\[\mathbf{z}=\mathbf{P}_{11}\mathbf{u}+\mathbf{P}_{12}\mathbf{d} \tag{12}\]  
\[\mathbf{y}=\mathbf{P}_{21}\mathbf{u}+\mathbf{P}_{22}\mathbf{d} \tag{13}\]  
The controller applies the control effort based on the measured variables, such that  
\[\mathbf{u}=\mathbf{K}(s)\,\mathbf{y} \tag{14}\]  
Upon eliminating \(u\) and \(y\) from Equations (12)-(14), the following relationship is obtained:  
\[\mathbf{z}=\mathbf{F}_{l}(\mathbf{P},\mathbf{K})\,\mathbf{d} \tag{15}\]  
where  
\[\mathbf{F}_{l}(\mathbf{P},\mathbf{K})=\mathbf{P}_{11}+\mathbf{P}_{12}\mathbf{K}(\mathbf{I}-\mathbf{P}_{22}\bm {K})^{-1}\mathbf{P}_{21} \tag{16}\]  
In Equations (15) and (16), \(\mathbf{F}_{l}(\mathbf{P},\mathbf{K})\) is known as the lower Linear Fractional Transformation (LFT), described by [PERSON] [14], which determines the system output \(z\), following disturbances. A sensible representation for the disturbance vector \(\mathbf{d}(s)\) can be obtained by using the second norm of the signal \(d(t)\) as follows (see, for example, [PERSON] and [PERSON] [15]):  
\[\left\|\mathbf{d}(t)\right\|_{2}=\sqrt{\sum_{i=1}^{n}\int_{0}^{\infty}\left|d_{i} (\tau)\right|^{2}d\tau} \tag{17}\]  
where \(n\) is number of inputs and outputs. Similarly, for the output vector \(\mathbf{z}(s)\), the second norm of \(z(t)\) would be as follows:  
\[\left\|\mathbf{z}(t)\right\|_{2}=\sqrt{\sum_{i=1}^{2n}\int_{0}^{\infty}\left|z_{i} (\tau)\right|^{2}d\tau} \tag{18}\]  
where \(2n\) is used when there are two output channels, and where, in some cases, there may be more than \(2n\) outputs. However, the question of interest is \"If we know how large the input disturbances are, how large are the outputs going to be?\" [PERSON] et al. [16]. Figure 3: General control configuration for \(H_{\infty}\) analysis. The answer is that an upper bound for the output can be determined.