# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 7 Design by Robust Control Toolbox  
The MATLAB toolbox for robust control [20] is based on \(\gamma\) iterations and operates on the state-space time domain, where \(\gamma\) is the coefficient of the weighting function in (36). The significant difference with our approach is that the parameters of the controller are fixed for each \(\gamma\) and can be obtained from the state-space realisation of the plant, i.e., \(\mathbf{P}(s)\), which is defined by Figure 3.  
The theoretical framework for the development of the toolbox initiated by [PERSON] et al. [21], who derived a formula for a family of \(H_{\infty}\) controllers for a given plant \(\mathbf{P}(s)\), can be described by a minimal state-space realisation, in four blocks, as follows:  
\[\mathbf{P}(s)= \begin{bmatrix}A&B_{1}&B_{2}\\ C_{1}&D_{11}&D_{12}\\ C_{2}&D_{21}&D_{22}\end{bmatrix}  \tag{51}\]  
The [PERSON] formulas [21] were based on several assumptions, the most significant being  
\[\mathbf{D}_{22}=\mathbf{0} \tag{52}\]  
Equation (52) does not include the general case. Later, [PERSON] et al. [22] derived new sets of formulas for the case \(\mathbf{D}_{22}\
eq\mathbf{0}\) with new assumptions, from which the robust control toolbox [20] in MATLAB was developed. The algorithm results in an observer-based controller, generally with a higher degree than the open-loop-system \(\mathbf{G}(s)\) model. The lowest possible degree of the controller is equal to the plant degree \(\mathbf{P}(s)\), which obviously depends on both of \(\mathbf{G}(s)\) and \(\mathbf{W}(s)\). This toolbox can also be employed for \(H_{\infty}\) mixed-sensitivity problems. Therefore, the results obtained would form a good comparison. For this purpose, we consider the same weighting matrix in (43) so that  
\[\mathbf{W}_{1}(s)\ = \begin{bmatrix}0.4\left(\frac{s+10}{s+1}\right)&0\\ 0&0.4\left(\frac{s+10}{s+1}\right)\end{bmatrix} \]  
and, to obtain a low-order, controller  
\[\mathbf{W}_{2}(s)\ = \begin{bmatrix}0.005&0\\ 0&0.005\end{bmatrix}  \tag{53}\]  
The following procedures have been employed in our design using the robust control toolbox:  
1. Start with a \(\gamma\) value (21) and convert all the weighting matrices (53) to a state-space form [23]. 2. Convert the open-loop system \(\mathbf{G}(s)\) (7)-(10) to a state-space form [23]. 3. Assemble \(\mathbf{G}(s)\) and \(\mathbf{W}(s)\) to obtain the plant, i.e., \(\mathbf{P}(s)\) (Figure 4 and (28) and (29)), augmented in state-space form [23]. 4. The conditions of the existence of an \(H_{\infty}\) optimal controller will be checked by the program, and if all the tests are passed, the state-space realisation of the controller will be produced by the program [20]. 5. When even one of the tests is failed, a new \(\gamma\) value is selected for the next trial. 6. Finally, convert the state-space form to obtain a transfer function matrix description of the resulting successful controller [23]. The conversion from state space to transfer function for multivariable systems herein is performed by the MFD toolbox in MATLAB [23]. If \(\gamma\) for this controller is too low, then the expected sensitivity function cannot be obtained by the controller and the performance would not be satisfactory. A starting value of \(\gamma\) = 0.4, to satisfy the weighting function in (43), was selected. The first iteration was successful; therefore, all four controllers have been evaluated from this unique weighting matrix.