# Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems
## 1 Introduction  
Active vibration control in rotors has been a well-known field of research during the last five decades. Due to the uncertainty of external disturbances, robust controllers seem to be suitable candidates for vibration suppression in rotors. However, the research on this subject, particularly for rotor vibration, is limited.  
Early investigations considering \(H_{\infty}\) control for short rotors showed the existence of a controller in the frequency domain [1, 2]. They have been used to control vibrations via Active Magnetic Bearings (AMBs). The experimental verification of such robust controllers has been conducted [3], and recently, effectiveness of the controllers for short rotors has been demonstrated [4].  
Apart from experimental attempts, a theoretical investigation of optimal controllers and robust-type vibration suppressors via AMBs has been performed for short rotors that are assumed rigid [5, 6]. For short and flexible rotors, optimal controllers have also been studied [7]. However, the multivariable model requirement for rotor vibration control (even for rigid short rotors) is studied and demonstrated in [8].  
For the vibration control of flexible and long rotors, an experimental and theoretical analysis is carried out in [9], and the multivariable nature of controllers is emphasized in [10]. Moreover, it has been shown that there exist controllers that suppress vibrations at resonance [11].  
Recently vibration suppression in long flexible rotors was studied, in which the existence of realizable controllers subject to any kind of disturbance was investigated and commented upon [12]. However, the comparative performance of robust vibration controllers in both the time and frequency domain for long and flexible rotors has not been carried out yet.  
In this paper, the long flexible rotor system that is described in [12] is considered for investigation. The \(H_{\infty}\) optimality criteria was selected for the purposes of comparison. Therefore, the \(H_{\infty}\) control problem is initially formulated. Thereafter, a mixed-sensitivity \(H_{\infty}\) criterion, which is an appropriate method for the investigation of disturbance rejection in vibration control problems, is described.  
An iterative algorithm is introduced, enabling robust controllers, which can also be \(H_{\infty}\) optimal, to be determined. It is shown that for the rotor system model, the resulting controller can be realizable with a low order. However, satisfying the \(H_{\infty}\) optimality condition does not guarantee an appropriate system response when operating under severe disturbances.  
Moreover, via a singular value analysis, it is concluded that the singular values for the sensitivity matrix are not the only factors to be considered for performance evaluation. Further reduction in the \(H_{\infty}\) norm for the \(K\!S\) matrix is required to decrease the control effort and avoid high amplitude oscillations. Finally, an appropriate system response is achievable, but it requires a high-order multivariable controller. The transfer function matrix of such a controller in the frequency domain and its complexity and feasibility are discussed and commented upon.