## 3 Active Control Design
### Numerical Simulations  
With the aim of analyzing the effects of the control on the nonlinear system, a first analysis is carried out for two values of \(\mu\) to show the impact of the linear and nonlinear contributions of the control vector in the dynamics of the closed-loop system. Based on the previous results without control presented in Section 2.2, the two specific following cases are chosen :  
* Case 1 for \(\mu=0.4\) with only one unstable mode detected by CEA and the generation of self-sustaining periodic vibrational responses;
* Case 2 for \(\mu=0.7\) with two unstable modes detected by CEA and a self-sustaining quasi-periodic motion. The impact of the choices of control gain is then studied using four different values of control gain \(\gamma\). Finally, the robustness and the limits of the controller depending on the noise-to-signal ratio is examined using four levels of signal-to-noise ratio to study the impact of measurement noise or poor boundary conditions on the dynamics of the controlled system. 3.1 Contributions of Linear and Nonlinear Parts in the Control Vector for Different Values of \(\mu\)  
The nonlinear responses and the associated limit cycles of the system with nonlinear control with \(\gamma=-0.1\) are presented in Figure 5a,b for \(\mu=0.4\) (i.e., the case with only one unstable pole) and in Figure 6a,b for \(\mu=0.7\) (i.e., the case with two unstable poles). In this first numerical study, \(\gamma\) is just chosen to stabilize unstable poles without adding further damping. In the two cases, the control allows to stabilize the unstable poles (see Figures 5b and 6b) and to decrease the level of the four degrees-of-freedom. Thus, the effectiveness of a controller to drastically reduce periodic or quasi-periodic self-sustained vibrations is demonstrated without any ambiguity. For the two cases proposed in this study, it clearly appears that the regime of rise in divergence of the oscillations does not have time to set up: the controller allows to stay with self-excited vibrations of very low level. The control vector applied on \(X_{1}\), \(Y_{1}\), \(X_{2}\) and \(Y_{2}\) and the associated poles chart are presented in Figure 7a,b for \(\mu=0.4\) and in Figure 8a,b for \(\mu=0.7\). The linear and nonlinear contributions of the control vector are presented in Figure 9a,b for \(\mu=0.4\) and in Figure 10a,b for \(\mu=0.7\). As a reminder, the linear part of the control vector can be written  
\[\mathbf{u}_{lin}=\mathbf{B}_{u}^{-1}\,\tilde{\mathbf{u}} \tag{16}\]  
and the nonlinear part can be written  
\[\mathbf{u}_{nonlin}=-\mathbf{B}_{u}^{-1}\,\mathbf{F}_{\mathbf{n} \mathbf{l}}(\mathbf{X}) \tag{17}\]  
These results show that the contribution of the nonlinear part of the control vector is negligible with respect to the linear part. This observation can be explained by the decrease of the amplitude of the system response when unstable poles are stabilized making the nonlinear force negligible. Moreover, as the control is started at t = 0 s, nonlinear phenomena do not have the time to affect drastically the response of the system and the controller is designed close to the the equilibrium point. Consequently, a simple linear control may be successfully applied on the nonlinear system. It is sufficient and as effective as a nonlinear control to reduce vibration amplitudes due to friction-induced vibration. In addition it will have the advantage from a practical point of view of being more easily implemented for an engineering problem in order to mitigate friction-induced vibration and noise such as squeal. Figure 5: Nonlinear controlled response with \(\gamma=-0.1\) and \(\mu=0.4\). (**a**) Displacements. (**b**) Limit cycles. Figure 6: Nonlinear controlled response with \(\gamma=-0.1\) and \(\mu=0.7\). (**a**) Displacements. (**b**) Limit cycles. Figure 7: Control vector **u** applied to the displacements and poles chart with \(\gamma=-0.1\) and \(\mu=0.4\). (**a**) Control vector. (**b**) Poles chart (+ uncontrolled poles; \(\circ\) controlled poles).