## 2 Methods
### Sensors and Actuators
Measurement data of the COR load case are taken for the calculation:  
\[\mathbf{O}(k)=\mathbf{S}_{xy}(k)\mathbf{S}_{xx}^{-1}(k) \tag{2}\]  
For the multi-tonal noise reduction problem, the observer matrix is evaluated at the five frequency lines corresponding to the COR frequencies. Alternative forms of \(\mathbf{O}\) having a diagonal structure or being a fully populated matrix are mentioned in Roure and Albarazzin [9]. The smart lining is equipped with two electrodynamic actuators of the type Visaton EX 45 S (10 W, 8 \(\Omega\), \(V_{max}=8.9\)\(V\), 0.06 kg) at the locations indicated in Figure 2a. The actuators are attached to the cabin-side surface of the lining with double-sided adhesive tape. This setup is suitable for the laboratory COR excitation with 113 dB SPL in front of the fuselage panel, but it is not powerful enough for a real COR load case with sound pressure levels of 130 dB [16]. The FRF from the actuators to the microphones \(\mathbf{\hat{G}}_{a}\) and the sound pressures at the microphone locations \(\mathbf{D}_{a}\) for the laboratory COR load case are used to estimate the required actuator number for the real case. Equation (3) provides the control voltage that is required to counteract the sound pressures at the microphone locations. The exclamation mark above the equal sign means that both sides should be equal:  
\[\mathbf{D}_{a}(k)\overset{!}{=}\mathbf{\hat{Y}_{a}}=\mathbf{\hat{G}}_{a}(k) \mathbf{U}(k)\rightarrow\mathbf{U}(k)=\left[\mathbf{\hat{G}}_{a}(k)^{H}\mathbf{ \hat{G}}_{a}(k)\right]^{-1}\mathbf{\hat{G}}_{a}(k)^{H}\mathbf{D}_{a}(k) \tag{3}\]The inverse Fourier transform of the control voltage spectrum \(\mathbf{U}\) containing the values from Equation (3) at the five CROR frequency lines yields the control voltage \(\mathbf{u}\). The peak amplitude of \(\mathbf{u}\) is \(0.924\,\mathrm{V}\), which is far below the maximum value of \(8.9\,\mathrm{V}\). Taking a safety factor of two, the actuator stroke margin is \(20\log_{10}(8.9/2/0.924)=13.7\) dB and a maximum CROR pressure level of \(113+13.7=126.7\) dB could be handled by the two actuators. A doubling of the actuator number from two to four provides additional \(6\) dB stroke and, hence, four Visaton EX 45 S (or similar) exciters are sufficient to ensure control authority for realistic CROR pressure levels of \(130\) dB. The locations of the two actuators are not optimized. However, an optimization study described in [PERSON] and [PERSON] [17] suggests that the chosen actuator locations are highly suitable. Due to the high damping of the structural parts and the coupled air cavity, the modal behavior of the system is weakly pronounced and the actuator optimization has a rather flat objective function. Another aspect which makes the performance of the smart lining less dependent on the actuator locations is that, unlike, for example, in [PERSON] et al. [18], the system is tuned to reduce the SPL locally in the acoustic near field instead of globally in the whole cavity. For the mentioned reasons, it is considered appropriate to position the actuators in a straightforward manner. A more elaborate actuator placement based on genetic optimization is described in [19]. Such an optimization could be used for the final implementation of a smart lining to maximise the performance for a given cabin layout.