## 2 Methods
### Sensors and Actuators  
The sensors and the actuators of the smart lining are defined using the measurement data of the two load cases. Since the CROR load case is deterministic, one remote sensor is sufficient to  
Figure 3: Locations of the virtual microphones on the planes \(\alpha\) and \(\beta\). Figure 2: Experimental setup in the transmission loss facility seen from the semi anechoic room. The locations of the actuators and the accelerometers are indicated in (a) and the microphone locations are indicated in (b). The accelerometers are indicated in blue, the actuators in red and the microphones in green. The accelerometers finally chosen as remote sensors are 2, 3, 7, 12, 14 and 15 (blue backed boxes). The microphones are positioned \(z=-0.5\) m, \(z=-1\) m or \(z=-1.5\) m in front of the lining along two vertical lines at \(x=0.4\) m and \(x=0.84\) m spanning two planes used for control performance evaluation. estimate the sound pressures at all microphone locations. However, the use of a single remote sensor is considered undesirable with regard to robustness because its failure would lock down the whole system. Furthermore, a larger number of remote sensors is needed to achieve a high signal coherence for the broadband load case. The multiple coherence function is used as a performance metric for the optimization of the number and the positions of the remote sensors. Equation (1) is applicable for systems with multiple inputs and a single output. Here we have up to 16 inputs from the remote sensors and one output from the 60 virtual microphones. The data from the broadband load case are taken for calculation. In Equation (1), \(\mathbf{x}\) is the vector of remote sensor (accelerometer) signals and \(y\) is the sound pressure at one of the 60 microphone positions. The spectra of these signals are denoted by \(\mathbf{X}\) and \(Y\):  
\[\bar{C}=\frac{1}{n_{2}-n_{1}+1}\sum_{k=n_{1}}^{n_{2}}\frac{\mathbf{S}_{xy}(k) \mathbf{S}_{xx}^{-1}(k)\mathbf{S}_{xy}^{H}(k)}{S_{yy}(k)} \tag{1}\]  
The multiple coherence function, which is a frequency dependent real number between zero and one, is averaged twice. The first average is taken over the discrete frequency \(k\) leading to \(\bar{C}\) as given by Equation (1) and the second average is taken over the 60 microphone positions leading to \(\bar{C}_{avg}\). [PERSON] et al. ([15], Equation (5)) propose a global coherence function that could be used for the calculation of a weighted average of the 60 multiple coherence values. The proposed scaling of the coherence with the corresponding autospectrum of the output signal implies that microphone locations associated with a higher autospectrum have a larger impact on the average coherence value. This means that locations closer to the lining surface generally would have a stronger influence on the optimization outcome. Here, the virtual microphone locations are not optimized and all signals are treated equally. The considered frequency bandwidth for Equation (1) is 50-500 Hz. The power spectral density (PSD) estimates \(\mathbf{S}_{xy}=E\{\mathbf{Y}\mathbf{X}^{H}\}\), \(\mathbf{S}_{xx}=E\{\mathbf{X}\mathbf{X}^{H}\}\) and \(S_{yy}=E\{YY^{H}\}\) are calculated with 79 spectral averages. A threshold value of \(\bar{C}_{avg}=0.8\) is considered a reasonable compromise between system complexity and prediction quality. At least six accelerometers are required to reach this value. This conclusion was drawn from initial calculation of \(\bar{C}\) for different heuristically chosen sensor combinations. The best combination of six sensors out of 16 is found by combinatorics. The complete optimization run requires \(16!/(10!6! )=8008\) calculations of \(\bar{C}_{avg}\). The best combination of six accelerometers has an average mean coherence of \(\bar{C}_{avg}=0.8021\) compared to the worst combination with \(\bar{C}_{avg}=0.7488\). According to Equation (2), the observer filter \(\mathbf{O}\) is defined as the FRF matrix of the remote sensor signals \(\mathbf{x}\) as inputs and the microphone signals \(\mathbf{y}\) as outputs.