## 2 Methods
### Simulation  
Figure 4 shows the scheme of the adaptive controller. The observer \(\mathbf{O}\) from Equation (2) is located in the upper left and the adaptive controller \(\mathbf{W}\) with least mean squares (LMS) algorithm in the lower left part. The right part of Figure 4 defines the output signals used for control performance evaluation. Estimated systems or signals are marked by a hat above their symbol to distinguish them from the true values. The simulation inputs \(\mathbf{d}_{s}\), \(x\) and \(\mathbf{d}_{a}\) (blocks highlighted in green) are measured in the laboratory with the CROR load case applied. The vector \(\mathbf{d}_{s}\) contains the signals of the remote sensors (accelerometers). The reference signal is denoted by \(x\). The vector \(\mathbf{d}_{a}\) contains the signals of \(20\) microphones measured at three distances from the lining (\(60\) time series in total). The block \(t\to F\) transforms the input signal of dimension \(N\) into an output signal of dimension \(N\times R\). In this block, the input signals are filtered through \(R\) parallel bandpass filters. The center frequencies of the pass-band of the bandpass filters correspond to the CROR frequencies. Here, with \(R=5\), the input dimension is multiplied by five. Each output element of the block \(t\to F\) is a complex tonal signal (phasor). Hence, the scalar \(x\) is transformed to the vector \(\mathbf{X}\) which contains the five complex reference signals \(X^{r}\) and the vector \(\mathbf{d}_{s}\) is mapped to the vector \(\mathbf{D}_{s}\) containing five sub-vectors of the complex structural disturbances \(\mathbf{D}_{s}^{r}\). Further details of the block \(t\to F\) are published in Misol ([6], Figure 6). The error signal vector \(\mathbf{e}_{a}\) is obtained from subtracting the estimated secondary pressure (anti-sound) signals \(\hat{\mathbf{y}}_{af}\) from the measured disturbance pressure signals \(\mathbf{d}_{a}\). This is true under the assumption that the estimated FRF matrix \(\hat{\mathbf{G}}_{af}\) matches the true (unknown) FRF matrix \(\mathbf{G}_{af}\). \(\hat{\mathbf{G}}_{af}\) describes the full FRF matrix from all actuators to all \(60\) virtual microphones and \(\hat{\mathbf{G}}_{a}\) describes the reduced FRF matrix from all actuators to the virtual error microphones. Furthermore, the actuator feedback on the remote sensors described by \(\mathbf{G}_{s}\) is assumed to be perfectly compensated inside the adaptive controller by an ideal plant model \(\hat{\mathbf{G}}_{s}\) (feedback compensation). Therefore, the blocks describing this feedback effect are omitted in the control scheme. Thus, the simulation results described in the next section must be considered as an upper limit for the SPL reduction which is possible for the system under consideration. In reality, the performance of the controller will be compromised due to imperfect plant models. A robustness study based on plant uncertainties and variances is required to assess the implications of the idealizing assumption. Such a study is beyond the scope of this work and remains a future task. The adaptive controller is implemented in the frequency domain. This is considered reasonable for a multi-tonal case. The applied complex filtered-x LMS algorithm is based on [PERSON] et al. [20]. In contrast to [PERSON] et al., the transformation of the input signals into the frequency domain is realized by the block \(t\to F\) as described above. By this kind of implementation, the signal delay associated with FFT analysis is avoided. This could be beneficial in case of non-stationary conditions as in climb or descent of an aircraft. It is beyond the scope of this paper to assess the benefits and drawbacks of the proposed method. Each actuator is controlled by one complex filter weight per frequency which makes the implementation of the adaptive finite impulse response (FIR) filter very efficient.