# A Four-Stage Method for Active Control with Online Feedback Path Modelling Using Control Signal
## 3 Simulations
The simulation parameters used for the proposed method are: \(\mu_{\mathrm{w}}=0.0001\), \(\mu_{\mathrm{f}}=0.03\), \(\mu_{\mathrm{l}}=0.01\), \(\delta=\varepsilon=0.00001\), and \(\beta=60\). For [PERSON]'s method \(\mu_{\mathrm{w}}=0.0001\), \(\mu_{\mathrm{f}}=0.001\), and \(\mu_{\mathrm{h}}=0.0001\). The step sizes are chosen by trial and error to make the system stable yet with the fastest convergence speed. Figure 6a depicts the modelling error obtained for the whole duration of control operation. One can notice that the proposed method maintains constant modelling error for the first \(50\) s because the feedback path is modelled before the control operation. After the detection of feedback path change at the \(50\)th second, the controller update is ceased, K\({}_{1}\) and K\({}_{3}\) are turned on and the other switches are turned off. The feedback path modelling is carried out from the \(50\)th second to the \(70\)th second, and reasonably accurate feedback path is obtained using the decorrelation filters. The feedback path remodelling in the fourth stage takes longer time compared to the feedback path modelling in the second stage because of the presence of the control filter, whose frequency response is not flat in general. The control operation is resumed after the \(70\)th second, K\({}_{1}\) and K\({}_{2}\) are turned on, the other switches are turned off and thus the modelling error is constant. But for [PERSON]'s method, both the control operation and feedback path modelling occur simultaneously, resulting in a continuous variation of modelling error. It is important to note that the proposed method with the decorrelation filters obtained approximately \(11\) dB less modelling error than the one without using the decorrelation filters and approximately \(6\) dB less modelling error than [PERSON]'s method. The learning curves for the control operation are depicted in Figure 6b, which shows the superiority of the proposed method with the decorrelation filters before and after the feedback path change. For the  
Figure 6: (a) Feedback path modelling error, (b) learning curve for the whole duration of control operation. first 50 s, the proposed method achieves faster convergence and lower steady-state residual error. The convergence of [PERSON]'s method is slow due to the fact that the control operation and feedback path modelling process interfere with each other. An inaccurate feedback path estimation filter leads to increased interferences in control operation, i.e., the presence of feedback component in reference signal affects the active control performance. After the 50 th second, the feedback path is changed and, therefore, the residual error increases sharply. From the 50 th second to the 70 th second, feedback path modelling is carried out neutralizing the feedback component, and hence the magnitude of the feedback compensated input signal to the control filer is reduced, consequently the error level reduces due to the existence of the fixed control filter. After the 70 th second, the control operation is resumed and the noise is further reduced. The improvement in control performance in the proposed method is attributed to the inclusion of the decorrelation filters to address the issue of biased estimation and the decoupling between the control operation and feedback path modelling. Using decorrelation filters in the proposed method increases the computational load, which include two identical decorrelation filtering, the coefficient update of the decorrelation filter and obtaining the prewhitening signal, but it has less computational burden compared to [PERSON]'s method using auxiliary noise (see Figure 5 of [10]), which uses an adaptive noise cancellation (ADNC) filter of length \(L_{\text{h}}\) and two separate filters for FBPM and FBPN. Furthermore, it is to be noted that the feedback path modelling and the control operation are not carried out simultaneously, leading to a reduced computational load.