# A Four-Stage Method for Active Control with Online Feedback Path Modelling Using Control Signal
## 1 Introduction  
In feedforward active noise control (ANC) systems, the control sound produced by the secondary source might propagate to the reference microphone, resulting in a contaminated reference signal. This acoustic feedback deteriorates the control performance and sometimes causes an instability issue [1; 2]. Several approaches have been proposed to overcome this issue. Directional microphones and loudspeakers can be used to avoid the acoustic feedback [3; 4]. However, the directivity is poor in the low-frequency range and the performance is limited. The use of non-acoustic sensors to obtain the reference signal avoids the acoustic feedback, but the existing one is only applicable for tonal noise control [5]. Signal processing techniques have been used to overcome the feedback issues. The infinite impulse response (IIR) based adaptive feedback neutralization methods have been proposed, but are constrained by the local minimum solution and stability issue [6; 7]. To address the stability issue of the IIR adaptive filtering, the lattice form IIR filter is introduced [8]. The most popular method for solving the feedback problem is the finite impulse response (FIR)-based feedback neutralization, where a fixed neutralization filter is used to subtract the feedback signal from the reference signal. In some applications, the feedback path is time varying, so online feedback path modelling is needed. Auxiliary noise generated by a white noise generator can be injected into the system for online feedback path modelling, but the injected signal is mixed with the residual noise and deteriorates the noise reduction performance [9]. A three adaptive filter-based method has been proposed in [10] to control both the predictable and broadband noise with online feedback path modelling using auxiliary noise, in which an adaptive control filter is used to update the control coefficients, a feedback path modelling filter is used to compensate the acoustic feedback, and the third filter is used to removethe disturbance during the modelling of the feedback path. Nevertheless, the method in [10] uses separate filters for feedback path modelling (FBPM) and feedback path neutralization (FBPN), which in turn increases the computational burden. A computationally efficient feedback path modelling and neutralization (FBPMN) method has been proposed for multichannel active noise control systems by combining the FBPM and FBPN filters into a single FBPMN filter [11]. A variable step size technique has been used for updating the coefficients of the FBPMN filter, which improves the performance of the online feedback path modelling [12]. However, in all the above methods, the injected auxiliary noise affects the noise reduction performance. A power scheduling strategy was imposed for the auxiliary noise used for modelling the feedback path to meet the conflicting requirement of faster convergence of feedback path modelling and lower steady-state residual error [13; 14]. A robust variable step size method for feedback path modelling and neutralization in a single channel narrowband system is proposed to achieve faster convergence [15]. Recently, a feedback path neutralization filter is used to compensate feedback and a time-varying gain is incorporated for generating the auxiliary noise [16]. Incorporating frequency weighted penalties, a robust controller can be designed with a fixed feedback neutralization filter to control noise in a ventilation duct system [17]. To avoid using auxiliary sound for feedback path modelling, in this paper we propose to use the control signal for online feedback path modelling and neutralization. A systematic method is proposed, which includes controller initialization, feedback path estimation, active control operation, and feedback path changing detection for maintaining the control operation. In the online feedback path modelling, de-correlation filters are used to increase the convergence rate and reduce the bias of the feedback path modelling.