Noise in a dynamic system is practically unavoidable. Today, such noise is commonly reduced using an active noise control (ANC) system with the filtered-x least mean square (FXLMS) algorithm. However, the performance of the ANC system with FXLMS algorithm is significantly impaired in nonlinear systems. Therefore, this paper develops an efficient nonlinear adaptive feedback neural controller (NAFNC) to eliminate narrowband noise for both linear and nonlinear ANC systems. The proposed controller is implemented to update its coefficients without prior offline training by neural network. Hence, the proposed method has rapid convergence rate as confirmed by simulation results. The proposed work also analyzes the stability and convergence of the proposed algorithm. Simulation results verify the effectiveness of the proposed method.
Noise is everywhere and affects us constantly. The usual approach to reduce noise uses sound-proof or sound-absorbing materials, called passive noise reduction, having the disadvantage of requiring bulky materials and being ineffective in cancelling low-frequency noise. The active noise control (ANC) method, introduced by Lueg in 1936 [
In recent years, several ANC approaches have been applied to vibration and noise reduction for vehicles. Gäbel et al. proposed a multichannel active control system to reduce vehicle interior noise on the road. The proposed algorithm reduced the level of structural vibration and vehicle interior noise under the operating conditions [
Usually, the ANC systems can be divided into two categories, the broadband and the narrowband ANC systems. The narrowband ANC (NANC) system includes two methods: feedforward and feedback ways [
Consequently, many studies proposed nonlinear ANC designs. The adaptive recursive Volterra controller and adaptive bilinear controller have been used with nonlinear systems [
Based on the above analysis, the FXLMS algorithm-based adaptive neural controller can update parameters online without offline training in advance. However, the feedforward neural controller approach is less effective than the feedback system [ The parameters of the NAFNC are updated online using the FXLMS algorithm The convergence and stability of the proposed method are ensured by analyzing the Lyapunov function The obtained results of the simulations were compared with the works in [ Simulation results verify the effectiveness of the proposed method in both linear and nonlinear ANC systems
This paper is organized as follows. Section
This section analyzes the algorithm of the conventional methods in ANC systems.
The ANC system with the FXLMS algorithm must estimate the secondary path to update the coefficients of the adaptive filter. Accordingly, the secondary path
The offline model.
Applying the convergence conditions of the mean square error (MSE) [
Figure
Conventional parallel form NANC model.
The filtered signal is
The FXLMS algorithm for updating the weights is
The feedback-based ANC model is shown in Figure
Narrowband feedback ANC model.
The antinoise signal
Based on the feedback ANC system, the adaptive neural network approach including input layer, hidden layer, and output layer was presented in [
Structure of previous controller in [
This section analyzes the NAFNC algorithm and considers its convergence and stability properties.
Firstly, using the parallel NANC structure, the proposed method is presented in Figure
The proposed method.
The error signal is given by the equation:
Figure
Structure of the presented NAFNC.
Layer 1:
Layer 2:
And the activation function at the
Substituting equation (
The output of layer 3 is
With
Therefore, we have
Comparing Figures
The NAFNC is designed with two FXLMS algorithms for rapid and stable convergence. The weight vectors
Therefore,
Substituting equation (
Equations (
Simulations are used to compare the performance of the proposed work with the previously presented method in [
The primary path
Error and noise signals in frequency domain, showing the error signals in the CNFANC method (blue), the previous method in [
Learning curve of MSE for the CNFANC method (blue), the previous method in [
In case 2, the secondary path remains as in case 1, but the nonlinear primary path is considered [
Noise and error signals in time domain, error signals in the CNFANC method (blue), the previous method in [
Learning curve of MSE for the CNFANC method (blue), the previous method in [
Case 3 considers the ANC system involving both nonlinear primary and secondary paths. The frequencies of the undesired narrowband noises are changed abruptly to see the responsiveness of the proposed method too. The nonlinear primary path has been modelled as a cascade of a linear filter and a nonlinear one. The former is defined as
Noise and error signals in time domain, error signals obtained using the CNFANC method (blue), the previous method in [
Learning curves of the MSE for the CNFANC method (blue), the previous method in [
These experiments confirm the noise reduction efficiency of the proposed method. The proposed method is effective not only for linear ANC systems but also for nonlinear ANC systems. In contrast, the CNFANC method is effective only for linear ANC systems, as in case 1, but not for nonlinear ANC systems as in cases 2 and 3. Moreover, the proposed method offers improved convergence speed and performance owing to the use of the adjustment parameter
This paper analyzed the NAFNC algorithm and considered its stability and convergence. Simulation results confirmed the superiority of the proposed method based on three simulations. Although the proposed method costed computation, it performed well for both linear and nonlinear conditions. In addition, the advantages of the proposed method included the fast and stable convergence. The addition of the feedback element in the hidden layer in the proposed method improved the system performance.
The data used to support the findings of this study are available within the article.
The authors declare that they have no conflicts of interest.