Laboratory Test of a Vehicle Active Noise-Control System Based on an Adaptive Step Size Algorithm

Abstract

NVH (noise, vibration, and harshness) is a key factor affecting vehicle comfort. Compared with traditional sound absorption and isolation methods, active noise control (ANC) offers a significant advantage in solving the problem of low- and medium-frequency noise from road surfaces. However, the classic filtered-x least mean squares (FxLMS) algorithm is ineffective in terms of adapting to different road noises to ensure a stable noise reduction effect when facing the complex and changeable noise environment of moving vehicles. Therefore, an adaptive step size algorithm (ASSFxLMS) is proposed in this paper, which can adjust the step size according to the size of the reference signal to ensure the stability of the adaptive process. In order to improve the performance of the algorithm, a particle swarm optimization algorithm is also used to automatically adjust the parameters, so that the step size of the adaptive algorithm always maintains a relatively ideal size. The simulated pulse noise of standard SαS distribution was used as the reference signal for the simulation. The simulation results show that compared with other algorithms, the proposed algorithm under different degrees of pulse noise conditions, noise reduction stability, and noise reduction amplitude are improved. In order to further verify the feasibility of the algorithm in vehicle road noise reduction, this paper also conducted a hardware-in-the-loop noise reduction experiment in the laboratory, employing the road noise data collected by the real vehicle. Under different interior noise conditions, the proposed active noise-control system has a maximum noise reduction effect of 12 dB for low-frequency noise below 100 Hz.

1. Introduction

At present, the NVH performance of vehicles is receiving a great deal of attention, and there is an increasing emphasis on noise reduction. The car’s interior noise in low-speed driving conditions mainly comes from the car and the impact of the pavement caused by vibration, due to acoustic transfer characteristics, traditional sound absorption, and vibration damping; while noise processing, such as passive noise reduction patterns, has good attenuation in the case of high-frequency noise [1,2,3], low-frequency noise will pass more easily through the sound insulation material to the human ear. In this case, traditional passive noise reduction has been unable to meet the needs of automobile users for noise reduction performance. Because active noise reduction has a good attenuation ability for low-frequency noise, active noise control systems in vehicles have become a hot research topic. The principle of active noise reduction is to form a quiet area in the local space via the superposition of two opposite sound waves, thus achieving the effect of noise reduction [4,5].

Because active noise control has a good effect on low-frequency noise, many researchers have studied active noise reduction technology in vehicles. A. Kinoshita et al. [6] developed a prototype ANC system and tested it on a real vehicle. The results showed that the system reduced the roar by more than 10 decibels. C. Bohn et al. proposed a control method for a time-varying fundamental frequency harmonic interference system, provided a non-adaptive method, and gave the results of the real-time measurement of automotive engine vibration by using an active control system, which showed that the noise in the passenger compartment of the vehicle was reduced [7]. Takahashi Akira et al. [8] proposed a new low-frequency road noise ANC control algorithm, enabling it to be combined with an engine roar noise ANC. The newly developed system is the world’s first commercial ANC and is used in sport utility vehicles and sedan models. Ji Guang Jiang et al. [9] established an adaptive active noise control strategy for vehicle interiors based on the FxLMS algorithm. The core circuit module of an active noise controller was designed, including the noise signal amplifier module, the main control unit, the audio power amplifier module, and the power module. The hardware of the controller was developed and the control software system was written. By using this system, the low-frequency noise heard by the left ear of the co-driver can be controlled under a steady state; the noise reduction amounts are 8.5 dB and 10.2 dB, respectively. Sun Guohua et al. [10] proposed an MFxLMS algorithm, combining the threshold of the reference signal and the error signal path to control impact road noise, and established a coupled vehicle system model, based on spectral substructure technology, to simulate the pulsed internal sound caused by road roughness. Numerical simulation results showed that the ANC system can deal with internal collision noise effectively. Tao Feng et al. [11] proposed a CSFxLMS algorithm that minimizes the influence of dynamic characteristics in different reference channels by introducing self-regulating parameters in each reference signal path. The simulation results show that the performance of the CSFxLMS algorithm is significantly better than the traditional FxLMS algorithm, and a reduction of about 5 dBA in the driver’s ear position can be achieved. Liping Zhu et al. [12] studied a multi-channel vehicle-wide broadband active noise control controller and proposed a practical non-causal inversion method for non-minimum phase secondary paths. The proposed controller can be used as an accurate predictor to estimate the maximum noise reduction that can be achieved. The authors verified the proposed causal optimal controller with measured data, and the results show that the proposed algorithm is superior to traditional algorithms and achieves significant broadband noise reduction in time-invariant systems. Hui Li et al. [13] used the finite element analysis method to conduct a modal analysis of the interior space and obtained the frequency response curves of the corresponding nodes in the ear. By establishing the adaptive neural network active noise control system Simulink model, the simulation experiment was conducted. The experimental results show that the neural network active control system has a good noise reduction effect in the frequency band of 0–50 Hz, and the average noise reduction is 4.3 dB. At 86 Hz, the maximum noise reduction is 9.8 dB, and the system can control the low-frequency noise generated under road excitation.

In the process of vehicle driving, it is easy to encounter impulse noise, such as in the speed bump, when the instantaneous vibration noise of the car is suddenly increased. For impulse noise, since there is no bounded second moment, the traditional FxLMS algorithm is no longer suitable for impulse noise. Many researchers have proposed various improved adaptive algorithms for impulse noise. Richard M. Leahy et al. [14] proposed a filtered x-minimum mean P-power algorithm (FxLMP), which is based on minimizing the fractional low-order moments existing in a stable distribution. The results show that the FxLMP algorithm, where p < α, has good robustness in terms of impulse noise. Xu Sun et al. [15] proposed a simplified FxLMS algorithm. If the amplitude of the reference signal is higher than a certain value set, the reference signal that is higher than this value is set to 0. Compared with FxLMS, this algorithm has better robustness, but the convergence rate is slow. Muhammad Tahir Akhtar et al. [16] proposed an improved normalized step size algorithm (MNFxLMS) as a solution to improve the performance of the FxLMP algorithm. Peng Li et al. [17] proposed an improved FxLMM pulse noise control algorithm. The results show that the FxLMM algorithm and the improved FxLMM algorithm are more robust than the FxLMS algorithm in suppressing large amplitude pulse noise. Yali Zhou et al. [18] proposed a universal step size normalization FxLMS algorithm and normalized the step size with a Gaussian distribution function. The algorithm requires neither parameter selection nor cost function selection. The simulation and experimental results show that the proposed scheme shows good performance for SαS pulse noise attenuation. Pucha Song et al. [19] proposed a convex combination method with the FxGMN algorithm for impulse noise. The simulation results show that, compared with other existing algorithms, the C-FxGMN algorithm can achieve better convergence speed and a greater noise reduction effect under different noise input conditions. Hao Meng et al. [20] proposed a new adaptive step size algorithm by adjusting the step size factor of reference signal X(n) and error signal e(n) and obtained a better coefficient via particle swarm optimization. The simulation results show that the algorithm has a fast convergence speed and strong stability. Feihong Gu et al. [21] proposed an improved normalized step size EFxatanLMS (NSSEFxatanLMS) algorithm, which adaptively adjusts the step size coefficient to an appropriate value by using a new time-varying normalized function. The simulation results show that the algorithm is effective against both Gaussian noise and impulse noise. Feng Pengxing et al. [22] proposed an adaptive algorithm combining self-normalization and the p-norm. The self-normalization method regenerates the filtered reference signal into a nonlinear function to transform the reference signal. The simulation and the experimental results show that the proposed self-P-normalization algorithm is more effective in suppressing the update weight of traditional algorithms.

In view of the interior noise of a complex noise environment, this paper proposes an adaptive step size FxLMS algorithm; this algorithm works according to the reference signal change step length. The reference signal has a large disturbance but the algorithm still has good robustness and convergence speed; by means of the stochastic particle swarm optimization algorithm for parameter optimization, the simulation results prove its excellent performance. The effectiveness of the proposed algorithm is further verified by noise reduction experiments.

The rest of this paper is structured as follows: in the second section, an adaptive step size algorithm (AASFxLMS) is proposed, and a particle swarm optimization algorithm is introduced to adjust the step size in the adaptive algorithm. In the third section, the computer simulation for noise reduction is performed for different noise situations, and the comparison is made with other algorithms. In the fourth section, the ANC experiment is conducted to further verify the effectiveness of the algorithm. Finally, the fifth section marks the conclusion of the article.

2. Methods

2.1. Active Noise Control Structure

Compared with the feedback active noise control system, the feed-forward active noise control system can suppress a larger spectrum of noise and is more suitable for the internal environment of the vehicle. Therefore, this paper adopts the feed-forward active noise control system as the basic structure of the active noise control system. Figure 1 shows the schematic diagram of the feed-forward active noise control system.

2.2. FxLMS Algorithm

As the core content of the feed-forward active control system, the adaptive algorithm follows the traditional feedforward FxLMS algorithm system principle shown in Figure 2, where X(n) is the input reference signal, P(z) and S(z) are the transfer functions of the primary and secondary path channels, respectively, Ŝ(z) is the estimate of the secondary path, S(z), and W(z) is the update weight controller. The d(n) is the noise signal in the cancellation area, e(n) is the error signal after cancellation, and y′(n) represents the cancellation signal.

Assuming that the number of filter taps is L, the input reference signal and weight coefficient at time n are expressed as vectors:

$$X(n)=\left[\begin{array}{cccc}x(n)& x(n-1)& \dots & x(n-L+1)\end{array}\right]$$

(1)

$$W(n)=\left[\begin{array}{cccc}{w}_1(n)& w_2(n)& \dots & w_L(n)\end{array}\right]$$

(2)

y(n) is the filter output signal:

$$y(n)=W(n)X^T(n)={\displaystyle \sum _{l=1}^L{w}_l}(n)x(n-l+1)$$

(3)

y(n) passes through the secondary channel to reach the position where the sound waves cancel, and the cancellation signal y′(n) is obtained:

$$y^{\prime }(n)=S(n)\ast y(n).$$

(4)

The error signal is obtained after the offset signal and noise signal are superimposed:

$$e(n)=d(n)-y^{\prime }(n)=d(n)-s(n)\ast W(n)X^T(n)$$

(5)

where ∗ is the convolution operation. According to the stochastic gradient descent method and the least mean square error criterion, the gradient of the l tap weight of the filter can be expressed as the partial derivative of the squared error signal, with respect to the l tap weight:

$${\nabla }_l=\frac{\partial e^2(n)}{\partial w_l(n)}=2\frac{\partial e(n)}{\partial w_l(n)}e(n)=2e(n)\ast S(n)x(n-l+1)=2e(n)x(n-l+1).$$

(6)

When using the FxLMS algorithm, the filter weights are updated as follows:

$$W(n+1)=W(n)+\frac{1}{2}\mu \cdot \nabla =W(n)+\mu e(n)X^{\prime }(n)$$

(7)

where μ is the step factor and X’(n) is the tap vector of the filtered reference signal.

2.3. FxLMS Algorithm with Adaptive Step Size

When there is a large amplitude transient increase in noise, the FxLMS algorithm will make the amplitude of the filter weight coefficient change too much, due to the sudden increase in the reference signal, which will lead to the instability of the system. If a small step factor is deliberately used to prevent such instability, the convergence of the adaptive process will be particularly slow. To solve the above problems, it is necessary to adjust the step factor so that the adaptive algorithm can adapt to the sudden increase in noise signal. Therefore, this paper proposes an adaptive step size algorithm based on a reference signal to adjust step size parameters, and the variable step size factor can be adjusted as follows:

$$u(n)=a{\left(c{\Vert X(n)\Vert }^2\right)}^2{e}^{-2c{\Vert X(n)\Vert }^2}$$

(8)

where ‖X(n)‖² is the square norm of X(n), and the values of parameters a and c have a significant influence on the control of step size. The influence of the different values of a and c on the step size u(n) is shown in Figure 3:

Obviously, a and c play a crucial role in the adjustment of step size u(n), and the different values of a and c will lead to different noise reduction effects on step size. When the adaptive step-size FxLMS algorithm is used, the filter weight coefficient is updated as follows:

$$W(n+1)=W(n)+u(n)e(n)X^{\prime }(n).$$

(9)

2.4. Adaptive Step Size Algorithm, Based on Particle Swarm Optimization

As parameters a and c play an important role in the process of adjusting the step size parameters, the values of a and c are correspondingly different for noise levels of different forms and amplitudes. Therefore, the particle swarm optimization algorithm is adopted in this study to select the parameter values in detail and further improve the convergence speed of the algorithm. The principle of an iterative variable step size algorithm based on particle swarm optimization is shown in Figure 4. For the key parameters a and c, which control the step size, the particle swarm optimization algorithm is adopted to optimize them, and the optimal value is selected for the given particle position range.

We set the values of parameters a and c as particles φ = [a c], then assume that the number of particles is p and the number of iterations of the FxLMS algorithm is k. Thus, the corresponding matrix of particles is initialized as:

$${\mathsf{\phi }}_P=\left[\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\\ ⋮& ⋮\\ a_p& c_p\end{array}\right]$$

(10)

Each row of the vector φ_P represents the corresponding parameter value of a particle, and the corresponding step size is initialized as:

$$u_P(k)={\left[\begin{array}{cccc}{u}_1(k)& u_2(k)& \cdots & u_p(k)\end{array}\right]}^T.$$

(11)

The number of filter taps is L, and the weight coefficient at time k is expressed in the form of a matrix:

$$W_P(k)=\left[\begin{array}{cccc}{w}_{11}& w_{12}& \dots & w_{1L}\\ w_{21}& w_{22}& \dots & w_{2L}\\ ⋮& ⋮& \ddots & ⋮\\ w_{p1}& w_{p2}& \dots & w_{pL}\end{array}\right].$$

(12)

The weight coefficient can be written as:

$$W_P(k+1)=W_P(k)+u_P(k)e_p^T(k)X_p(k)$$

(13)

where X_p(k) is a matrix consisting of p identical X(k) vectors with p rows and L columns. The output signal of the filter is:

$$y_P(k)=W_P(k)X^T(k)={\left[\begin{array}{cccc}{y}_1(k)& y_2(k)& \cdots & y_p(k)\end{array}\right]}^T.$$

(14)

The error signal corresponding to each particle is obtained by the superposition of the offset signal and the primary channel signal:

$$e_p(k)=d_p(k)-{y^{\prime }}_p(k).$$

(15)

In this way, we can obtain the error signal corresponding to each particle. In this paper, the mean square error is adopted as the objective function of the particle swarm optimization algorithm, and the mean square error of K iterations is selected to update the position and velocity of the particle. The expression is as follows:

$$\psi =\frac{1}{K}{{\displaystyle \sum _{k=1}^K\left\{e(k)\right\}}}^2.$$

(16)

In the first r iterations of the particle swarm optimization algorithm, we will find the particle with the minimum mean square error value as the current individual optimal solution, φ_pb(r). If the mean square error of the next iteration of the particle φ(r + 1) is smaller, φ(r + 1) will replace φ_pb(r) as the individual optimal solution, φ_pb(r + 1). At this time, the particles in the whole population where the minimum MSE value is found are taken as the optimal solution of the whole population and denoted as φ_gb(r). If the particles in the next iteration particle swarm find a smaller MSE value, then φgb(r + 1) replaces φ_gb(r) as the optimal solution for the whole population. The updated speed and position can be written as:

$$v_i(r)=\omega \cdot v_i(r-1)+r_1{m}_1\left\{{\phi }_{pb}(r)-\phi (r)\right\}+r_2{m}_2\left\{{\phi }_{gb}(r)-\phi (r)\right\}$$

(17)

$${\phi }_i(r+1)={\phi }_i(r)+v_i(r).$$

(18)

In the above equation, ω is the inertia factor; the larger its value, the stronger the global optimization ability, and the weaker the local optimization ability. m₁ and m₂ are the learning factors. The velocity and position of the particles are updated in the given range. The value of the total optimal solution a_gb, output when the update times r_a reach the maximum, is used as the adjustment parameter of the step factor. The adaptive algorithm can obtain better convergence performance by using this step parameter. The step factor obtained by the particle swarm optimization algorithm is shown in Equation (19):

$$u(n)=a_{gb}{(c_{gb}\Vert X(n)\Vert )}^2{e}^{-2c_{gb}\Vert X(n)\Vert }.$$

(19)

3. Simulation Analysis

Using the simulated noise signal as the reference signal of the active control algorithm, the proposed algorithm is compared with other known active noise control algorithms to analyze the performance of the proposed active noise reduction algorithm. The primary channel P(z) and secondary channel S(z) adopted in this simulation are both from the real physical channel model in the noise suppression laboratory. This offline modeling adopts an FIR (finite impulse response) filter with a length of 128 to establish the primary channel model P(z) and secondary channel model S(z). Figure 5 shows the impact response curves of the primary channel and secondary channel.

Three kinds of analog signals, as adopted by the authors of [20], are used as reference signals, namely, mild impulsive noise (α = 1.7), medium impulsive noise (α = 1.5), and highly impulsive noise (α = 1.3). The average noise reduction (ANR) can be used to observe the performance of the studied algorithm more intuitively. The average noise reduction [23] is defined as follows:

$$ANR(n)=20{\log }_{10}\left(\frac{A_e(n)}{A_d(n)}\right)$$

(20)

$$A_e(n)=\lambda A_e(n-1)+(1-\lambda )\left|\mathrm{e}(n)\right|$$

(21)

$$A_d(n)=\lambda A_d(n-1)+(1-\lambda )\left|\mathrm{d}(n)\right|.$$

(22)

λ is a coefficient with a value range of (0, 1). The closer λ is to 1, the smoother the average noise reduction curve will be. For noise with a large amplitude change, the value of coefficient λ should be as large as possible; in this simulation, λ values are 0.99.

Example 1: The standard SαS distribution pulse noise is used as the reference signal, its characteristic index being α = 1.7. For the proposed AASFxLMSPSO algorithm, different particle numbers have a significant impact on the performance and complexity of the algorithm. Given particle movement ranges of a_limt = [0.001, 0.01] and c_limt = [0.02, 0.1], MATLAB software was used to simulate the noise reduction of the AASFxLMSPSO algorithm with different particle numbers, and the influence of the different particle numbers on the performance of the algorithm was compared. Figure 6 shows the simulation result. It can be seen from the figure that the particle number has little influence on the noise reduction performance when p = 5, p = 20, and p = 50. In the scenario where increasing the number of particles would greatly increase the algorithm complexity, the number of particles was set as p = 5 for subsequent analysis.

Then, using the α = 1.7 pulse noise as a reference signal, several different adaptive algorithms were simulated to verify the performance of the proposed algorithms. As shown in Figure 7, when the reference signal of FxLMS increases, the convergence rate of the FxLMS algorithm will increase over a short time, and the overall convergence rate is very slow. The MNFxLMS algorithm declines steadily, and the convergence rate is slow. The convergence speed and stability of the NASFSxLMS algorithm and NASFSxLMSPSO algorithm demonstrated a good performance compared with the previous two algorithms. The proposed AASFxLMS algorithm showed better performance in terms of convergence speed and stability. After particle swarm optimization, the noise reduction performance of the AASFxLMSPSO algorithm was further improved, and the noise reduction effect of 20 dB was achieved in about 2000 iterations.

Table 1. The key parameters of each algorithm in the MATLAB simulation of Example 1.

FxLMS	u = 0.00005
MNFxLMS	p = 1.59, µ = 2.5 × 10−3
NASFSxLMS	ρ = 0.25, γ = 1.6
NASFSxLMSPSO	ρ = 0.2514, γ = 1.3826, ω = 0.6, C1 = 2, C2 = 2, p =50, δ = 0.000001, m = 10, G = 100
AASFxLMS	a = 0.001, c = 0.05
AASFxLMSPSO	alimt = [0.0005, 0.002], alimt = [0.04, 0.08], K = 1000, ω = 0.6, m1 = 2, m2 = 2, p =5, r = 10

shows the key parameters of the simulation process.

In Example 2, an SαS pulse noise where α = 1.5 is used as the reference signal. The noise reduction effect is shown in Figure 8. The FxLMS signal, which originally converged slowly, decreased rapidly when the reference signal increased. However, the FxLMS algorithm became unstable when it encountered a large impact. The MNFxLMS algorithm is stable, but the convergence speed is slow. The convergence speed and stability of the NASFSxLMS algorithm and NASFSxLMSPSO algorithm both demonstrated good performance. However, the performance of the proposed AASFxLMS algorithm had significant advantages; the convergence speed of the proposed algorithm after particle swarm optimization was further improved, and the noise reduction effect of 20 dB was reached when the number of iterations reached 4000 times.

Table 2. The key parameters of each algorithm in the MATLAB simulation of Example 2.

FxLMS	u = 0.00005
MNFxLMS	p = 1.59, µ = 2.0 × 10−3
NASFSxLMS	ρ = 0.3, γ = 1.2
NASFSxLMSPSO	ρ = 0.4524, γ = 1.1458, ω = 0.6, C1 = 2, C2 = 2, p =50, m = 10, G = 100
AASFxLMS	a = 0.0005, c = 0.05
AASFxLMSPSO	alimt = [0.0001, 0.001], climt = [0.02, 0.06], K = 1000, ω = 0.6, m1 = 2, m2 = 2, p =5, r = 10

shows the key parameters of the simulation process.

Example 3: SαS pulse noise with α = 1.3 is used as the reference signal. The noise reduction effect is shown in Figure 9. With the further increase of impact amplitude, the FxLMS algorithm becomes extremely unstable and diverges when the reference signal suddenly increases. The MNFxLMS algorithm is stable when the signal suddenly increases, but the convergence rate is slow. NASFSxLMS algorithm convergence speed is slow, NASFSxLMSPSO algorithm convergence speed is relatively fast. The proposed AASFxLMS algorithm has a fast convergence speed and remains stable in the face of the sudden increase of reference signal noise reduction performance. After particle swarm optimization, the convergence speed and stability of AASFxLMSPSO are further improved.

Table 3. The key parameters of each algorithm in MATLAB simulation of Example 3.

FxLMS	u = 0.00005
MNFxLMS	p = 1.59, µ = 2.0 × 10−3
NASFSxLMS	ρ = 0.3, γ = 1.2
NASFSxLMSPSO	ρ = 0.3843, γ = 1.2868, ω = 0.6, C1 = 2, C2 = 2, p =50, m = 10, G = 100
AASFxLMS	a = 0.0005, c = 0.05
AASFxLMSPSO	alimt = [0.0001, 0.001], climt = [0.02, 0.04], K = 1000, ω = 0.6, m1 = 2, m2 = 2, p =5, r = 10

shows the key parameters of the simulation process.

By comparing the simulated SαS pulse noise of three different characteristic indices, the proposed algorithms all showed better noise reduction performance under different impact amplitudes, which verifies the performance of the proposed algorithms.

4. Laboratory Experiment

4.1. Vehicle Internal Noise Acquisition

In order to better verify the noise reduction performance of the active noise control system proposed in this paper in terms of the internal noise of the vehicle, the actual noise that the driver can hear during the driving of the car should be collected before the ANC experiment. In this experiment, a Volkswagen Lavida 182CP2 car was used and an iec711 318-4 artificial ear was fixed in the co-driver headrest, then noise signals under different working conditions were collected. Figure 10 shows the positioning relationship of the artificial ear in the headrest of the passenger seat. With the window closed, four kinds of noise signals were collected under different working conditions, respectively, 30 km/h, 40 km/h, 50 km/h, and 60 km/h at uniform speeds, among which the first three working conditions incorporated a speed bump. Figure 11 shows the noise signal collected by the artificial ear closest to the window.

4.2. The ANC Experiments

In order to further verify the performance of the AASFxLMS algorithm proposed in this paper, an ANC experiment was conducted in an anechoic chamber via an ANC experiment. The proposed ANC system is implemented using MATLAB and Simulink, together with dSPACE. On the basis of Simulink modeling, a TLC code is used to compile the Simulink model into the C programming language and generate the model. The model is read by DspaceControldesk, so that the ANC noise reduction process can be completed. The experimental principle is shown in Figure 12. The experimental equipment used in this experiment was mainly a dspace1401 board, with parameters of 8 Ω, a 30 W speaker, an iec711 318-4 artificial ear, a B&K1704 pre-amplifier power supply, and a computer. Through the primary loudspeaker used to play different road noises, the iec711 318-4 artificial ear measured the before and after noise reduction frequency response and the corresponding time domain curve. The hardware-in-the-loop experimental platform, built according to ANC experimental principles, is shown in Figure 13.

In this experiment, the signals collected on one side of the artificial ear were used as error signals, and the signals collected by the artificial ear before and after noise reduction were compared. The traditional FxLMS algorithm was compared with the proposed AASFxLMS algorithm. The duration of the primary noise played by the primary speaker was 15 s. Due to the short lag of dSPACE measurement data, the data from the first few seconds of dSPACE operation cannot be observed. The error signal collected at this time was timed from the 6th second until the noise reduction process was completed, and a total of 10 s of noise reduction time was recorded. The frequency resolution of the frequency response graph was 0.1. In addition, the L_att of noise attenuation in reference [24] is adopted as the overall noise reduction amount, which is defined as:

$$L_{\mathrm{att}}=-10{\log }_{10}[\mathrm{Tr}(S_{ee})/\mathrm{Tr}(S_{dd})]$$

(23)

where Tr (∙) is the trace of the matrix. S_dd and S_ee are the power spectral density matrix of the signal before and after noise reduction, respectively.

Working condition 1: The car runs at a constant speed of 30 km/h and passes over the speed bump. The experimental results of the vehicle’s internal noise, used as a reference signal under the first working condition, are shown in Figure 14. As can be seen from the time-domain response diagram, the proposed algorithm has a certain suppression effect on noise, and it also has a good suppression effect when encountering a sudden increase in signal, while the FxLMS algorithm has almost no suppression effect on noise. According to the frequency response diagram, the proposed algorithm has an obvious noise reduction effect when the frequency range is 20–100 Hz, the noise reduction amplitude is about 3–10 dB, and the noise reduction effect is 10 dB, near 45 Hz. The overall noise reduction is 3.1 dB, in the frequency range of 20–400 Hz. The FxLMS algorithm has a noise reduction effect of 1–2 dB in the 20–100 Hz frequency band but has a noise enhancement effect when the frequency is above 100 Hz. The overall noise reduction is 0.3 dB in the frequency range of 20–400 Hz.

Working condition 2: The car runs at a constant speed of 40 km/h and passes over the speed bump. The experimental results of the vehicle’s internal noise as a reference signal in the second working condition are shown in Figure 15. The time-domain response diagram shows that the proposed algorithm significantly reduces the amplitude of sound pressure after noise reduction, but still shows obvious suppression when the noise signal suddenly increases, and the amplitude of the FxLMS algorithm does not change significantly before and after noise reduction. It can be seen from the frequency response diagram that the proposed algorithm has an obvious noise reduction effect in the frequency range of 30–135 Hz, and the noise reduction range is about 3–8 dB, among which the maximum noise reduction effect is near 50 Hz. The overall noise reduction is 2.8 dB in the frequency range of 20–400 Hz. The FxLMS algorithm has a noise reduction effect of 1–2 dB in the range of 30–90 Hz, but the noise is enhanced in the frequency band above 90 Hz. The overall noise reduction is 0.1 dB in the frequency range of 20–400 Hz.

Working condition 3: The car runs at a constant speed of 50 km/h and passes over the speed bump. The experimental results of the vehicle’s internal noise as a reference signal in the second working condition are shown in Figure 16. The time-domain response diagram shows that the proposed algorithm significantly reduces the amplitude of sound pressure after noise reduction, which has a good inhibition effect on impact noise. The amplitude of the FxLMS algorithm does not change significantly before and after noise reduction. It can be seen from the frequency response diagram that the proposed algorithm has an obvious noise reduction effect in the frequency range of 25–140 Hz, and the noise reduction range is about 3–12 dB, among which the maximum noise reduction effect is near 40 Hz. The overall noise reduction is 3.1 dB, in the frequency range of 20–400 Hz. The FxLMS algorithm has a noise reduction effect of 1–2 dB in the range of 30–80 Hz, but the noise is greatly enhanced in the frequency band above 80 Hz. The overall noise reduction is −0.8 dB in the frequency range of 20–400 Hz.

Working condition 4: The car is driving at a constant speed of 60 km/h. The experimental results of the vehicle’s internal noise as a reference signal in the fourth working condition are shown in Figure 17. The time-domain response diagram shows that the proposed algorithm significantly reduces the amplitude of sound pressure after noise reduction, while the amplitude of the FxLMS algorithm does not change significantly before and after noise reduction. It can be seen from the frequency response diagram that the proposed algorithm has an obvious noise reduction effect in the frequency range of 20–130 Hz, and the noise reduction range is about 3–12 dB, among which the maximum noise reduction effect is near 40 Hz. The overall noise reduction is 3.1 dB in the frequency range of 20–400 Hz. The FxLMS algorithm has a 1 dB noise reduction effect in the range of 40–50 Hz, but the noise is enhanced in the range of 120–140 Hz. The overall noise reduction is −0.3 dB in the frequency range of 20–400 Hz.

5. Discussion and Conclusions

In this paper, an adaptive step size FxLMS algorithm (AASFxLMS) and an adaptive step size FxLMS algorithm (AASFxLMSPSO) with particle swarm optimization are proposed. Through mathematical modeling, simulations, and experiments, the following conclusions are drawn:

(1)

The AASFxLMS algorithm proposed in this paper adjusts the step size according to the size of the reference signal, which can effectively cope with the impact noise signal and further improve its performance via particle swarm optimization.

(2)

The MATLAB simulation results show that, compared with the FxLMS algorithm, MNFxLMS algorithm, NASFSxLMS algorithm, and NASFSxLMSPSO algorithm, the proposed AASFxLMS algorithm and AASFxLMSPSO algorithm have better noise reduction performance. In the case of faster convergence, the steady-state error is lower.

(3)

As shown by the collection of the interior noise signal and the hardware-in-the-loop test in the noise suppression room, the noise reduction amplitude can reach 12 dB, which verifies the reliability and effectiveness of the proposed algorithm in terms of interior noise active control.

(4)

In this paper, the noise reduction effect of the proposed algorithm is verified only in the laboratory. The most necessary future development is a test conducted in a car running over speed bumps.

(5)

In addition to the classical particle swarm optimization algorithm to optimize parameters a and c, the process can also be realized by using a neural network algorithm, genetic algorithm, ant colony algorithm, or bee colony algorithm.

(6)

As another future development direction, the potential applications of the in-vehicle active noise control system proposed in this paper can also include aircraft, ships, offices, and other indoor application scenarios.