Multi-Objective Optimal Control Method for Interior Vehicle Personal Sound Zone Generation Free

Through the reasonable arrangement of a speaker array in the car cabin, the personal sound zone (PSZ) system meets different people's listening requirements without causing interference to others. When the speaker arrangement and external interference remain unchanged, the control algorithm determines the effect of PSZ. At present, many PSZ control algorithms have been proposed. For example, the acoustic contrast control (ACC) [1], the pressure matching (PM) [2], the ACC–PM fusion algorithm [3], the variable span trade-off filters [4–6], the mode matching method [7], and the amplitude matching method [8]. To evaluate the effectiveness of the PSZ system, most scholars use acoustic contrast (AC) to indicate the energy difference between bright and dark zones, signal distortion (SD) to indicate the error between the reproduced sound field and the desired sound field, and array effort (AE) to indicate the efficiency of the sound field reproduction system [9,10].

Many methods for solving the balancing of two indicators have been developed. For example, the proposed least-squares method [11] optimizes the SD while constraining the AC. In addition, Hu et al. [12] proposed a constraint optimization method to optimize the AC under the condition that the SD is constrained. However, these proposed constrained optimization methods can only be applied to achieve a balance between two indicators under some specific constraints. These methods also do not consider the problem that the actual speaker power will be constrained, thus failing to obtain the better effect of PSZ.

Cheng et al. [13] proposed the weighted sum method (WSM) and ε-constraint method (ECM) to further study the balance of AC, SD, and AE. However, this method can only find the Pareto-optimal point that is located on the boundary of the convex envelope of the target set, and there may be a situation where the optimal solution cannot be found. Cheer and Elliott [14] used a mixed speaker array to generate PSZ but did not consider the effect of speaker position or used it to optimize the overall performance of the PSZ. Peng et al. [15] proposed a method for position and number optimization of mixed speaker arrays, but the effect on PSZ is unclear. The method proposed by Zhao and Burnett [16] and Gao et al. [17] cannot further optimize the interior vehicle PSZ system with the fixed speaker position and quantities.

Stuart et al. [18] and Thipmaungprom [19] pointed out that in full-range speakers, driving multiple frequencies through the same speaker unit simultaneously tends to produce intermodulation distortion. This distortion alters the signal spectrum and waveform, affecting the quality and accuracy of the signal. The mixed speaker system can reduce the distortion by assigning different frequencies to different speaker units, resulting in clearer sound.

In this article, PSZ-MOGA is proposed to achieve the dynamic balance of AC, SD, and AE, and the accuracy of sound field reproduction is further improved by using the mixed speaker array.

Section 2 presents the theory and methodology of PSZ control. Section 3 is the method and theory proposed in this article. Section 4 presents the simulation part. An experimental part is presented in Sec. 5. Section 6 is the conclusion.

The schematic diagram of the generation interior vehicle PSZ based on the speaker array is shown in Fig. 1. In Fig. 1, the main driver's area is the bright area, which will reproduce the target sound field in this area. The right side of the back row is the dark area, which will not be affected by the reproduction sound field of the bright area.

Considering the three optimization objectives AC, SD, and AE, higher AC often requires higher speaker power and poorer reconstruction accuracy. At some frequencies, better reconstruction accuracy also means that the speaker power consumption needs to be increased; thus, these three evaluation indicators are contradictory. The desired effect is to make the AC as large as possible and the SD and array power consumption AE as small as possible.

As shown in Eq. (5), under the condition that the acoustic transfer function does not change (i.e., the position of the loudspeaker and the external interference do not change), the speakers' weight vectors play a decisive role in the reproduction effect of the sound field. PSZ-MOGA is proposed to achieve a better effect of PSZ. Under the constraint of the speaker's power, the algorithm uses continuous iteration to narrow the range and then finds the speakers' weight vectors that can make AC, SD, and AE reach Pareto optimality. The flow of the PSZ-MOGA is shown in Table 1.

Non-dominated Genetic Algorithm II (NSGA-II) [21] has better performance in solving multiobjective optimization problems, but it also suffers from slow convergence and may fall into local optimal solutions.

In this article, we make the following three improvements to the NSGA, and the flow of the improved NSGA-II is shown in Fig. 2.

• (1) Further optimization of the generated new populations

The PSZ-MOGA aims to achieve a balance between AC, SD, and AE under the constraints of the loudspeaker array power, so the PSZ-MOGA should at least obtain an SD superior to the ACC algorithm, an AC superior to the PM algorithm, and an AE superior to both. As shown in Fig. 4, the average value of SD and AE in the calculation results of the ACC and PM algorithm in this article is about 0.4 and 0, respectively, so AC should be more than 11 dB [22], and SD and AE need to be less than 0.4 and 0, respectively. This limits the search space of the algorithm, improves its accuracy and convergence speed, and reduces the number of iterations.

• (2) Dynamically update parameters that cannot be changed once selected

  • The optimal part of the merged population needs to be selected each time to generate a new population. However, the number of selectable populations will decrease since the populations that do not meet the AC restriction are removed. Therefore, the crossover and mutation probabilities are dynamically updated to ensure the diversity of the populations.

  • The solutions at the end of each sort are not necessarily uniformly distributed, and dynamic updating of crossover and mutation probabilities improves population coverage and avoids the local optimization problem.

The Topsis algorithm evaluates the advantages and disadvantages of the alternatives by calculating the distance of each alternative from the ideal and least ideal solutions and ultimately determines the best choice.

Simulations are carried out to verify the effectiveness of the proposed PSZ-MOGA. The PSZ-MOGA is applicable to any arrangement of bright and dark zones and speakers in the car cabin. The speaker arrangement scheme used in this simulation is shown in Fig. 3. Taking the common symmetrical distribution of 10 speakers as an example, with the main driver's seat as the bright zone and the right side of the rear row as the dark zone, 18 microphones are arranged in the bright and dark zones, totaling 36 control points, to construct the PSZ system. The No. 5 speaker is selected as the reference speaker to generate the reference sound field for calculating the AE. The ideal sound field is set to have zero sound pressure in dark zones. Set the frequency response range to 60–4000 Hz and the sampling frequency to 10,240 Hz, and set the constraints, $|S|2≤η×3×|Sr|2$⁠, $0<η≤1$⁠, for the loudspeaker array with the single speaker signal not exceeding 3 V. The initial number of populations is set to 80, the maximum number of iterations is 100, and the initial crossover and mutation probabilities are 0.8 and 0.5, respectively. The target weights of AC, SD, and AE in the Topsis algorithm are set to 0.5, 0.3, and 0.2, respectively.

The PSZ-MOGA method proposed is compared with the ACC method and the PM method, using the same number and position of speakers. Using 10 speakers, the AC, SD, and AE values are shown in Fig. 4. From Fig. 4, it can be seen that the ACC method achieves the best AC, but this comes at the cost of significant SD. The PM method exhibits less SD and better sound quality, but its acoustic contrast effect is less pronounced. From Refs. [1,2], it can be seen that S for the ACC and PM methods is selected with the determination of the transfer function, and therefore, no constraints can be imposed on the AE, so they are not ideal for AE. In contrast, the PSZ-MOGA method can find the Pareto-optimal curves of AC, SD, and AE in fewer than 100 iterations. The optimal solution is determined by the preset weights to improve AC while keeping SD as low as possible under the speaker power constraints.

Two multiobjective optimization methods, WSM and ECM, have also been proposed in Ref. [13], and since both methods can achieve the same optimization results, the WSM method is used to compare with the PSZ-MOGA method. The results are shown in Fig. 5. Although the WSM method has better SD around 1500–2500 Hz, this comes at the expense of AE, and PSZ-MOGA gives higher AC at a similar SD and AE.

According to Ref. [13], the WSM method can only find the optimal solution on the convex Pareto front, but the Pareto front of the multiobjective optimization problem of AC, SD, and AE is not always convex in all frequency bands, so there is a situation that the optimal solution cannot be found. Moreover, the selection of weights only reflects the trend of the optimization objective, and the relationship with the AC, SD, and AE values is not clear in advance. The performance of the ECM method relies on the setting of $ε$ constraint, and unreasonable $ε$ constraint values can lead to a less-than-optimal solution or an infeasible optimization problem, whereas the constraint values of AC, SD, and AE for arbitrary environments are unknown beforehand, and repeated attempts to determine the constraint values are required. The selection of weights for the PSZ-MOGA is directly related to the value of the optimization objective, and a more comprehensive Pareto optimal solution can be found without constraining the optimization objective. And when the number of optimization objectives increases, the PSZ-MOGA method is still applicable. However, PSZ-MOGA increases the computation time compared to these two methods.

In the process of sound field reproduction in a car cabin, different speaker arrays yield varying reproduction effects. To achieve more effective multiobjective optimization control of PSZ, a sound field control method based on a mixed speaker array is explored. The accuracy of the reproduced sound field in different zones by separating the target frequency bands and controlling them independently with different speaker arrays is optimized.

As shown in Fig. 6, $fLi$ and $fHi$$(i=1,2,3)$⁠, $fiL$ and $fiH$$(i=1,2,3)$⁠, and $fL$ and $fH$ represent the upper and lower bounds of the target optimization frequency range of different frequency bands, working frequency range of different speakers, and needed optimization frequency range, respectively. The different frequency bands should not interfere with each other to avoid affecting the sound field reproduction of other bands. In this article, the frequency bands are defined as 60–100 Hz for low frequency, 100–500 Hz for middle frequency, and 500–4000 Hz for high frequency.

After completing the frequency band division, a total of 2 low-frequency speakers, 4 mid-frequency speakers, and 4 high-frequency speakers (10 speakers in total) were used for sound field reproduction.

Inspired by Ref. [15], the gain $‖Gl‖2$ of the transfer function of the loudspeaker indicates the controllability of the lth candidate loudspeaker position to the loudspeaker array, the larger $‖Gl‖2$ indicates that the loudspeaker in this position has a greater influence on the sound field reproduction should be retained, and the controllability of the wide band loudspeaker candidate position is the sum of the single frequency controllability of the position.

The controllability of the 10 candidate positions in the low-, mid-, and high-frequency bands was calculated separately, and the positions of the 2 low-frequency loudspeakers, 4 mid-frequency loudspeakers, and 4 high-frequency loudspeakers were determined according to the magnitude of $Cl$⁠. The results are shown in Table 2, and the arrangement of the mixed speaker array is shown in Fig. 7.

According to this mixed speaker array arrangement, the PSZ-MOGA is used for multiobjective optimization, and the optimized reproduction results are shown in Fig. 8. Inspired by Refs. [20,24], the AC is reduced in the low- and mid-frequency parts due to the limitation of the number of loudspeakers, while at high frequencies the directivity of the loudspeakers is increased, so that the AC is not affected significantly. The use of mixed loudspeaker arrays offers significant advantages in the high-frequency section, with only four speakers, and gives control that approximates the PSZ of a full-range loudspeaker array and has a better SD overall.

The experiments were conducted in a car within a semianechoic chamber. The main driver's headrest was designated as the acoustically bright area, and the rear right seat headrest as the acoustically dark area. The mixed speaker array shown in Fig. 7 is used for sound field reproduction. Eighteen microphones (PCB-378B02) were placed in each area to measure the impulse response. Each measurement lasted 10 s. The input and output signals were processed using the BEHRINGER X32 RACK, and the LMS SCADAS system was used to collect the time-domain sound pressure signals. Frequency-domain sound pressure signals were obtained via Fourier transform to calculate the AC between the bright and dark zones. The control frequency range was 60–4000 Hz, the sampling frequency was 8192 Hz, and the frequency resolution was set to 1 Hz. The process of experimental validation is shown in Fig. 9.

The proposed method is compared with the WSM method in Ref. [13], and the parameters are set the same as in the simulation. Figure 10 shows that the PSZ-MOGA method achieves better SD and AE, and AC is almost the same.

To achieve better sound field reproduction while generating interior vehicle PSZ, the balance of AC, SD, and AE is studied. The PSZ-MOGA multiobjective optimization method outperforms existing algorithms and effectively addresses the balance problem among AC, SD, and AE. The performance of the reproduced sound field can be effectively controlled by simply increasing and decreasing the weights, which provides better flexibility. Using a mixed speaker array optimizes the frequency response of each band, reduces intermodulation distortion, improves overall sound quality, and provides better sound field reproduction performance compared to traditional speaker array. The proposed method is not restricted by the number of speakers, arrangement, and the configuration of bright and dark zones.

• The Project of National Natural Science Foundation of China (No. 51905331).

• Young Scientific Research Team Cultivation Program of SUES (QNTD202112).

• Technical Service Platform for Vibration and Noise Evaluation and Control of New Energy Vehicles (No. 18DZ2295900).

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.