## 1 Introduction  
Based on multizone sound field reconstruction technologies, a personal audio system can be used to concentrate sound effects in a listening area without affecting listeners in other areas. A bright zone and a dark zone can be generated by multizone sound field reconstruction techniques. The desired sound pressure field could be generated in the bright zone, while the sound pressure level is attenuated in the dark zone. Personal audio systems can be used in game systems, audio system exhibits, home theaters, car audio systems, etc. [1]. With the development of sound field reconstruction technology, more and more people pay attention to personal audio systems. Many researchers have put forward many new research methods and techniques for a personal audio system. The acoustic contrast control (ACC) method was introduced by [PERSON] et al., which solves loudspeakers' allocation coefficients by maximizing the acoustic contrast between the bright zone and the dark zone [2]. However, the ACC method does not control the phase, and its energy is unevenly distributed in the bright zone, which will affect the subjective listening experience of listeners. The pressure matching (PM) method can solve these problems by minimizing the error between the desired sound pressure and the reconstructed sound pressure at control points [3]. Additionally, the planarity control (PC) method was proposed [4], which filters the sound pressure at the control points spatially, and maximizes the energy of the bright zone under the constraint of the dark zone energy based on the super-directional beamforming. The advantage of the PM method and the PC method is that they can generate a directional sound field in the range of the bright zone, while the disadvantage of the PM method and the PC method is that the acoustic contrasts generated by them are smaller than that of the ACC method. [PERSON] et al. introduced a method that maximizes the acoustic energy difference, which changes the focus of the ACC method by focusing on the ratio of the mean square sound pressure difference between the bright and dark zones. The comparison experiments with the ACC method show that this method can maintain the sound pressure difference between two listening zones and improve the radiation efficiency of sound space. In addition, this method is simple to calculate [5]. [PERSON] et al. tried to take advantage of the ACC method and the PM method and proposed the ACC-PM method by combining these two methods. By selecting the appropriate adjustment factor, the ACC-PM method can obtain the tradeoff effect between the bright zone reconstruction error and acoustic contrast [6]. Based on the PM approach, [PERSON] et al. proposed a beamforming approach to control the tradeoff between listener position sound field quality and directivity performance. This proposed method selects control points that contribute to the PM cost function in a frequency-dependent manner according to the angular distance between control points relative to a given wavelength [7]. Based on the PM method, [PERSON] et al. proposed the eigen decomposition pseudoinverse method (EDPM) [8]. This method constructs a regularization strategy to solve the pseudoinverse of ill-conditioned matrices, replacing the Tikhonov regularization method traditionally used in the PM method. This method is used to improve the performance at dark points without generating artifacts at bright points. In view of the robustness and regularization of the ACC method, [PERSON] et al. proposed that robustness could be enhanced by imposing limits on array effort. Robustness can generally be improved by a regularization method, but good regularization parameters are often difficult to choose [9]. [PERSON] et al. proposed a robust sound zone reconstruction design framework that solves the parameters by acoustic modeling. However, they did not examine the relationship between different loudspeaker directivities and robust optimization and explore the accuracy required for robust ACC acoustic modeling [10].