# Directional Active Noise Control with a Local Minimax Error Criterion
## 1 Introduction  
Currently, acoustic noise problems are very serious as more noise sources, such as car engines, fans, transformers, and compressors, emerge. Noise control technology can be divided into two categories: passive and active noise control [1]. Passive control, such as enclosures, barriers, and silencers, are no longer practically effective at low frequencies. Active noise control (ANC) can offset the primary noise source, by adding a secondary source, which has an equal amplitude and opposite phase at the control point based on the principle of superposition. ANC has shown the great control performance at low frequencies. Until now, ANC has been successfully applied in many areas. Studies have shown that ANC can be used to control the noise associated with air-craft fuselages, automobile interiors, and open windows [2; 3; 4; 5; 6; 7]. ANC systems can be divided into two categories based on two control strategies, feedforward control and feedback control [8]. Feedforward control is more robust than the feedback control since feedforward control has a coherent reference signal as the input of the controller, which is more stable in terms of control performance [9]. However, possible acoustic feedback from secondary sources to reference sensors arises in the feedforward control, destroying the noise reduction effect. Traditionally, the use of non-acoustic sensors or directional microphones and speakers can avoid the acoustic feedback phenomenon. A neutralization method has been proposed to solve this problemby subtracting the acoustic feedback from the signal detected by the reference microphone. Like the infinite impulse response (IIR) controller, the neutralization method can generate a pole, producing an instability in the control system [10]. A complex noise field requires a multi-channel strategy to meet the desired demands of noise reduction. Traditionally, when the mean squared error (MSE) criterion is adopted to calculate optimal filter coefficients, a multi-channel system can achieve a good performance in terms of noise reduction. However, the residual acoustic field will generate an unpredictable directivity, due to the large differences between the maximum and minimum levels of the residual sound field [11]. Here, we can expect the noise reductions at all error points and hope for better reductions at certain locations. Previous work has reported much effort dedicated to active directivity control of the radiated sound. These methods are roughly divided into two categories: methods that use special speakers and methods that choose an appropriate error criterion. A parametric loudspeaker array (PLA) is a special kind of speaker that generates a directional sound beam from the interaction of ultrasonic beams in the sound control field [12]. When we adopt a PLA in multi-channel ANC, the computational complexity of the algorithm is greatly reduced, since crosstalk secondary path models can be removed [13]. [PERSON] used a PLA as the secondary source to avoid spillover phenomena and achieved noise control when an obstacle existed between the PLA and target points [14]. In addition, a moving directional ANC system was achieved by [PERSON] [15]. The system adopted a steerable PLA based on a phased array rather than mechanically rotating the PLA, enabling the generation of a moving quiet zone. A problem with the PLA arrangement is nonlinear distortions, which have been the primary technical barriers to promoting PLAs' application [16]. Although an ANC system with a PLA can mitigate sound locally, the system has little influence on the circumferential field. [PERSON] and [PERSON] adopted a weighted sum of the squared sound pressures within an angle at error points as a cost function and achieved a directivity control from the near field to the far field [17]. [PERSON] and [PERSON] also proposed using a similar weighted sum of error signals based on the spatial Fourier transform as the error criterion [18].