## 4 Simulation Results and Discussion
### Simulation Results of the Control Perormance in Far Field for Each Method  
In the fourth simulation, a comparison of the directivity for each method before and after active noise control at far-field observing points from 400 Hz to 1000 Hz is performed. As seen in Figure 1, 360 uniformly placed far-field observing points were located at a distance of 1000a from the center of the cylinder (the radius of the cylinder was \(\mathrm{a}=0.38\) m). Before active noise control, the sound pressure for one far-field observing point can be calculated by \(P_{ob\_far\_ANCoff}=Z_{pob\_far\_sca}q_{p}\). The source strength of the primary field is \(q_{p}=1\) m\({}^{3}\)s\({}^{-1}\). The scattered acoustic transfer function at one far-field observing point \(Z_{pob\_far\_sca}\) can be calculated using Equation (1). After active noise control, the sound pressure for one far-field observing point can be calculated by \(P_{ob\_far\_ANCon}=Z_{pob\_far\_sca}q_{p}+\mathbf{Z}_{sob\_far\_q}\). The source strength vector (dimension \(NL\times 1\)) of the selected secondary loudspeakers \(\mathbf{q}_{s}\) is calculated by each method introduced in Sections 2 and 3. The matrix (dimension \(1\times NL\)) of the transfer functions form \(NL\) selected secondary loudspeakers to one far-field observing point \(\mathbf{Z}_{sob\_far}\) can be obtained using Equation (2). The absolute values of the sound pressure before active noise control for 360 far-field observing points are shown as the red dotted lines in Figure 5. The absolute values of the sound pressure after active noise control for 360 far-field observing points are shown as the blue solid lines in Figure 5. The frequencies range from 400 Hz to 1000 Hz at a frequency interval of 300 Hz. The first row shows the directivity patterns for Uni-RLS algorithm, the second row shows the directivity patterns for CLASSSO-RLS algorithm, the third row shows the directivity patterns for LASSO-RLS-\(\gamma_{C}\) algorithm, and the fourth row shows the directivity patterns for LASSO-RLS-\(\gamma_{NL}\) algorithm. Figure 5: Comparison of the directivity for each method before and after active noise control at far-field observing points from 400 Hz to 1000 Hz: **(a)** Uni-RLS (400 Hz); **(b)** Uni-RLS (700 Hz); **(c)** Uni-RLS (1000 Hz); **(d)** CLASSOR-RLS (400 Hz); **(e)** CLASSOR-RLS (700 Hz); **(f)** CLASSOR-RLS (1000 Hz); **(g)** LASSO-RLS-\(\gamma_{\rm C}\) (400 Hz); **(h)** LASSO-RLS-\(\gamma_{\rm C}\) (700 Hz); **(i)** LASSO-RLS-\(\gamma_{\rm C}\) (1000 Hz); **(j)** LASSO-RLS-\(\gamma_{\rm NL}\) (400 Hz); **(k)** LASSO-RLS-\(\gamma_{NL}\) (700 Hz); and **(l)** LASSO-RLS-\(\gamma_{NL}\) (1000 Hz). The red dotted line corresponds to the control off and the blue solid line corresponds to the control on. As seen in Figure 5, in the directions where the target areas are located, the blue solid line is smaller than the red dotted line for each method. It illustrates that all algorithms have good noise reduction in the directions of the forward-scattered sound and backward-scattered sound target areas. However, as shown in Figure 5a-c,g-l, in the directions with an angle of 90\({}^{\circ}\) from the incident direction, the blue solid lines are bigger than the red dotted lines. It illustrates that Uni-RLS algorithm, LASSO-RLS-\(\gamma_{\text{C}}\) method and LASSO-RLS-\(\gamma_{NL}\) method exhibit sound amplification after active noise control in the two directions and the sound amplification becomes more apparent with increasing frequency. This is due to the fact that as the frequency increases, the secondary loudspeakers need more power to provide good noise reduction in the target areas. However, as the power increases, the secondary loudspeakers produce larger sound amplification outside the target areas. In contrast, as shown in Figure 5d-f, in the directions with an angle of 90\({}^{\circ}\) from the incident direction, the blue solid lines and the red dotted lines have approximately the same values.