###### Abstract  
Due to the complexity of the coupling between the active noise control (ANC) controller and secondary path estimator, performance analysis of the system becomes particularly difficult. At present, the performance analysis of the system is often based on the fact that the secondary path tends to be stable, and the secondary path fitting error is minimal. However, in the early stage of system operation, or when the secondary path changes suddenly, the secondary path fitting error is significant, which easily causes divergence of the system control. It is still unable to guarantee the step-size bounds of convergence stability. Therefore, factor decomposition was used to analyze the mean weight behavior in this study. This strategy emphasizes the influence of secondary path modeling (SPM) error. The mean square behavior was evaluated using the energy conservation relationship. According to the established theoretical model, the convergence condition of the system was derived and the upper bound of step size suitable for all stages of system operation was obtained. The simulation and experimental results show that the ANC system is quite stable and robust under extreme conditions and has an obvious noise reduction effect in a specific range of open space, which can reach about 20 dB noise reduction. active noise control; factor decomposition; online secondary path modeling; stability condition; quiet space +
Footnote †: journal: Applied  
**Applied**  
**Acticle**  
**A Novel Narrowband Active Noise Control System with Online Secondary Path Modeling Based on Factor Decomposition and Application in Open Space**  
**[PERSON], [PERSON], [PERSON] and [PERSON]**  
Key Laboratory for Optoelectronic Technology and System of the Education Ministry of China,  
College of Optoelectronic Engineering, Chongqing University, Chongqing 400044, China;  
[EMAIL_ADDRESS] (Y.L. ): [EMAIL_ADDRESS] (Z.L. ): [EMAIL_ADDRESS] (Q.X.) **Correspondence: [EMAIL_ADDRESS]**  
**Key Laboratory for Optoelectronic Technology and System of the Education Ministry of China,  
College of Optoelectronic Engineering, Chongqing University, Chongqing 400044, China;  
[EMAIL_ADDRESS] (Y.L. ): [EMAIL_ADDRESS] (Z.L. ): [EMAIL_ADDRESS] (Q.X. )**  
**Key Laboratory for Optoelectronic Technology and System of the Education Ministry of China,  
College of Optoelectronic Engineering, Chongqing University, Chongqing 400044, China;  
[EMAIL_ADDRESS] (Y.L. ): 20142604@cqu. technology is typically used to estimate the secondary path [7, 8]. To compensate for the influence of the secondary path model (SPM), as an improvement of the least mean square algorithm, the filtered-x least mean square (FxLMS) algorithm is the most popular adaptive algorithm [9, 10]. In the FxLMS algorithm, the reference signal must be filtered using the secondary path response function, and then the control filter coefficients must be updated. Therefore, it is necessary to fit the secondary path. The fitting methods of the secondary path are divided into online and offline fittings. In practical use, the adjustment of the power amplifier gain, the change in environment, and the aging of devices will cause the change in secondary path response function. Therefore, it is necessary to perform online fitting on the secondary path, and the FxLMS algorithm can maintain stability when the relative error of the secondary path is \(\pm 90^{\circ}\)[11, 12]. [PERSON] et al. first proposed the use of auxiliary noise to perform online fitting on the secondary path [13]. This marked a new journey to pursue a more efficient ANC system. In recent years, different strategies have been used to improve the convergence speed and steady-state performance of secondary path online fitting [14, 15, 16, 17, 18, 19, 20]. In the process of secondary path modeling (SPM), due to the interaction between the fitting errors of SPM and ANC filters, the FxLMS algorithm is more complicated than the least mean square (LMS) system.