## 1 Introduction  
In the field of commercial headphones, ANC has become a common application [1]. Controllers in this realm are classified into two categories based on the time variability of their filtering coefficients: fixed controllers and adaptive controllers. Fixed controllers contribute to reduced computational demands and extended battery life, thereby improving the standby time of headphones. Recently, [PERSON] et al. [2] found that compared to an adaptive controller, a fixed controller achieved satisfactory noise reduction performance and signal tracking quality, which are highly valuable merits in practical applications. ANC controllers are mainly divided into feedforward, feedback, and hybrid controllers in terms of their control structures. Notably, the feedback control strategies have been paid particular attention in a long history of research [3]. The introduction of feedback mechanisms leads to two primary issues: firstly, according to the Bode sensitivity integral theorem, reducing low-frequency noise often results in the amplification of noise at other frequencies, a phenomenon commonly known as the \"waterbed effect\" [4]; secondly, the stability of the system varies across different usage environments, primarily owing to the impact of secondary path uncertainty [5, 6] on system stability. Therefore, the basic principle of designing a feedback controller for ANC headphones is to maximize the low-frequency noise reduction performance while robustly maintaining the stability of the system and limit the noise enhancement at high frequencies. [PERSON] [7] pointed out that owing to the additional delays in a digital system, the bandwidth often becomes limited. The controllers for ANC headphones were initially implemented using analog circuits, with various design methods proposed by researchers such as [PERSON] and [PERSON] [8], [PERSON] [9], and [PERSON] et al. [10; 11]. However, analog circuits have drawbacks such as significant discrete component errors and temperature drift, deteriorating the noise reduction performance and even system stability, so precise and low-power controller designs are needed in practical applications. The uncertainty of the secondary path exceeds the deviation of the controller, often leading to stability issues. The modeling errors of the secondary path may deteriorate the stability bound of the system [12]. [PERSON] and [PERSON] [13] first attempted to use random noise for online identification of a system's secondary path, while [PERSON] [14] proposed a prediction method applicable for both online and offline identification. However, the method of online identification with added noise is not suitable for ANC headphones as the identification error signal may contaminate the error signal of the noise reduction system itself. Given the close proximity of the headphone system to the human ear, even minor noise can be perceived as discomforting, and the presence of identification noise is unacceptable. [PERSON] discussed using music or speech signals present in the headphone system as identification signals for the secondary path [15]. More recently, [PERSON] et al. [16] proposed a method to control the identification error energy, and the system stability was significantly improved, although further research is required for headphone applications. In practice, the transfer function of a headphone's secondary path is usually obtained through offline identification and is then constrained by disk [17]. The uncertainty in the secondary path manifests mainly in the variance between a single headphone's use by a single subject or a dummy head, in terms of wearing and removing the headphones, and the differences in multiple wearings by one or multiple subjects. [PERSON] et al. [18] noted that putting on and taking off headphones introduces large variance in the transfer functions of the secondary path, affecting the convergence of the adaptive system, and this variance can be reduced using the internal model control (IMC) feedback control implemented with analog circuits.