## 1 Introduction  
With the increasing demand of high-accuracy pointing for LOS (line of sight), growing attention has been paid to the MISP [1, 2], which places magnetic bearings in the inner gimbal of the conventional gimbal system to achieve dual-stage stabilization [3, 4]. A variety of factors affect the performance of the MISP, and it is of great significance to improve the control precision of the inertial stabilization platform. [PERSON] [5] established and the modeled the maglev dual-stage systems and applied different control strategies. In the study in [6], a new adaptive controller versus an H\(\infty\) control was proposed on a magnetic levitation. In [7, 8], an extended state observer (ESO) was adopted to estimate the disturbances, and the disturbances were compensated in real time to achieve a high precision compound control. [PERSON] [9] designed a neural network backstepping sliding mode controller to estimate the nonlinear friction between the frames and three-axis vehicle inertial stabilization platform system, which is helpful for the real-time estimation and compensation on the nonlinear friction. The MISP is driven by a motor gear, which is an elastic structure. The existence of an elastic structure leads to mechanical resonance [10]. When the resonant frequency is high enough to exceed the control bandwidth, it has little effect on the control system. When the resonant frequency is low, the performance of the control system deteriorates, owing to the mechanical resonance. In the MISP, the ratio of the stiffness between the drive shafting and the gimbal frame is low, and the equivalent stiffness changes with gear clearance. Therefore, the resonance frequency is variable [11]. It is necessary to take measures to suppress mechanical resonance. The studies in [12, 13] presented a method of employing a disturbance observer to estimate the torque brought by resonance, which could be fed back to the input and compensated. Filtering is a common method of resonance suppression. The conventional notch filter joins between the output of the speed loop and the input of the current loop and can significantly suppress the resonance at a constant frequency [14, 15, 16] but is unable to effectively restrain the resonance at variable frequencies. Several studies [17; 18; 19] have been conducted on adaptive notch filters to realize resonance suppression. In [17], a fractional-order low-pass filter substituted the integer-order low-pass filter to achieve a better tradeoff between robustness and resonance suppression. The study in [18] proposed a novel excitation signal for a vibration test to improve the system performance and locate the resonance frequency to adjust the frequency parameter of the notch filter in time. The study in [19] proposed an adaptive infinite impulse response (IIR) filter based on the recursive least squares algorithm, managed to estimate the resonance frequency, and adjusted the filter parameter online. The RLS algorithm is suitable for parameter identification and signal processing. Many scholars have focused on improving the recursive least squares algorithm [20; 21; 22; 23]. In [20], a recursive extended least squares (RELS) algorithm was presented to provide an unbiased estimation of model parameters online. The performance of the RLS algorithm is governed by the forgetting factor, which influences tracking capability and misadjustment. The study in [21] used the output error to control the forgetting factor and proposed a novel modified RLS algorithm to strengthen the accuracy of sparse system identification. The studies in [22; 23] put forward variable forgetting factor RLS algorithms, and they took control of the value of the forgetting factor to make the algorithm robust for both stationary and nonstationary input signals, achieving fast convergence and low misadjustment.