## 1 Introduction  
The inner-formation technology, also known as drag-free control technology [1; 2], enables the achievement of pure gravitational orbits crucial in space physics and other domains by effectively compensating for non-conservative forces acting on spacecraft, making it extensively utilized in space science experimental platforms [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. This control method utilizes external actuation to counteract relative motion deviations between the external and internal spacecraft, thereby achieving compensation for interference from the external spacecraft. Due to its significant developmental potential, scholars have conducted extensive research on the inner-formation system [14; 15; 16; 17; 18] and drag-free spacecraft [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. In the inner-formation control system, the research focus lies in investigating the dynamics of relative motion between internal and external spacecraft, followed by designing control strategies. However, when studying relative motion characteristics, it is often challenging to directly measure the velocity and acceleration of relative motion as well as external interference sources and uncertainties due to limited sensor capabilities. To address this issue, a modified Unscented Kalman Filter algorithm is employed for estimating the relative motion between the inner satellite and outer satellite in [14]. Additionally, [33] proposes an estimation method for the relative motion state and disturbance of test masses in drag-free satellites based on the self-recurrent wavelet neural network. Moreover, observers can be utilized to analyze the dynamic characteristics of the system. An extended state observer not only estimates the system state but also quantifies additional interference sources and uncertainties. The extended state observer was initially introduced in reference [34], and subsequently developed further with a design approach that incorporates high gain parameters for system gain configuration. Reference [35] provides a comprehensive explanation on designing high-gain extended state observers, which have gained significant attention due to their superior observation efficiency. For a two-test-mass inner-formation drag-free system, an observer-based Model Predictive Control (MPC) is proposed, demonstratingits feasibility for deep space exploration according to simulation results [36]. Furthermore, active disturbance rejection control incorporating an extended state observer is investigated for achieving frequency domain performance requirements in gravitational detection missions [37]. However, these estimation algorithms encounter a limitation in that they fail to consider the constraints imposed by system information transmission resources when utilizing system output information for state estimation. The system acquires such information through continuous or periodic sampling methods. By adopting event-driven estimation, this issue can be addressed and improved upon. Event-triggered method determines the sampling time of the system by establishing specific triggering conditions, thereby enabling high-frequency sampling during periods requiring more frequent measurements and low-frequency sampling during more stable stages, thus optimizing resource allocation for sampling [38]. Since its inception, event-based estimation has made significant advancements and found applications across diverse domains. In [39], the event-triggered method is considered for a state feedback controller with an input-to-state stability (ISS) result, and the minimum sampling interval is obtained by a Comparison Lemma [40]. The event-triggered method is introduced to spacecraft systems and underwater systems in [41] and [42], respectively; the event-based controller shows its advantage with non-periodic sampling.