## 1 Introduction  
Owning to the aperture obscuration caused by secondary mirrors (SM) and their supporting structure, on-axis reflective astronomical telescopes [1; 2] not only lose light energy but also reduce the modulation transfer function (MTF) value at medium and low frequencies, so that the observation performances of optical systems are limited. On the basis of the on-axis reflective structure, the optical path structure with a field of view (FOV) off-axis, pupil off-axis, or a combination of the two is proposed to solve this problem [3; 4; 5]. Off-axis astronomical telescopes have the characteristics that their stray light is easy to control, their scattering property is low, and filled the pupil for wavefront sensing; these characteristics cause them to have a small and sharp diffraction pattern and high spatial resolution [6]. Compared with on-axis astronomical telescopes, when off-axis astronomical telescopes are in orbit, due to their special optical path structure, the slight perturbation caused by factors such as gravity, thermal stress, and mechanical oscillation will affect their imaging quality more easily. This brings challenges for active optical systems to complete wavefront correction in orbit. The correction process of telescope perturbation parameters (rigid-body alignment errors and figure errors) is an important link in the active optical system [7; 8]. The correction results of perturbation parameters can be divided into restoration correction and compensation correction, according to different ideas of wave aberration correction. Restoration correction is obtained according to the idea of restoring the telescope to the nominal design, which requires the active optical system to be able to adjust all optical elements to achieve wave aberration correction. Compensation correction is obtained according to the idea of mutual compensation of optical elements, which requires the active optical system to adjust some of the optical elements to achieve wave aberration correction. However, in practical engineering, if the idea of restoration correction is followed, it is necessary to equip all optical components with corresponding multi-degree-of-freedom platforms (position and orientation adjustment) and force actuators (mirror surface figure adjustment). This approach is relatively expensive for space-based optical systems that have strict weight and volume constraints, both in construction cost and the structural stability of maintaining the system [9]. In addition, the correction efficiency (real-time) of an active optical system would be influenced by too many adjustment devices [10]. Therefore, studying the corresponding compensation correction methods is necessary to reduce the complexity of the active optical system and improve the correction efficiency. For pupil-offset off-axis telescopes with designed tilts and decenters, discussed in this paper [11; 12], the influence of different optical elements on wave aberrations has a strong coupling effect, which makes aberration compensation of this type of optical system easier to achieve. To correct the wave aberration of telescopes, the corresponding adjustment values (perturbation parameters) of optical elements should be determined. At present, the methods for solving these adjustment values mainly consist of traditional numerical methods and analytical methods based on nodal aberration theory (NAT). The numerical methods [13; 14; 15; 16] do not consider field dependence of net aberrations induced by perturbation parameters when constructing a solution model for a specific optical system, and cannot analyze the coupling effect between different aberration fields from an analytical point of view, which makes it difficult to reasonably use this coupling effect to construct an optimal compensation correction model.