## 2 Problem Formulation
### Kinematics and Dynamics Models  
The mathematical model of satellite attitude motion contains a kinematics model and a dynamics model [27]. The coordinate systems used in this paper are first introduced, where the Earth-centered inertial coordinate system is \(O_{i}X_{i}Y_{i}Z_{i}\), and the micro/nano satellite centroid inertial spindle coordinate system is \(O_{b}X_{b}Y_{b}Z_{b}\). The symbol \(\mathbf{Q}_{bi}=\left[q_{0}\quad\mathbf{q}^{\mathrm{T}}\right]^{\mathrm{T}}= \left[q_{0}\quad q_{1}\quad q_{2}\quad q_{3}\right]^{\mathrm{T}}\) is used to denote the attitude from \(O_{b}X_{b}Y_{b}Z_{b}\) to \(O_{i}X_{i}Y_{i}Z_{i}\). The symbol \(\boldsymbol{\omega}_{bi}^{b}=\left[\omega_{x}\quad\omega_{y}\quad\omega_{z} \right]^{\mathrm{T}}\) is used to denote the angular rate of the satellite relative to the inertial space, and the superscript \(b\) means that the vector is represented in the coordinate system \(O_{b}X_{b}Y_{b}Z_{b}\).  
The kinematics equation of the system is as follows and describes the relationship between the attitude quaternion \(\mathbf{Q}_{bi}\) and the angular rate \(\boldsymbol{\omega}_{bi}^{b}\):  
\[\dot{\mathbf{Q}}_{bi} = \frac{1}{2}\mathbf{H}(\mathbf{Q}_{bi}\ )\left[0\quad\boldsymbol{ \omega}_{bi}^{b}\mathrm{T}\right]^{\mathrm{T}}, \tag{1}\]  
Figure 1: Structure diagram of considered ADCS.  
where the square matrix \(H(Q_{bi})\) is an orthogonal matrix, which satisfies \(H(Q_{bi})H(Q_{bi})^{\mathrm{T}}=I_{\mathbf{\lambda}}\):  
\[H(Q_{bi})\ =\ \begin{bmatrix}q_{0}&-q_{1}&-q_{2}&-q_{3}\\ q_{1}&q_{0}&-q_{3}&q_{2}\\ q_{2}&q_{3}&q_{0}&-q_{1}\\ q_{3}&-q_{2}&q_{1}&q_{0}\end{bmatrix}. \tag{2}\]  
In this paper, the micro/nano satellite is regarded as a rigid body. According to the angular momentum theorem, the satellite satisfies the following equation [27]:  
\[J\dot{\mathbf{\omega}}_{bi}^{b}+\mathbf{\omega}_{bi}^{b}\times(J\mathbf{\omega}_{bi}^{b}) =\mathbf{T}_{MT}+\mathbf{d}, \tag{3}\]  
where \(\mathbf{J}\) is the rotational inertia matrix of the satellite, and it is a positive definite matrix; \(\mathbf{\omega}_{bi}^{b}\) is the angular rate of the satellite relative to the inertial space; \(\mathbf{T}_{MT}\) is the control torque of the 3-axis MTQs, and \(\mathbf{d}\) is the sum of the external disturbance torques acting on the satellite. This is the dynamics equation of the satellite, which mainly describes the relationship between the angular rate \(\mathbf{\omega}_{bi}^{b}\) and \(\mathbf{T}_{MT}\), \(\mathbf{d}\). For the satellite centroid inertial spindle coordinate system \(O_{b}X_{b}Y_{b}Z_{b}\), the rotational inertia matrix \(\mathbf{J}\) has the following form:  
\[J\!=\mathrm{diag}\left[J_{x},J_{y},J_{z}\right], \tag{4}\]  
where \(J_{x}\), \(J_{y}\), and \(J_{z}\) are the rotational inertia of the satellite around \(O_{b}X_{b}\), \(O_{b}Y_{b}\), and \(O_{b}Z_{b}\) axes, respectively.  
This paper focuses on attitude control under bounded external disturbance torques. Therefore, \(\mathbf{d}\) satisfies the following assumption:  
**Assumption 1**: _The sum of the external disturbance torques \(\mathbf{d}\) in (3) is bounded; i.e., there always exists a constant \(\varepsilon\in\Re_{+}\) such that \(\|\mathbf{d}\|\leq\varepsilon\)._