## 2 Mathematical Description of the Problem  
A satellite attitude determination system is the basis for the attitude control of a satellite. Its task is to process information measured by the attitude sensors to obtain the attitude of the satellite body coordinate system relative to the orbit coordinate system [29; 30]. The most commonly used attitude sensors include star sensors, sun sensors, Earth sensors, and gyroscopes [30; 31]. A model of satellite dynamics can be equivalently transformed into three independent axes: the roll, pitch, and yaw. The pitch angular velocity of a satellite is a fast variable in comparison with the other two axes. In this paper, a satellite pitch-axis attitude determination system based on an \"infrared Earth sensor + gyroscope\" is taken as an example for analysis. The infrared Earth sensor and the gyroscope are used to measure the attitude angle and angular rate of the satellite, respectively, and the combined use of the two can achieve high-precision combined attitude determination. Considering that the attitude angle/angular velocity of the satellite's pitch axis (Y-axis) is decoupled from the roll and yaw axes (X-axis and Z-axis), and in order to reduce the model's dimensions and simplify the problem, the discrete-form state-space model of the satellite attitude determination system on the pitch axis is given below, as shown in Equation (1). \[\left\{ \begin{bmatrix}\theta(k+1)\\ d_{y}(k+1)\\ b_{y}(k+1)\end{bmatrix} = \begin{bmatrix}1&-dt&-dt\\ 0&1-\frac{1}{t_{y}}dt&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\theta(k)\\ d_{y}(k)\\ b_{y}(k)\end{bmatrix} + \begin{bmatrix}dt&0&0\\ 0&dt&0\\ 0&0&dt\end{bmatrix}\begin{bmatrix}\omega_{0}+g_{y}\\ 0\\ 0\end{bmatrix} + \begin{bmatrix}dt&0\\ 0&0\\ 0&0\end{bmatrix}\begin{bmatrix}g_{f}\\ \theta_{f}\end{bmatrix} \\ + \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}n_{y}(k)\\ n_{by}(k)\\ n_{dy}(k)\end{bmatrix} \\ \theta(k)= \begin{bmatrix}1&0&0\end{bmatrix}\begin{bmatrix}\theta(k)\\ d_{y}(k)\\ b_{y}(k)\end{bmatrix} + \begin{bmatrix}0&1\\ 0&0\\ 0&0\end{bmatrix}\begin{bmatrix}\delta f\\ \theta_{f}\end{bmatrix} +n_{\theta}(k)\right. \tag{1}\]The mathematical model of the satellite attitude determination system of the infrared Earth sensor + gyroscope is expressed as above, where \(\theta\) and \(\omega_{0}\) represent the attitude angle and orbital angular velocity of the satellite, respectively; \(g_{y}\) represents the pitch-axis angular rate of the satellite, which is the output of the gyroscope; \(d_{y}\) and \(b_{y}\) are the exponentially correlated drift and constant drift of the gyroscope that are caused by the small amount of torque interference on it; \(\tau_{y}\) is the time constant of the gyroscope. \(g_{f}\) stands for the fault vector, and \(n_{y}/n_{by}/n_{dy}\) are the noises of the gyroscope's output and two drifts, respectively, which are all in the form of Gaussian white noise. \(\theta_{f}\) and \(n_{\theta}\) represent the fault vector and Gaussian white noise of the measured attitude angle output by the infrared Earth sensor, respectively. \(dt\) represents the sampling time interval. The values of the related parameters are: \(dt\ =\ 0.1\) s, and the time constant \(\tau_{y}=1/11\) (dimensionless); the orbital angular velocity \(\omega_{0}=0.06\) rad/s, and the Gaussian white noises \(n_{y}\sim N(0,10^{-6})\), \(n_{by}\sim N(0,10^{-6})\), \(n_{\theta}\sim N(0,10^{-6})\). Here, \(\sim\)\(N\) is a Gaussian distribution symbol, and the distribution's parameters are shown in the parentheses. Equation (1) can be simplified as follows:  
\[ \begin{cases}x(k+1)=Ax(k)+Bu(k)+B_{f}f_{a}(k)+B_{w}w(k)\\ y(k)=Cx(k)+Du(k)+D_{f}f_{s}(k)+D_{v}v(k)\end{cases}  \tag{2}\]  
In fact, state \(x\), input \(u\), and output \(y\) may be influenced by the coupling of interferences \(w\) and \(v\), as well as by fault \(f\). In order to separate the influence of state \(x\) and decouple the fault from the interference, the following equation was constructed in [32]. \[Lz(k)=Hx(k+1)+Ff(k)+Ee(k) \tag{3}\]  
where  
\[