## 4 Simulation Experiment  
Summarizing the content in Section 2, the algorithm for the quantitative evaluation method for diagnosability is provided in Algorithm 1:  
```
0: State-Space Model of a Dynamic System:
1. Pretreatment: \(Lz(k)=Hx(k+1)+Ff(k)+Ee(k)\)
2. Pretreatment: \(N_{H}Lz=N_{H}Ff+N_{H}Ee\)
3. Manifold parameter: Mean value \(\mu(\theta)=N_{H}Ff\), Variance \(C(\theta)=N_{H}Ee\)
4. Fisher Information Metric \(G(\theta)=g_{\alpha\beta}(\theta)=f(\mu(\theta),C(\theta))\)
5. Fisher Information Distance (FID) \(D_{F}(\theta_{1},\theta_{2})=\int_{t_{1}}^{t_{2}}\biggl{(}\sqrt{\left(\frac{d \theta^{T}}{dt}\right)G(\theta)\frac{d\theta}{dt}}\biggr{)}dt\)
6. Detectability Index \(F_{D}(f_{i})=D_{F}(\theta_{i},\theta_{0})\)
7. Isolability Index \(F_{I}\bigl{(}f_{i},f_{i}^{{}^{\prime}}\bigr{)}=D_{F}\bigl{(}\theta_{i},\theta _{i}^{{}^{\prime}}\bigr{)}\)
8. Import specific fault/faults to obtain the detectability/isolability index
```  
**Algorithm 1**The quantitative evaluation method for diagnosability on an information manifold  
To verify the effectiveness of the proposed algorithm, in this section, for the satellite pitch-axis attitude determination system, which is shown in Equation (1), a simulated experiment on diagnosability evaluation is carried out in a joint mathematical and Matlab simulation platform. As the derived manifold parameters \(\mu(\theta),C(\theta)\) already contain the information of the satellite pitch-axis attitude determination system (Equation (1)), as shown in Equation (10), \(f_{gy}\) is the fault vector of the gyroscope, and \(f_{hy}\) is the fault vector of the infrared Earth sensor. Each fault vector contains a nonlinearity related to its sensor output characteristics. We set the initial value to \(f_{gy}(0)=0,f_{gy}{}^{\prime}(0)=0.1,f_{hy}(0)=1,f_{hy}{}^{\prime}(0)=0.1\). The time-varying geodesics for the two fault components are obtained, as shown in Figure 1.  
The FID of \(f_{gy}(k)\) is 1.55741, and the FID of \(f_{hy}(k)\) is 1.25, which means that  
\[F_{D}\bigl{(}f_{gy}\bigr{)}=1.55741\]  
Figure 1: Development of single faults \(f_{gy}\) and \(f_{hy}\) on a fault information manifold over time. It can be found that the geodesic images of the \(f_{gy}(k)\) fault component and the \(f_{hy}(k)\) fault component also reflect this difference; the derivatives of the geodesics of the two fault components are different. It should be noted that the two geodesics in Figure 1 only reflect the development and variation of a single fault itself on the manifold. For the two-dimensional fault information manifold (including two fault components) studied in this paper, there are three possible fault types: two single-fault cases and one compound fault case. A fault information manifold with a curvature is a typical surface that is hard to display. To intuitively demonstrate its shape and study its properties, this paper then \"transforms the surface into a plane\" through geodesics, then researches the fault diagnosability problem on it. In this paper, by taking the two fault components as the X-axis and Y-axis, respectively, the two-dimensional fault manifold surface can be mapped in the Cartesian coordinate system. In this coordinate system, the X-axis (\(f_{gy}\)) and Y-axis (\(f_{hy}\)) represent two single-fault cases (a gyroscope fault and infrared sensor fault), while the first quadrant represents a compound fault case. In the same type of fault case, different points in the coordinate system indicate a fault with different parameters. Plotting the geodesics, the figure shows the path taken by a group of composite fault geodesics with a departure velocity of 1 that are located at the fault manifold coordinate (4,4) in the same time period. The compound fault parameters are set as follows: \(f_{gy}(0)=4,f_{hy}(0)=4,f_{gy}^{\prime}(0)=1*\sin[2\pi*\frac{j}{MX}],fhy^{ \prime}(0)=1*\cos[2\pi*\frac{j}{MX}]\). We set the number of geodesic lines to \(\text{MX}=128\).