## 4 Simulation Experiment
A common alternative to the information distance is the KLD (Kullback-Leibler divergence). \[ \begin{split}& KLD[p(x|\theta_{i})||p(x|\theta_{j})]=\int p(x| \theta_{i})\ln\frac{p(x|\theta_{i})}{p(x|\theta_{j})}\,dx\\ &=E\{\ln p(x|\theta_{i})-\ln p(x|\theta_{j})\}\end{split}  \tag{28}\]  
The relationship between the KLD and differential Fisher information distance is:  
\[ds^{2}=2 KLD[p(x|\theta)||p(x|\theta+d\theta)] \tag{29}\]  
The KLD provides a measure of the distance between two points on a manifold, but the KLD cannot give the shortest path between two points, which means that the KLD does not contain information about the structure of the manifold. This is also one of the most significant differences between the KLD and the Fisher information distance. At the same time, the KLD is not a true distance measure because it does not satisfy the symmetry and triangle inequality of the distance definition. Since the KLD of the two distributions is equivalent to the maximum likelihood estimate between them, it can be used to evaluate the detectability of faults \(f_{i}\) and the isolability between faults \(f_{i}\) and \(f_{j}\), which are calculated as follows:  
\[Detectability(f_{i})=KLD[p(x|\theta_{i})||0] \tag{30}\]  
\[Isolability(f_{1},f_{2})=KLD[p(x|\theta_{1})||p(x|\theta_{2})] \tag{31}\]  
The numerical results of the detectability and isolability indexes of the three types of faults based on the KLD with the same settings are shown in the following. It can been seen from Table 2 that though the diagnosability indexes obtained with the KLD can realize the evaluation of fault detectability and isolability, the evaluation value of each index appears to be chaotic and irregular. Because the KLD is asymmetric, the value of the isolability evaluation between \(f_{i}\) and \(f_{j}\) is unequal to the value of the isolability evaluation between \(f_{i}\) and \(f_{i}\). The problem of asymmetry in the fault isolability index is inevitable for a diagnosability evaluation based on the KLD [27]. Similarly, the diagnosability evaluation method based on the Bhattacharyya coefficient (BC) has the same problem [37]. The FID is a significant concept in information and statistical theory. It has made some achievements in theoretical and applied research on signal processing, target tracking, path planning, and other fields. In the field of fault diagnosis, the results obtained with the diagnosability method proposed in this paper have a similar tendency to that of the results obtained with the traditional KLD method, which demonstrates their correctness on the other side. At the same time, the method presented in this paper contains accurate fault information, and there is no problem of asymmetry in the isolability index. However, for the method proposed in this paper, the diagnosability problem is transferred to a fault information manifold for research, and the real distance measurement of the FID is used to design detectability and isolability indexes, so there is no such problem of asymmetry. The design of the indexes with the FID is relatively more scientific and is conducive to the development of subsequent research on diagnosability (such as fault diagnosis, diagnosability optimization, etc. ). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fault** & \(Det\) & \(Iso(,f_{gy})\) & \(Iso(,f_{hy})\) & \(Iso(,f_{1})\) \\ \hline \(f_{gy}\) & 43.722891 & 0 & 43.209558 & 11,304 \\ \(f_{hy}\) & 9949.144 & 47.283433 & 0 & 9931.571 \\ \(f_{1}\) & 8673.768471 & 11,312.5 & 9796.349 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Diagnosability indexes based on the KLD for the three types of faults. Otherwise, through comparison with Table 1, we find that there is a certain relationship between the diagnosability evaluation values based on the FIDs of the three types of faults. It is manifested as: The isolability evaluation value of the single faults \(f_{gy}\) and \(f_{hy}\) is equal to the sum of the detectability evaluation values of \(f_{gy}\) and \(f_{hy}\).