## 3 Quantitative Diagnosability Evaluation Based on the Fisher Information Distance
As the parameter \(\theta^{\prime}\) approaches \(\theta\), FIM measures the ability to distinguish between two adjacent parameters \(\theta^{\prime}\) and \(\theta\) from the data \(x\). This equation can be rewritten in parametric form as:  
\[g_{\alpha\beta}(\theta)=\left[\frac{\partial\mu(\theta)}{\partial\theta_{\alpha }}\right]^{T}\!C^{-1}(\theta)\!\left[\frac{\partial\mu(\theta)}{\partial\theta _{\beta}}\right]+0.5*tr\!\left[C^{-1}(\theta)\frac{\partial C(\theta)}{ \partial\theta_{\alpha}}*C^{-1}(\theta)\frac{\partial C(\theta)}{\partial \theta_{\beta}}\right] \tag{12}\]  
According to the Equation (12), the information metric \(g_{\alpha\beta}(\theta)\) of the attitude determination system studied under the fault set is Equation (13):  
\[g_{\alpha\beta}(\theta)=\left[ \begin{array}{cc}\frac{19.899}{400(1+f_{yy}{}^ {2}(k))^{2}}&-\frac{248.733}{1000(1+f_{yy}{}^{*}(k))\sqrt{f_{yy}}}\\ -\frac{248.733}{1000(1+f_{yy}{}^{*}(k))\sqrt{f_{yy}}(k)}&\frac{1,243.643}{1000 /f_{yy}(k)}\end{array} \right] \tag{13}\]  
Its determinant is:  
\[\frac{49.707}{2,000,000(1+f_{yy}{}^{2}(k))^{2}f_{hy}(k)} \tag{14}\]  
On a manifold, since space is curved, to define the distance between two points on the manifold, the length of the curve connecting the two points on the manifold should first be defined. The differential distance between two points (or two distributions) \(p(x|\theta)\) and \(p(x|\theta+d\theta)\) on a manifold can be expressed by the metric:  
\[ds^{2}=\Sigma_{\alpha\beta}g_{\alpha\beta}d\theta_{\alpha}d\theta_{\beta}=d \theta^{T}G(\theta)d\theta \tag{15}\]  
Considering that \(\theta(t)\in\Theta\) is the curve connecting \(\theta_{1}=\theta(t_{1})\) and \(\theta_{2}=\theta(t_{2}),t_{1}\leq t\leq t_{2}\), this curve can be described as a parametric equation with a single free parameter \(t\). The distance between the distributions \(p(x|\theta_{1})\) and \(p(x|\theta_{2})\) on the statistical manifold is defined by the curve \(\theta(t)\)[34]:  
\[D_{F}(\theta_{1},\theta_{2})=\int_{t_{1}}^{t_{2}}\!\left(\sqrt{\left(\frac{d \theta^{T}}{dt}\right)G(\theta)\frac{d\theta}{dt}}\right)dt \tag{16}\]  
where \(\triangleq\) stands for \"define as\". This integral distance depends on the choice of the curve \(\theta(t)\). Generally, the minimum value of curves for all possible connections is defined as the integral distance between the two distributions, and it is called the Fisher information distance (FID). The Fisher information distance between the distributions \(p(x|\theta_{1})\) and \(p(x|\theta_{2})\) is expressed as [35]:  
\[D_{F}(\theta_{1},\theta_{2})=\min_{\theta(t):\theta(t_{1})=\theta_{1},\theta(t _{2})=\theta_{2}}\int_{t_{1}}^{t_{2}}\!\left(\sqrt{\left(\frac{d\theta^{T}}{dt }\right)G(\theta)\frac{d\theta}{dt}}\right)dt. \tag{17}\]  
The curve with the smallest distance defined above is actually the geodesic that connects two points on the manifold, while the Fisher information distance is the length of the shortest geodesic connecting the two points. Geodesics are generalizations of straight lines on manifolds in Euclidean space, and the Fisher information distance is a generalization of the Euclidean distance on manifolds. The FID satisfies the property of the distance definition, and it is symmetrical for all \(\theta_{1},\theta_{2}\in\Theta\):  
\[D_{F}(\theta_{1},\theta_{2})=D_{F}(\theta_{2},\theta_{1}) \tag{18}\]  
For \(\theta_{1},\theta_{2},\theta_{3}\in\Theta\), the FID satisfies the triangle inequality:  
\[D_{F}(\theta_{1},\theta_{2})+D_{F}(\theta_{2},\theta_{3})\geq D_{F}(\theta_{1},\theta_{3}) \tag{19}\]Based on the above studies, the following detectability evaluation index is designed:  
\[F_{D}(f_{i})=D_{F}(\theta_{i},\theta_{0}) \tag{20}\]  
where \(\theta_{0}\) represents the parameters of the fault manifold in the normal state or fault-free state, and \(\theta_{i}=f_{i}(k)\), \(i=1,2,\ldots n\), represents the parameters of the fault manifold at time \(k\) in the case of a fault \(f_{i}\). \(g_{\alpha\beta}(\theta)\) is an all-zero matrix in the case of the fault-free state.