## 3 Quantitative Diagnosability Evaluation Based on the Fisher Information Distance  
Parameter vectors usually exist in an abstract manifold, and the manifolds corresponding to a real system usually have a complex topology. Consider the probability distribution parameter family \(S=\left\{p(x|\theta)\right\}\), where \(x\) is a random variable, \(\theta=\left[\theta_{1},\ldots,\theta_{n}\right]^{T}\) is an \(n\)-dimensional parameter vector with a particular distribution, and \(S\) is a statistical manifold with the (local) coordinate system \(\theta\). For a particular \(\theta\) that belongs to the parameter space \(\Theta\in R^{n}\), the measured and observed value of x belongs to the sampling space \(X\in R^{n}\)then, each \(p(x|\theta)\) corresponds to an actual probability distribution, that is, each probability distribution \(p(x|\theta)\) corresponds to a point on the statistical manifold S.  
Take the vector \(N_{H}Ff\) containing the fault information as the mean value \(\mu(\theta)\) of the fault manifold and take the interference vector \(N_{H}Ee\) as the variance \(C(\theta)\) of the fault manifold, namely:  
\[N_{H}Ff=N_{H} \begin{bmatrix}B_{f}&0\\ 0&D_{f}\end{bmatrix}\begin{bmatrix}f(k)\\ f(k)\end{bmatrix} =N_{H} \begin{bmatrix}0.1&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}  \tag{5}\]  
\[N_{H}Ee=N_{H} \begin{bmatrix}B_{w}&0\\ 0&D_{v}\end{bmatrix}\begin{bmatrix}w(k)\\ v(k)\end{bmatrix} =N_{H} \begin{bmatrix}w(k)\\ v(k)\end{bmatrix}  \tag{6}\]  
The values of coefficient matrices \(B_{f},D_{f},B_{w},D_{v}\) in Equation (1) are known; now, only \(N_{H}\) needs to be determined, and \(N_{H}\) is the left orthogonal basis for the null space of \(H\), which means that \(N_{H}H\ =\ 0\). The matrix \(H\) is determined by the system matrix \(A\) and the output matrix \(C\) of Equation (1) [32]. Extending the system to a full-dimensional observable system, let \(C=I_{3}\); then,  
\[H= \begin{bmatrix}A\\ C\end{bmatrix} = \begin{bmatrix}1&-0.1&-0.1\\ 0&-0.990909&0\\ 0&0&1\\ 1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}  \tag{7}\]  
Then,  
\[\mu(\theta)= \begin{bmatrix}-0.0705&0.7050\\ 0.0015&-0.0148\\ 0.0015&-0.0146\end{bmatrix}\begin{bmatrix}\arctan f_{gy}(k)\\ \sqrt{f_{hy}}(k)\end{bmatrix} = \begin{bmatrix}-0.0705\arctan f_{gy}(k)+0.705\sqrt {f_{hy}(k)}\\ 0.0015\arctan f_{gy}(k)-0.0148\sqrt{f_{hy}(k)}\\ 0.0015\arctan f_{gy}(k)-0.0146\sqrt{f_{hy}(k)}\end{bmatrix}  \tag{8}\]  
\[C(\theta)= \begin{bmatrix}N\begin{pmatrix}0,10^{-4}\\ 0&N\end{pmatrix}0\\ 0&0&N\begin{pmatrix}0,10^{-4}\\ 0&0\end{pmatrix}\\ 0&0&N\begin{pmatrix}0,10^{-4}\end{pmatrix}\end{bmatrix}  \tag{9}\]  
Different values of the parameter \(\theta\) represent different types of faults. Considering the nonlinearity caused by the sensor measurement conversion, the settings in this paper are as follows:  
\[\theta=f^{T}(k)= \begin{bmatrix}g_{f}&\theta_{f}\end{bmatrix} = \begin{bmatrix} \arctan f_{gy}(k)&\sqrt{f_{hy}(k)}\end{bmatrix}  \tag{10}\]  
The manifold of a fault system is obtained, and the fault probability distribution \(p(x|\theta)\) and parameterized probability distribution family \(S=\{p(x|\theta)\}\) constitute an n-dimensional statistical manifold. Here, \(\theta\) is the distribution parameter (and the fault parameter vector), \(\mu(\theta)\) is the mean value, and \(C(\theta)\) is the variance. Different faults have different \(\mu(\theta)\) and \(C(\theta)\) in this manifold. In this statistical manifold, \(\theta\) is the coordinate system and is global. In statistical manifolds, a Fisher information matrix (FIM) is the unique Riemannian geometric metric tensor for the parameterized probability distribution family [33], and it is expressed as \((G(\theta)=[g_{\alpha\beta}(\theta)]\). The FIM is given by the following:  
\[g_{\alpha\beta}(\theta)=E\{\frac{\partial\log p(x|\theta)}{\partial\theta_{ \alpha}},\frac{\partial\log p(x|\theta)}{\partial\theta_{\beta}}\} \tag{11}\]where \(E\) is the mathematically expected value.