## 3 Observability of the Navigation System
### Observable Degree Analysis
Define \(\mathbf{D}_{k,k+m}=\left(\mathbf{I}+\mathbf{P}_{k}\mathbf{L}_{k,k+m}\right)^{-1}\) as the calculation matrix of the observable degree, and its diagonal element is the observable degree of the corresponding state variable, namely  
\[\eta_{k,i}=\left(\mathbf{D}_{k,k+m}\right)_{i,i} \tag{18}\]  
where \(\left(\mathbf{D}_{k,k+m}\right)_{i,i}\) is the \(i\)-th main diagonal element of \(\mathbf{D}_{k,k+m}\), \(\eta_{k,i}\) is the observable degree of the \(i\)-th state variable at the \(k\)-th moment. The larger \(\eta_{k,i}\) is, the weaker the observability of the \(i\)-th state variable, and the smaller \(\eta_{k,i}\) is, the stronger the observability of the \(i\)-th state variable at that time. **4. Optimized Filter Algorithm for Navigation System Based on the Observable Degree**  
As the observability is related to the filter accuracy tightly, one of the most efficient methods to increase the accuracy is to optimize the filter algorithm based on the observability. In this paper, a novel Adaptive extend Kalman Filter (AKF) base on the novel observability criterion is proposed for the navigation system of high-orbit satellites. The framework is summarized in Figure 1. The details of the method will be reported in the following part. The traditional process of EKF is reported as follows. The one-step prediction of the state vector is  
\[\mathbf{\hat{X}}_{k/k-1}=f\left(\mathbf{\hat{X}}_{k-1}\right). \tag{19}\]  
Figure 1: The framework of navigation filter optimization algorithm proposed in the paper. The error covariance matrix of \(\hat{\mathbf{X}}_{k/k-1}\) is  
\[P_{k/k-1}=F_{k/k-1}P_{k-1}F_{k/k-1}^{\mathrm{T}}+\mathbf{Q}_{k-1}. \tag{20}\]  
The filter gain matrix is  
\[\mathbf{K}_{k}=P_{k/k-1}\mathbf{H}_{k}^{\mathrm{T}}\Big{(}H_{k}P_{k/k-1}\mathbf{H}_{k}^{ \mathrm{T}}+\mathbf{R}_{k}\Big{)}^{-1}. \tag{21}\]  
The prediction of state vector \(\hat{\mathbf{X}}_{k}\) is  
\[\hat{\mathbf{X}}_{k}=\hat{\mathbf{X}}_{k/k-1}+\mathbf{K}_{k}\left(\mathbf{Z}_{k}-h\left(\hat{ \mathbf{X}}_{k/k-1}\right)\right). \tag{22}\]  
The error covariance matrix of \(\hat{\mathbf{X}}_{k}\) is  
\[\mathbf{J}_{k}=\left(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k}\right)^{ \mathrm{T}}\mathbf{P}_{k}^{-1}(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k})+\sum \limits_{i=k}^{k+m}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right)^{ \mathrm{T}}\mathbf{R}_{i}^{-1}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right) \tag{11}\]0  
Recent works contribute a variety of the frameworks of the adaptive Kalman Filter, in the paper, we choose the framework with an adjusting factor \(\mathbf{\psi}_{k}\) in front of the filter gain \(\mathbf{K}_{k}\) in Equation (21). Then the adaptive filter gain matrix can be written as  
\[\mathbf{J}_{k}=\left(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k}\right)^{ \mathrm{T}}\mathbf{P}_{k}^{-1}(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k})+\sum \limits_{i=k}^{k+m}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right)^{ \mathrm{T}}\mathbf{R}_{i}^{-1}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right) \tag{11}\]1  
where \(\mathbf{\psi}_{k}\) is a diagonal matrix. The value of the \(i\)-th diagonal element of \(\mathbf{\psi}_{k}\) is the adjusting factor of the corresponding \(i\)-th state variable. Increasing the value of the adaptive adjustment factor \(\mathbf{\psi}_{k}\) will change the value of the original gain matrix which can be considered as a balance parameter between the one-step predicted value of the state variable and the observed value. When the navigation system suffers a weak observable degree, in order to suppress the influence of the weak observable degree, adjusting the weighting method can be used to correct the bias of predicted state variables. As we discussed above, the observability plays an important role in the navigation system. The adjusting factor should also be designed for each state variable in order to suppress the influence of weak observable degree.