## 3 Observability of the Navigation System
### Observable Degree Analysis  
Due to the inconsistency of the information validity between different state variables, in order to more accurately pass the state variable information to the subsequent filter, the observable degree of each state component of the system needs to be given. The method of the observable degree of each state variable in this article is given below. In the filter process, the filter estimation of state variables is a sequential process, and there is a relationship between different moments. Regarding the estimated value of the state variable at the previous moment as a random variable, the estimated error covariance of the random variable will affect the prediction accuracy of the filter at the current moment and subsequent moments. The traditional filter only quotes the measurement information at various previous moments to estimate the state value at the current moment, which is not enough for a stable sequence process. The measurement information at subsequent moments can also provide useful information for the current state estimation. Therefore, we set an objective function to minimize the error value of the current state variable estimation and the subsequent measurement information, in order to determine the best estimate at the current moment. Based on the above optimization ideas, we give an optimal estimation problem as follows. At the \(k\)-th moment, we give a set of measured values \(\mathbf{Z}_{k},\mathbf{Z}_{k+1},\cdots\mathbf{Z}_{k+m}\) from \(k\) to \(k+m\) to find the optimal estimation \(\mathbf{\hat{X}}_{k,k+m}\) of \(\mathbf{X}_{k}\) that minimizes the loss function (10) [37]. \[\mathbf{J}_{k}=\left(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k}\right)^{ \mathrm{T}}\mathbf{P}_{k}^{-1}(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k})+\sum \limits_{i=k}^{k+m}\mathbf{Z}_{i}^{\mathrm{T}}\mathbf{R}_{i}^{-1}\mathbf{Z}_{i} \tag{10}\]  
where \(\overline{\mathbf{X}}_{k}\) is the expectation of the state variable at the \(k\)-th moment, \(\mathbf{P}_{k}\) is the estimation error covariance matrix at the \(k\)-th moment. \(\mathbf{\hat{Z}}_{i}=\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\) is the error between the measured value and the measured estimated value at the \(i\) (\(i=k,\cdots,k+m\))-th moment, \(\mathbf{R}_{i}\) is the covariance matrix of the measurement error at the \(i\)-th moment. Substitute \(\mathbf{\hat{Z}}_{i}=\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\) into the right side of Equation (10), we can get:  
\[\mathbf{J}_{k}=\left(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k}\right)^{ \mathrm{T}}\mathbf{P}_{k}^{-1}(\overline{\mathbf{X}}_{k}-\mathbf{X}_{k})+\sum \limits_{i=k}^{k+m}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right)^{ \mathrm{T}}\mathbf{R}_{i}^{-1}\left(\mathbf{Z}_{i}-\mathbf{H}_{i}\mathbf{X}_{i}\right) \tag{11}\]  
Substitute Formula (10) into (11), we get  
After simplification, we can get:  
\[\mathbf{\tilde{X}}_{k,k+m}=\left(\mathbf{I}+\mathbf{P}_{k}\mathbf{L}_{k,k+m}\right)^{-1}\left( \mathbf{\tilde{X}}_{k,k}-\mathbf{P}_{k}\sum_{i=k}^{k+m}\mathbf{F}_{i,k}^{\mathrm{T}}\mathbf{H}_{ i}^{\mathrm{T}}\mathbf{R}_{i}^{-1}\mathbf{v}_{i}\right) \tag{17}\]  
It can be seen from Equation (17) that the state estimation error obtained is proportional to \(\left(\mathbf{I}+\mathbf{P}_{k}\mathbf{L}_{k,k+m}\right)^{-1}\), and \(\left(\mathbf{I}+\mathbf{P}_{k}\mathbf{L}_{k,k+m}\right)^{-1}\) represents the ratio of the state estimation error value obtained at the \(k\)-th moment from the previous moment to the state estimation error estimated by the measurement at the later moment, and it can be expressed as the error loss of the state estimation.