## 3 Observability of the Navigation System
### Observability Qualitative Analysis  
For the navigation model (4), according to the state equation, we can get  
\[\mathbf{X}_{i}=\mathbf{F}_{i,i-1}\mathbf{F}_{i-1,i-2}\cdots\mathbf{F}_{k+1,k}\mathbf{X}_{k}+ \overline{\mathbf{w}}_{i}=\mathbf{F}_{i,k}\mathbf{X}_{k}+\overline{\mathbf{w}}_{i} \tag{6}\]  
where \(\overline{\mathbf{w}}_{i}=\sum\limits_{j=k+1}^{i}\mathbf{F}_{i,j}\mathbf{w}_{j-1},i=k,k+1,...,k+m\), \(\mathbf{F}_{i,j}=\prod\limits_{s=1}^{i-j}\mathbf{F}_{i-s+1,i-s}\). According to the observability definition, given \(m\) measurement data, \(m\) measurement equations are combined, and Equations (4) and (6) can be obtained. We can get  
\[\begin{cases}\mathbf{Z}_{k}=\mathbf{H}_{k}\mathbf{X}_{k}+\mathbf{v}_{k}\\ \mathbf{Z}_{k+1}=\mathbf{H}_{k+1}\mathbf{X}_{k+1}=\mathbf{H}_{k+1}\left(\mathbf{F}_{k+1,k}\mathbf{X}_ {k}+\overline{\mathbf{w}}_{k+1}\right)+\mathbf{v}_{k+1}=\mathbf{H}_{k+1}\mathbf{F}_{k+1,k}\bm {X}_{k}+\mathbf{H}_{k+1}\overline{\mathbf{w}}_{k+1}+\mathbf{v}_{k+1}\\ \mathbf{Z}_{k+2}=\mathbf{H}_{k+2}\mathbf{X}_{k+2}\text{=}\mathbf{H}_{k+2}\left(\mathbf{F}_{k+2,k} \mathbf{X}_{k}+\overline{\mathbf{w}}_{k+2}\right)+\mathbf{v}_{k+2}=\mathbf{H}_{k+2}\mathbf{F}_{k+2,k}\mathbf{X}_{k}+\mathbf{H}_{k+2}\overline{\mathbf{w}}_{k+2}+\mathbf{v}_{k+2}\\ \vdots\\ \mathbf{Z}_{k+m-1}=\mathbf{H}_{k+m-1}\mathbf{X}_{k+m-1}=\mathbf{H}_{k+m-1}\left(\mathbf{F}_{k+m-1, k}\mathbf{X}_{k}+\overline{\mathbf{w}}_{k+m-1}\right)+\mathbf{v}_{k+m-1}\\ =\mathbf{H}_{k+m-1}\mathbf{F}_{k+m-1,k}\mathbf{X}_{k}+\mathbf{H}_{k+m-1}\overline{\mathbf{w}}_{k+ m-1}+\mathbf{v}_{k+m-1}\end{cases} \tag{7}\]  
Let \(\mathbf{Z}_{k,m}=\left[\mathbf{Z}_{k}^{\text{T}},\mathbf{Z}_{k+1}^{\text{T}},\cdots,\mathbf{ Z}_{k+m-1}^{\text{T}}\right]^{\text{T}}\), \(\mathbf{O}_{k,m}=\left[\mathbf{H}_{k}^{\text{T}},\mathbf{F}_{k+1,k}^{\text{T}}\mathbf{H}_{k}^ {\text{T}},\cdots\mathbf{F}_{k+m-1,k}^{\text{T}}\mathbf{H}_{k+m-1}^{\text{T}}\right]^{ \text{T}}\), \(\mathbf{V}_{k,m}=\left[\mathbf{v}_{k^{\prime}}^{\text{T}}(\mathbf{H}_{k+1}\overline{\mathbf{w}}_ {k+1}+\mathbf{v}_{k+1})^{\text{T}},\cdots,(\mathbf{H}_{k+m-1}\overline{\mathbf{w}}_{k+m-1}+ \mathbf{v}_{k+m-1})^{\text{T}}\right]^{\text{T}}\) Then the Equation (6) can be transferred to  
\[\mathbf{Z}_{k,m}=\mathbf{O}_{k,m}\mathbf{X}_{k}+\mathbf{V}_{k,m} \tag{8}\]Based on the weighted least squares, we can get  
\[\mathbf{O}_{k,m}^{\mathrm{T}}\mathbf{G}_{k,m}^{-1}\mathbf{Z}_{k,m}=\mathbf{O}_{k, m}^{\mathrm{T}}\mathbf{G}_{k,m}^{-1}\mathbf{O}_{k,m}\mathbf{X}_{k} \tag{9}\]  
where \(\mathbf{G}_{k,m}\) is the covariance matrix of \(V_{k,m}\). To make the above equation have a unique solution, \(\mathbf{O}_{k,m}^{\mathrm{T}}\mathbf{G}_{k,m}{}^{-1}\mathbf{O}_{v}\) should be nonsingular. Let \(\mathbf{\ }B_{k,m}=\mathbf{O}_{k,m}^{\mathrm{T}}\mathbf{G}^{-1}\mathbf{O}_{k,m}= \sum\limits_{i=k}^{k+m-1}\mathbf{F}_{i,k}^{\mathrm{T}}\mathbf{H}_{i}^{\mathrm{ T}}\left(\sum\limits_{j=k+1}^{i}\mathbf{H}_{i}\mathbf{F}_{i,j}\mathbf{Q}_{j-1} \mathbf{F}_{i,j}^{\mathrm{T}}\mathbf{H}_{i}^{\mathrm{T}}+\mathbf{R}_{i} \right)^{-1}\mathbf{H}_{i}\mathbf{F}_{i,k}\), it is defined as the observability discrimination matrix. If the system is observable, since \(\mathbf{B}_{k,m}\) is a positive semi-definite matrix, \(\mathbf{B}_{k,m}\) being nonsingular is equivalent to that being a positive definite matrix, that is, \(\mathbf{B}_{k,m}>0\). Calculate the condition number of \(\mathbf{B}_{k,m}\) in different periods, \(\mathit{cond}\left(\mathbf{B}_{k,m}\right)=\sigma_{\mathrm{max}}/\sigma_{ \mathrm{min}}\), where \(\sigma_{\mathrm{max}}\) and \(\sigma_{\mathrm{min}}\) are the largest and the smallest singular values of the matrix respectively. If the condition number is large, it indicates that the overall observability of the system at that moment is weak; if the condition number is small, it indicates that the overall observability of the system at that moment is strong.