# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 7 Conclusions
Thus, \(c_{\text{min}}\) will be determined with the help of \(\lambda_{\mathbf{\Psi\Theta}}^{\text{max}}\) and the maximum eigenvalue of \(\mathbf{\Psi}_{\text{max}}\mathbf{\Theta}\), \(\lambda_{\mathbf{\Psi},\text{max}}^{\text{max}}\). Because \(\mathbf{\Psi}\), \(\mathbf{\Psi}_{\text{max}}\), and \(\mathbf{\Theta}~{}(=(c\mathbf{I}~{}+~{}\mathbf{\Psi}_{\text{max}})^{-1})\) are block diagonal matrices, \(\lambda_{\mathbf{\Psi\Theta}}^{\text{max}}\) is \(\max_{i}\{\



It can be easily shown that for any \(c>0\), \(\lambda^{\max}_{\overline{\
u},\max\mathbf{\Theta}}=\lambda^{\max}_{\overline{ \
u},\max}/(c\ +\lambda^{\max}_{\overline{\
u},\max})\ <1\). On the other hand, \(\lambda^{\max}_{\overline{\
u}\mathbf{\Theta}}=1/(\lambda^{\min}_{\mathbf{D}} \ +\ 1)\), where \(\mathbf{D}=(c\ \mathbf{I}\ +\ \Delta\mathbf{V})\mathbf{V}^{-1}\) and \(\Delta\overline{\
u}=\overline{\
u}_{\max}\ \mathbf{\cdot}\ \mathbf{V}\). So, for \(\lambda^{\max}_{\overline{\
u}\mathbf{\Theta}}<1\), \(\lambda^{\min}_{\mathbf{D}}\) should be greater than zero. From the structure, it is evident that \(\mathbf{D}\) is a block diagonal matrix. Its eigenvalues are the same as those of \(\mathbf{V}^{-1/2}(c\ \mathbf{I}\ +\ \Delta\mathbf{V})\mathbf{V}^{-1/2}\) and hence real. Since with the modeled \(a_{\ell q}\) of Equation (20), the trace of \(\Delta\mathbf{V}\) (or the sum of eigenvalues) is zero, its minimum eigenvalue is negative. To ensure \(\lambda^{\min}_{\mathbf{D}}>0\), the minimum \(c\ (c^{\prime})\) is proposed to be the absolute value of the minimum eigenvalue of \(\Delta\mathbf{V}\) or \(\left|\min_{i}\lambda^{\min}_{\Delta\mathbf{V}^{i},\,\text{sub}}\right|\) plus a non-zero number 0.01. \(\Delta\mathbf{V}^{i}_{\text{sub}}\) is \((\mathbf{V}^{i}_{\text{sub},\,\max}-\mathbf{V}^{i}_{\text{sub}})\), and \(\lambda^{\min}_{\Delta\mathbf{V}^{i},\text{sub}}\) is its minimum eigenvalue. The first two subplots of Figure 1 illustrate the proposed \(c^{\prime}\) for GPS and NavIC satellites as a function of initial elevation angle over \(M\) epochs and with \(M\) as a running parameter. As \(\mathbf{D}\) is a block diagonal matrix, the third subplot of the figure actually shows variations of \(\lambda^{\min}_{\mathbf{D}}\) with elevation angle and different \(M\). It is apparent that with the selected \(c^{\prime}\), \(\lambda^{\min}_{\mathbf{D}}\) is always greater than zero. Although results are shown for decreasing elevation angles (i.e., \(\theta^{(\ell+1)}(i)\ =\ \theta^{(\ell)}(i)\ -\ |\frac{\
u^{\text{set}}_{m}+\
u^{ \text{user}}_{m}}{r^{(\ell)}_{su}}|\times\frac{180}{\pi}\)), similar plots can be obtained when \(\theta\) increases over \(M\) epochs. This can be verified with the Matlab program, matlab_program_appendix_figure.m, of the Supplementary Materials. The maximum value of all \(c^{\prime}\)s for a given \(M\) considering both GPS and NavIC and all elevation angles is denoted as \(c^{\prime\prime}\). In addition, another \(c^{\prime}\) denoted as \(c^{\prime\prime}\) is calculated as before, but with \(\mathbf{\Psi}^{i}_{\text{sub},\,\min}\) (or \(\mathbf{\Psi}_{\min}\)) in place of \(\mathbf{\Psi}^{i}_{\text{sub}}\) (or \(\mathbf{\Psi}\)). \(\mathbf{\Psi}^{i}_{\text{sub},\,\min}\) is formed the same way as that of \(\mathbf{\Psi}^{i}_{\text{sub},\,\max}\), but with \(a_{\min}\). The maximum of \(c^{\prime\prime}\) and \(c^{\prime\prime}\) is \(c_{\min}\) for a given \(M\). During the adaptive calculation, the initial guess of \(c\) for \(a^{\prime}_{1}\) should be \(c_{\min}\) or more. With the selected \(c_{\min}\), \(\lambda^{\min}_{\mathbf{D}}\) is greater than \(c_{\min}/\lambda^{\max}_{\overline{\
u},\,\max}\) for all elevation angles and \(M\leq 20\) (see matlab_program_appendix_figure.m), which ensures that \(\lambda^{\max}_{\overline{\
u}\mathbf{\Theta}}<\lambda^{\max}_{\overline{\
u},\max\mathbf{\Theta}}\). It can also be shown that for a \(c\) higher than \(c_{\min}\), this inequality holds.