## 3 Steady-State Error Modeling and Analysis
### Steady-State Error Modeling of INS-Aided Second-Order PLLs
Therefore, using the tracking error caused by \(\delta\!V_{{}_{N}}(0)\) to express the steady-state error of INS-aided PLLs caused by the INS bias-type error sources:  
\[\theta_{{}_{\mathit{bias}}}=360\,\frac{\delta\!V_{{}_{N}}(0)}{\lambda\cdot e \cdot 1.89\,B_{{}_{n}}}(^{\circ}) \tag{10}\]  
where \(e\) is the Euler number and \(\lambda\) is the carrier wavelength. Since the INS scale factor-type errors are modeled as random constants, the steady-state error could be derived from its error transfer function. When the acceleration dynamics \(\Delta\!\mathfrak{R}\) appear, the INS scale factor-type errors bring frequency ramp estimation errors \(K_{a}\cdot\Delta\!\mathfrak{R}\) into the second-order PLL, which can generate a steady-state error. Therefore, the steady-state error of the INS-aided PLL caused by the INS scale factor-type errors can be expressed as:  
\[\theta_{{}_{K_{a}}}=\frac{360}{2\pi}\frac{K_{a}\cdot\Delta\!\mathfrak{R}}{\left( 1.89\,B_{{}_{n}}\right)^{2}}(^{\circ}) \tag{11}\]  
When the vehicle movement changes, the jitter brought into the second-order PLL caused by the aiding delay is the frequency step stimulus, which does not generate steady-state tracking error. Since the phase error \(\theta_{{}_{\mathit{bias}}}\) is uncorrelated with that due to thermal noise and oscillator noise, the steady-state error caused by \(\theta_{{}_{\mathit{bias}}}\) is the geometric sum relationship to that caused by thermal noise and oscillator noise. Therefore, compared with Equation (3), the one-sigma rule threshold for the INS-aided PLL tracking loop is:  
\[\sigma_{{}_{\mathit{Add\_PLL}}}=\sqrt{\sigma_{{}_{\mathit{tPLL}}}^{2}+\sigma _{{}_{\mathit{r}}}^{2}+\sigma_{{}_{\mathit{r}},t}^{2}+\theta_{{}_{\mathit{ bias}}}^{2}}+\theta_{{}_{\mathit{Ka}}}\,/\,3\leq 15(^{\circ}) \tag{12}\]  
The steady-state error model of the INS-aided second-order PLL can be given based on Equations (4)-(12), and it is expressed as:\[\sigma_{{}_{\mathit{Ad.-PLL}}}=\frac{180}{\pi} \begin{cases}\frac{B_{a}}{C\,/\,N_{ 0}}\left(1+\frac{1}{2T_{\mathit{coh.}}\cdot C\,/\,N_{0}}\right)+\frac{\pi^{2} \int_{0}^{2}K_{g}^{2}G_{g}}{2.67\,B_{a}}\\ \left.+2\pi^{2}\int_{0}^{2}\left(\frac{\pi^{2}\,h_{-1}}{\sqrt{2}\left(1.89B_{a} \right)^{3}}+\frac{\pi h_{-1}}{4\left(1.89B_{a}\right)^{2}}+\frac{h_{0}}{4 \sqrt{2}\left(1.89B_{a}\right)}+\frac{4\pi^{2}\delta V_{-3}^{2}\left(0\right)}{ \lambda^{2}e^{2}\left(1.89B_{a}\right)^{2}}+\frac{60K_{g}\Delta\Re}{\pi\left(1.8 9B_{a}\right)^{2}}\right.\end{cases}  \tag{13}\]  
Just as the parameter optimization idea of the normal PLL, if the GNSS signal strength and the vehicle dynamics are known, the optimal bandwidth of the INS-aided PLLs could be calculated based on Equation (13). Comparing Equation (13) with Equation (8), after the INS aiding, the effect of the INS bias-type errors is added into the steady-state error model; but the steady-state error caused by the dynamic stress error is replaced by the INS scale factor-type errors, which is only a small portion of the original dynamic stress error. In the next part, the steady-state error characteristic of PLLs before and after INS aiding is compared in detail based on the steady-state error models. Meanwhile, the optimal parameters of the normal PLLs and the INS-aided PLLs could be obtained. Moreover, thanks to the detailed model of the INS branch, the INS initial velocity error after correction of GNSS/INS integration \(\delta V_{N}^{\prime}(0)\) as the main factor of all of the INS bias-type errors is established in relation to the carrier phase tracking error shown in Equation (10), which has never been discussed in previous works.