## 3 Steady-State Error Modeling and Analysis
### Steady-State Error Analysis  
In order to analyze the INS effect on the PLLs, a typical low-grade INS with the model of MTI-G and a typical medium-grade INS with the model of FSAS are selected as illustrations [24, 25], with their initial velocity errors \(\delta V_{N}(0)\) of about 0.02 m/s and 0.005 m/s, respectively, based on GNSS real-time kinematic (RTK)/INS integration testing. In addition, an oven controlled crystal oscillator (OCXO) is used with the parameters listed in Table 1.  
Since the INS bias-type errors are extra added error sources to the PLL, their impact to the steady-state error should be analyzed (under static conditions). Assuming that the acceleration \(\Delta\Re\) equals zero, Figures 6 and 7 show the steady-state tracking errors due to each individual error source and the total effect based on their steady-state error models (Equations (8) and (13)). Figure 6 depicts the relationship between the bandwidth and the steady-state error of the low-grade INS (MTI-G)-aided PLLs with an integration time of 1 ms and 30-dB-Hz signal power under static conditions. When the horizontal axis represents the noise bandwidth \(\mathit{B_{n}}\) (\(\mathit{B_{n}}\) =0.53\(\mathit{\alpha_{n}}\)) of the PLL, the vertical axis is the steady-state phase tracking error in degrees. Similar to the trend of the steady-state error due to OCXO errors \(\sigma_{rv}\), \(\sigma_{r,t}\), the steady-state error due to the INS bias-type errors \(\mathit{\theta_{bias}}\) rapidly decreases with the widening of the bandwidth. Hence, compared with the normal PLL's steady-state error \(\sigma_{\mathit{PLL}}\), the low-grade INS PLL's steady-state error \(\sigma_{\mathit{Aid.-PLL}}\) almost has no increase with a wide \begin{table}
\begin{tabular}{c c c c c} \hline \(K_{g}\) & \(G_{g}\) & \(h_{-2}\) & \(h_{-1}\) & \(h_{0}\) \\ \hline
1e\(-\)10 (1/g) & 0.05 (g\({}^{2}\)/Hz) & 2.51e\(-\)22 & 2.51e\(-\)23 & 2.51e\(-\)26 \\ \hline \end{tabular}
\end{table} 
Table 1: OCXO parameters. bandwidth. Only when the bandwidth is too narrow, \(\mathbf{\sigma}_{\mathit{idid\_PLL}}\) is just a little larger (less than one degree) than \(\mathbf{\sigma}_{\mathit{PLL}}\). Therefore, when the acceleration equals zero (under static condition), the low-grade INS does not obviously increase the PLL's steady-state error, unless the bandwidth is excessively narrow. Compared with Figure 6, Figure 7 shows the relationship of the bandwidth and the steady-state error of the medium-grade INS FSAS-aided PLL. The INS bias-type errors caused steady-state error \(\mathbf{\theta}_{\mathit{bus}}\) of the PLL, with the medium-grade INS aiding being less than that with the low-grade INS aiding, especially when the bandwidth is narrow. Since the curve of \(\mathbf{\sigma}_{\mathit{idPL}}\) is in good agreement with that of \(\mathbf{\sigma}_{\mathit{PLL}}\), the medium-grade INS does not obviously increase the static steady-state errors, even if  
Figure 6: Tracking errors of MEMS INS-aided PLL under static conditions. MTI-G is a model of a typical low-grade INS. Figure 7: Tracking errors of medium-grade INS-aided PLL under static conditions. FSAS is a model of a typical medium-grade INS. the bandwidth is excessively narrow. Therefore, if the PLL is aided by a medium-grade INS, the effect of \(\mathbf{\theta}_{\textit{bias}}\) on the total phase tracking error can be ignored. Under dynamic conditions, the steady-state error caused by the dynamic stress should be considered. When the vehicle acceleration is 9.8 m/s\({}^{2}\), the carrier to noise ratio (CNR) is 40 dB-Hz and the integration time is 1 ms, the relationships of the steady-state tracking errors and the bandwidths caused by normal second-order PLL, MTI-G-aided PLL and FSAS-aided PLL are as shown in Figure 8. It can be seen that the optimal bandwidth of the normal PLL is wider than 25 Hz, and its minimum tracking error is larger than five degrees; the optimal bandwidth of PLL with MTI-G or FSAS aiding is narrower than 5 Hz with the minimum tracking error being no more than three degrees.