# A Method for Autonomous Generation of High-Precision Time Scales for Navigation Constellations
## 5 Simulation Results and Analysis
### Experimental Analysis of Time-Frequency Steering Algorithm  
According to the physical parameters of the time scale TA(Rb) and TA(Cs), the fitness function for optimal parameter selection introduced in the previous section can be constructed through Equations (17), (24), (25) and (41).The SSB phase noise spectral densities of TA(Cs) and TA(Rb) are \(L(f)_{TA(Cs)}\) and \(L(f)_{TA(kb)}\); the intersection point of \(L(f)_{TA(Cs)}\) and \(L(f)_{TA(Rb)}\) is \(f(L(f))=5.4236\times 10^{-6}\)(Hz). Therefore, the optimization range of the genetic algorithm is \(\left[f_{down},f_{up}\right]=\left[5.4\times 10^{-6},5.5\times 10^{-6}\right] \left(\mathrm{Hz}\right)\). The minimum resolution for optimization is \(1\times 10^{-11}\mathrm{Hz}\). Normalize the noise variance parameter \(R\) to \(Q_{33}\); the genetic algorithm iteratively adjusts the normalized \(R\) and can approach the minimum value of the fitness function. The iterative convergence of the optimal fitness value is shown in Figure 9.  
The selection range of parameter \(R/Q_{33}\) in Figure 10 is \(\left[10^{20},10^{30}\right]\). The green mark represents the fitness value corresponding to the initial value of the population, and the red circle represents the minimum value of the fitness function obtained by the genetic algorithm. Because the minimum resolution of the fitness function is preset, when the emergence of the optimal population minimizes the fitness function value to 0, it indicates that the deviation between the dynamic debugging value \(f(H)\) and the fixed reference value \(f(L(f))\) exceeds the expected minimum resolution. Using time scale TA(Cs) and TA(Rb) data, select a group of optimal noise variance parameters obtained by genetic algorithm as group 1, make two types of intersection points in the frequency domain, and make statistics on the frequency deviation of the two types of intersection points, compared with the parameter groups 2-5 under the time-frequency  
Figure 10: Comparison chart of frequency domain intersection points under different parameter pair selection values. Figure 9: Convergence graph of genetic algorithm fitness value. steering mode of the existing discrete debugging and approximate parameter selection; the comparison results are shown in Figure 10 and Table 1. Obviously, the improved parameter selection method combined with the genetic algorithm has better deviation accuracy. Substituting the optimal DPLL gain into the closed-loop system transfer function (24), we get:  
\[H(z)=\frac{0.0101z^{2}-0.0201z+0.01}{0.9899z^{3}-2.9596z^{2}+2.9496z-0.9799}\]  
We can solve a set of poles, \(0.9975\pm 0.0043\)i and \(0.9949\), because two conjugate complex poles and one real pole are in the unit circle of the Z plane; the DPLL system determined by the optimal DPLL gain is stable. Figures 11 and 12 have shown the frequency stability and steering error of the output signal after being driven under different parameter selection situations. Parameter group 1 \(TA(Rb)_{1}\) is the time scale output by the time-frequency steering algorithm optimized by the genetic algorithm. Its frequency stability does not appear to be biased and unbalanced, and it can better take advantage of the frequency stability of each sampling interval period and its time. The steering error is in the middle of the balance. However, the frequency stability swing of the output signal corresponding to the parameter groups 2-4 corresponding to the existing manual repeated debugging method is obviously larger, and is the best overall under the balance of advantages that cannot be more accurately understood. Figures 10-12 and Table 1 also jointly characterize the influence of the time-frequency steer value determined by the five parameter control groups on the driving effect. When the noise variance parameter group is normalized by \(Q_{33}\), the larger \(R\) will affect the dynamic debugging value \(f(H)\) and deviate from the fixed reference value \(f(H)\) to the left.