# A Method for Autonomous Generation of High-Precision Time Scales for Navigation Constellations
## 4 Time-Frequency Driving Algorithm Principle
### Algorithm Principle of Equivalent DPLL Steering Control System  
The time-frequency steering system adopts the Kalman filter shown in Equation (6), and the three-dimensional state-space model of phase difference, frequency difference and frequency drift established is [19; 22]:  
\[\left\{ \begin{array}{l}X_{k+1}=A\cdot X_{k}+J_{k}\\ Z_{k}=H\cdot X_{k}+w_{k}\end{array} \right. \tag{18}\]  
In Equation (18), the first equation is the state equation, \(X_{k}= \begin{bmatrix}x_{k}&y_{k}&d_{k}\end{bmatrix} ^{T}\), \(x_{k}\), \(y_{k}\) and \(d_{k}\) respectively represent the three state variables of phase, frequency, and frequency change rate. \(J_{k}= \begin{bmatrix}0&0&u_{k}\end{bmatrix} ^{T}\), \(u_{k}\) is process noise. \[A= \begin{bmatrix}1&T&\frac{1}{2}\cdot T^{2}\\ 0&1&T\\ 0&0&1\end{bmatrix}  \tag{19}\]  
where \(A\) is the state transition matrix, \(T\) is the sampling time interval. The second equation in Equation (12) is the measurement equation, \(Z_{k}\) is the measured value, \(w_{k}\) is the measurement noise, \(H= \begin{bmatrix}1&0&0\end{bmatrix} \) is a linear connection matrix, which describes the phase deviation process between the to-be-driven frequency standard and the driving reference frequency standard in the time-frequency steering system. The covariance of the process noise \(u_{k}\) is:  
\[Q=E \begin{bmatrix}J_{k}\cdot J_{k}^{T}\end{bmatrix} = \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&E\begin{bmatrix}u_{k}^{2}\end{bmatrix}\end{bmatrix} = \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&Q_{33}\end{bmatrix}  \tag{20}\]  
where \(u_{k}\sim N(0,Q_{33})\). The covariance of measurement noise \(w_{k}\) is \(R=E \begin{bmatrix}w_{k}^{2}\end{bmatrix} \), \(w_{k}\sim N(0,R)\). Equations (21) and (22) represent the input-output relationship between the measured value \(Z_{k}\) and the estimated value \(\hat{x}_{k}\) of the phase state variable in the steady-state Kalman filter. \[\widetilde{X}=\frac{G^{\prime}(z)}{1-K_{s1}+G^{\prime}(z)}\cdot\widetilde{Z} \tag{21}\]\[G^{\prime}(z)=\frac{Ks_{1}\cdot\left(1-z^{-1}\right)^{2}+\left(Ks_{2}\cdot T+ \frac{1}{2}\cdot Ks_{3}\cdot T^{2}\right)\cdot z^{-1}\cdot\left(1-z^{-1}\right)+ Ks_{3}\cdot T^{2}\cdot z^{-2}}{\left(1-z^{-1}\right)^{3}} \tag{22}\]  
where \(\widetilde{X},\widetilde{Z}\) are the z-transform forms of \(\hat{x}_{k}\) and \(Z_{k}\), respectively. Through Equation (21), the closed-loop system transfer function structure consistent with the third-order type-three DPLL can be obtained [23], which is a further extension of the two-dimensional state-space model Kalman filter [22]. In order to ensure the normal and orderly output of the time-frequency steer value in the loop filter, it is considered to add a delay device \(z^{-1}\). From Equation (21) and the delay device, the open-loop system transfer function \(G(z)\), closed-loop system transfer function \(H(z)\) and closed-loop error transfer function \(He(z)\) equivalent to the DPLL system can be obtained, which are expressed as Equations (23)-(25):  
\[G(z)=\frac{z^{-1}}{1-K_{s11}}\cdot G^{\prime}(z) \tag{23}\]  
\[H(z)=\frac{G(z)}{1+G(z)} \tag{24}\]  
\[He(z)=\frac{1}{1+G(z)} \tag{25}\]  
The Kalman filter guarantees the stability of the system because of its complete observability, and the equivalent DPLL with a delay is represented by Equations (23) and (24); in addition, (25) has also been verified by simulation. The noise variance parameters \(Q_{22}\) and \(R\) are only used to determine the steady-state Kalman gain \(K_{s}\) equivalent to the DPLL gain when constructing the equivalent DPLL, [PERSON] [22] derived and verified the approximate relationship between the noise variance parameters \(Q_{22}\), \(R\) and the steady-state Kalman gain \(K_{s}\) in the two-dimensional Kalman filter model. By analyzing the five steps of Kalman filtering, \(P_{k,k}\) is the estimated covariance matrix, and \(P_{k,k-1}\) is the predicted covariance matrix.