CN104898404B - The digital phase-locked loop atomic clock for being equivalent to two-state variable Kalman filter controls method - Google Patents

Abstract

A kind of digital phase-locked loop atomic clock for being equivalent to two-state variable Kalman filter controls method, by a DPLL for being equivalent to two-state variable Kalman filter, hydrogen clock is controlled with caesium clock.The closed loop system transmission function of the DPLL has been derived first, it was demonstrated that the DPLL is 2 class DPLL of second order.Then the Kalman gains approximately determined by process-noise variance and observation noise variance ratio have been derived.Then, from the closed loop transfer function of DPLL, the ratio, and loop gain, the sampling interval, this 3 parameters determined the performance of the DPLL.Further, The present invention gives the method for a simple and effective Selecting All Parameters.By equivalent transformation, obtain being equivalent to the analog phase-locked look (APLL) of the DPLL.In a frequency domain, by adjusting parameter, make the intersection frequency of the ssystem transfer function and error transfer function of the APLL of equal value be equal to the intersection frequency of the SSB phase noise curves of hydrogen clock and caesium clock, finally obtain optimized parameter.Which is effectively controlled to hydrogen clock with caesium clock, controls performance very good.

Description

Digital phase-locked loop atomic clock control method equivalent to two-state variable Kalman filter

Technical Field

The invention relates to the field of time frequency and signal processing, in particular to a digital phase-locked loop atomic clock control method which is equivalent to a two-state variable Kalman filter.

Background

Atomic clock steering technology plays an important role in time-keeping laboratories and satellite navigation systems. There are two main purposes of handling atomic clocks: firstly, the time of the driven atomic clock is synchronized with the time of the reference atomic clock, and the time deviation between the driven atomic clock and the reference atomic clock is reduced as much as possible; and secondly, the long-term stability of the driven atomic clock is improved.

Typical atomic clock manipulation methods include two broad categories: an open loop steering method and a closed loop steering method. The core of the open loop driving method is to design a reasonable prediction algorithm. The closed loop control method mainly comprises the following steps: a linear quadratic (LGQ) control method, a switch (Bang-Bang) control method, and a Digital Phase Locked Loop (DPLL) method. However, the parameters of these closed-loop control methods are not easily selected, and generally, for each specific application, a large number of parameters are compared through a large number of simulation experiments to select an optimal set of parameters. In addition, for the DPLL method, if the parameters are not properly selected, not only the handling performance cannot be guaranteed, but also the control system is unstable.

Disclosure of Invention

In order to solve the problems in the prior art, the invention aims to provide a digital phase-locked loop atomic clock control method which is equivalent to a two-state variable Kalman filter.

The principle of the invention is as follows: the hydrogen clock is steered with the cesium clock through a DPLL equivalent to a two-state variable Kalman filter. The invention derives the closed-loop system transfer function of the DPLL, and proves that the DPLL is a second-order class-2 DPLL. The present invention derives a Kalman gain determined approximately by the ratio of the process noise variance and the observed noise variance. Thus, as can be seen from the closed loop transfer function of the DPLL, the 3 parameters of the ratio, as well as the loop gain and the sampling interval, determine the performance of the DPLL. Furthermore, the invention provides a simple and effective method for selecting parameters. Through the equivalent transformation, an Analog Phase Locked Loop (APLL) equivalent to the DPLL is obtained. In the frequency domain, by adjusting parameters, the intersection frequency of the system transfer function and the error transfer function of the equivalent APLL is equal to the intersection frequency of SSB phase noise curves of a hydrogen clock and a cesium clock, and finally the optimal parameters are obtained. Briefly, a reference atomic clock (or timescale) is used to steer the atomic clock (or timescale) such that the steered atomic clock (or timescale) remains synchronized with the reference atomic clock (or timescale).

The technical scheme adopted by the invention is as follows:

a digital phase-locked loop atomic clock control method equivalent to a two-state variable Kalman filter adopts the following steps:

s.1, deriving a transfer function of a DPLL equivalent to a two-state variable Kalman filter;

s.1.1 use DPLL and cesium bell to drive hydrogen bell. Fig. 1 depicts the principle structure of the DPLL, summarized as: acquiring the time deviation of the cesium clock and the hydrogen clock after control; filtering the time deviation by using a loop filter to filter high-frequency noise; and then adjusting the time deviation of the hydrogen clock by using the filtered time deviation to obtain the time deviation of the hydrogen clock after driving. In fig. 1: cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, symbol Steered Hm represents the output signal of the hydrogen clock being driven, and LF represents the loop filter.

In the Z domain, the transfer function of the loop filter LF of the DPLL is denoted g (Z); the output of the DPLL is then:

it is clear that,anda system transfer function and an error transfer function of the DPLL respectively; cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, and symbol Steered Hm represents the output signal of the hydrogen clock being driven.

By z ═ e^j2πf·T(where f represents frequency and T represents sampling period) to equate the DPLL to a corresponding APLL; thus, equation (2) is expressed in terms of SSB phase noise, written as:

wherein, symbol L represents SSB phase noise, symbol G represents transfer function; subscript Cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, and symbol Steered Hm represents the output signal of the hydrogen clock being driven;

s.1.2, deriving expressions of a system transfer function and an error transfer function of the DPLL equivalent to the two-state variable Kalman filter according to the system transfer function and the error transfer function of the step S.1.1;

finally, the system transfer function of the DPLL is expressed as:

wherein, K₀Is the loop gain. Ks₁₁And Ks₂₁Respectively, the steady state gains of the Kalman filters; t is the sampling interval.

The error transfer function of the DPLL is expressed as:

to this endThe system transfer function and the error transfer function of the DPLL equivalent to a two-state variable Kalman filter are obtained. Represented by formula (22) and formula (23): parameter (Ks)₁₁,Ks₂₁,T,K₀) Determines the performance of the DPLL.

S.2, deducing Ks in the system transfer function and the error transfer function according to the system transfer function and the error transfer function of the DPLL obtained in the step S.1₁₁And Ks₂₁By process noise variance Q₂₂Observation noise variance R, and sampling interval T.

When the Kalman filter enters a steady state, obtaining

And

from equations (30) and (35), the Kalman gain is given by Q₂₂And R.

Let Q₂₂＝1s²Equation (30) and equation (35) are substituted for equation (22) and equation (23) to obtain a system transfer function and an error transfer function, which are respectively expressed as:

and

as can be seen from equations (36) and (37): when Q is₂₂＝1s²When, the parameters (R, T, K)₀) Determines the performance of the DPLL. Substituting equations (36) and (37) for equation (1) yields the relation between DPLL input and output:

s.3, determining parameters (R, T, K) according to the system transfer function and the error transfer function obtained in the step S.2₀) Optimizing the frequency stability of the DPLL output specifically comprises the following steps:

first, the phase noise of the cesium clock and the hydrogen clock is expressed as:

and

wherein the symbol L represents SSB phase noise, f₀Is the carrier frequency, h_iFor noise figure, f is the sideband frequency, i is used to indicate the noise type.

By solving equation L_Cs(f)-L_Hm(f) The frequency at the intersection of the cesium clock phase noise curve and the hydrogen clock phase noise curve can be found as f' 0.

Secondly, changing z to e^j2πf·TThe system transfer function and the error transfer function of the equivalent APLL are obtained by substituting equations (36) and (37), which are respectively expressed as:

and

from equations (41) and (42), the amplitude-frequency responses 20log | h (f) and 20log | he (f) of the system transfer function and the error transfer function are obtained, and the frequency at the intersection of these two curves is obtained and is denoted as f ".

Third, for a given T, K is given first₀Set a value (for convenience of analysis, it can be set as an integer of 1, 2, etc. first, and then gradually set as other values) where (T, K)₀) Under the condition, the value of f ' is changed by changing the value of R, so that f ' is equal to f '. At this time, the frequencies of the two intersections are equal. In this case, given T, the corresponding R can be determined. Thus, a set of parameters, denoted (R', K), is obtained that makes the frequencies of the two intersections equal₀'). When the frequencies of the two intersection points are equal, the DPLL filters short-term noise of the cesium clock and long-term noise of the hydrogen clock to the maximum extent, so that the phase noise of the output of the DPLL is optimal, and the frequency stability of the output of the DPLL is optimal at the moment.

Fourthly, for a given T, K is changed₀Using the same method, a number of sets of optimum parameters (R', K) can be obtained₀'). These parameters (R', K)₀') are all optimal parameters, all ensure that the frequency stability of the DPLL output is optimal.

Specifically, in step s.1.2 of the present invention, the derivation process of equations (22) and (23) is as follows:

for a two-state variable system, the state equation is expressed as:

wherein x is_kAnd y_kIs two state variables, T is the sampling interval, u_kIs process noise;

the observation equation is expressed as:

z_k＝x_k+w_k(4)

wherein z is_kAs an observed quantity, w_kTo observe noise;

the two equations, the state equation and the observation equation, are expressed in the form of a matrix as:

wherein s is_k＝[x_ky_k]^T；J_k＝[0u_k]^T；H＝[10]；

The variances of the process noise and the observation noise are respectively: r ═ E [ w_k ²]，Wherein symbol E represents a mathematical expectation;

the Kalman filter is described by the following 5 equations:

P_k,k-1＝φP_k-1,k-1φ^T+Q (7)

K_k＝P_k,k-1·H^T(H·P_k,k-1·H^T+R)^-1(8)

P_k,k＝(I-K_k·H)·P_k,k-1(10)

wherein,is an estimate of the state variable at time k, P_k,kIs to estimate the error matrix of the received signal,is a single step prediction value, P, of the state variable at time k-1 to time k_k,k-1Is a single step prediction error matrix, K_kIs a Kalman gain matrix;

it can be demonstrated that the system defined by equation (5) is fully observable, so P_k,k-1、P_k,kAnd K_kAll converge to P_k,k、P_k,k-1And K_kRespectively, are recorded as: ps, Ps^-And Ks;

from equations (6) and (9), when the Kalman filter enters the steady state, there are:

wherein,is an estimate of the state variable x at time k,is an estimate of the state variable y at time k, Ks_ijAn element representing the ith row and the jth column in the Ks matrix;

defining:

by substituting formula (12) for formula (11), we obtain:

equation (13) is expressed in the Z domain as:

wherein X, Y and V are respectivelyv_kZ-transformation of (a);

obtained from formula (14):

from equations (14) and (15), equation (12) is represented in the Z domain as:

wherein Z represents Z_kZ-transformation of (a);

from formula (16), we obtain:

defining:

from the formulae (15), (17) and (18) to give

Obviously, z is_kIs an input to a Kalman filter, andis the output of the Kalman filter; equation (19) thus shows the relationship between the steady state Kalman filter input and output.

The system transfer function is then expressed as:

from equation (20), the transfer function of the Kalman filter is the same as the closed loop system transfer function of the second-order class 2 DPLL. Thus, the Kalman filter is equivalent to a second order class 2 DPLL.

Due to Ks₁₁1, then equation (20) is written approximately as:

for the DPLL to work properly, a delay z is introduced^-1The system transfer function of the DPLL is then expressed as:

wherein, K₀Is the loop gain;

the error transfer function of the DPLL is expressed as:

thus, a system transfer function and an error transfer function equivalent to the DPLL of the two-state variable Kalman filter are obtained.

In s.2 of the present invention, the derivation process of equations (30) and (35) is as follows:

when the Kalman filter enters steady state, equation (7) is written as:

equation (8) is written as:

equation (10) is written as:

in equations (25) to (27), subscript ij denotes an element corresponding to ith row and jth column of the matrix;

from formula (25), formula (26), formula (2), formula (27), formula 3, we obtain:

from formula (28) and formula (26), formula 2, we obtain:

due to the fact thatIs obtained from formula (29)

From formulae (25) to (27) to give

From formula (31) to yield

Due to Ks₂₁·T＜＜Ks₁₁From formula (32), we obtain:

by substituting formula (33) for formula (26), the following can be obtained:

from equation (34), we obtain:

the present invention can be applied to: using one atomic clock (such as cesium clock) to drive another atomic clock (such as hydrogen clock); using an atomic clock to drive a Numerical Control Oscillator (NCO); using a paper time scale (for example: TA (k)) to drive an atomic clock; steering one paper time scale (e.g., UTC) over another paper time scale (e.g., TA (k)); an atomic clock (e.g., UTC (NTSC)) is used to drive a paper time scale (e.g., BDT).

In each of the above applications, there is one reference atomic clock (or time scale), and one driven atomic clock (or time scale). In these applications, the reference atomic clocks (or time scales) are: an atomic clock (such as cesium clock), an atomic clock, a paper time scale, and an atomic clock; the atomic clocks (or time scales) driven are respectively: another atomic clock (e.g. hydrogen clock), a numerically controlled oscillator, an atomic clock, another paper time scale, and a paper time scale.

The principles and procedures of the present invention are the same in all of these applications. Technical solution the present invention is described in the context of using cesium bell to drive hydrogen bell. Indeed, the present invention can be considered to be applicable in the context of other applications as described above, provided that the cesium clock is replaced with other reference atomic clocks (or time scales), and the hydrogen clock is replaced with other driven atomic clocks (or time scales).

The present invention uses a cesium clock to steer the hydrogen clock through a DPLL equivalent to a two-state variable Kalman filter. The invention derives the closed-loop system transfer function of the DPLL, and proves that the DPLL is a second-order class-2 DPLL. The present invention derives a Kalman gain determined approximately by the ratio of the process noise variance and the observed noise variance. Thus, as can be seen from the closed loop transfer function of the DPLL, the 3 parameters of the ratio, as well as the loop gain and the sampling interval, determine the performance of the DPLL. Furthermore, the invention provides a simple and effective method for selecting parameters. Through the equivalent transformation, an Analog Phase Locked Loop (APLL) equivalent to the DPLL is obtained. In the frequency domain, by adjusting parameters, the intersection frequency of the system transfer function and the error transfer function of the equivalent APLL is equal to the intersection frequency of SSB phase noise curves of a hydrogen clock and a cesium clock, and finally the optimal parameters are obtained.

The invention has the beneficial effects that:

1. the cesium clock can be effectively used for controlling the hydrogen clock, and the control performance (control error and frequency stability) is very good;

2. the parameter selection method of the invention is relatively simple.

Drawings

FIG. 1 illustrates the implementation principle of the present invention;

FIG. 2 illustrates the algorithm principle of the present invention;

FIG. 3 illustrates SSB phase noise curves for a hydrogen clock and a cesium clock

FIG. 4 illustrates the magnitude-frequency response of the system transfer function and the error transfer function of the DPLL;

FIG. 5 illustrates SSB phase noise curves for a hydrogen clock, a cesium clock, and a steered hydrogen clock;

FIG. 6 illustrates the time difference between a hydrogen clock, a cesium clock, and a hydrogen clock being driven when there is no drift in the hydrogen clock;

FIG. 7 illustrates the Allan bias of the hydrogen clock, cesium clock, and hydrogen clock being driven when there is no drift in the hydrogen clock;

FIG. 8 illustrates the Allan deviation of hydrogen bell, cesium bell, and hydrogen bell under control when there is a drift in hydrogen bell;

FIG. 9 illustrates the ride error when the hydrogen clock has no drift;

Detailed Description

The following description of the embodiments of the present invention will be made with reference to the accompanying drawings.

The invention relates to a digital phase-locked loop atomic clock control method equivalent to a two-state variable Kalman filter, which comprises the following steps.

S.1, deriving a transfer function of a DPLL equivalent to a two-state variable Kalman filter;

s.1.1 a schematic diagram of the implementation of the present invention is shown in fig. 1. The principle of using DPLL with cesium clock to drive the hydrogen clock is depicted in fig. 1. In the figure, symbol Cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, symbol steeedhm represents the output signal of the hydrogen clock, and LF represents the loop filter.

The performance of the DPLL is analyzed in the Z-domain, and the transfer function of the loop filter LF of the DPLL is denoted as g (Z). The output of the DPLL is then:

it is clear that,andthe system transfer function and the error transfer function of the DPLL, respectively.

By z ═ e^j2πf·T(where f represents frequency and T represents sampling period) may be equated to a corresponding APLL. Thus, equation (2) is expressed in terms of SSB phase noise, written as:

wherein, symbol L represents SSB phase noise, symbol G represents transfer function; the symbol Cs in the subscript represents the output signal of the cesium clock, the symbol Hm represents the output signal of the hydrogen clock, and the symbol steeedhm represents the output signal of the hydrogen clock;

s.1.2 derives expressions of the system transfer function and the error transfer function of the DPLL equivalent to the two-state variable Kalman filter from the system transfer function and the error transfer function of step s.1.1.

For a two-state variable system, the state equation is expressed as:

wherein x is_kAnd y_kIs two state variables, T is the sampling interval, u_kIs process noise.

The observation equation is expressed as:

z_k＝x_k+w_k(4)

wherein z is_kAs an observed quantity, w_kTo observe the noise.

These two equations are expressed in matrix form as:

wherein s is_k＝[x_ky_k]^T；J_k＝[0u_k]^T；H＝[10]。

The variances of the process noise and the observation noise are respectively: r ═ E [ w_k ²]，Where the symbol E represents the mathematical expectation.

The Kalman filter can be described with the following 5 steps:

P_k,k-1＝φP_k-1,k-1φ^T+Q (7)

K_k＝P_k,k-1·H^T(H·P_k,k-1·H^T+R)^-1(8)

P_k,k＝(I-K_k·H)·P_k,k-1(10)

wherein,is an estimate of the state variable at time k, P_k,kIs to estimate the error matrix of the received signal,is a single step prediction value, P, of the state variable at time k-1 to time k_k,k-1Is a single step prediction error matrix, K_kIs a Kalman gain matrix.

It can be demonstrated that the system defined by equation (5) is fully observable, hence P_k,k-1、P_k,kAnd K_kAre all converged. Handle P_k,k、P_k,k-1And K_kRespectively, are recorded as: ps, Ps^-And Ks.

From equations (6) and (9), when the Kalman filter enters the steady state, there are:

wherein,is an estimate of the state variable x at time k,is an estimate of the state variable y at time k, Ks_ijRepresenting the element in the ith row and the jth column of the Ks matrix.

Defining:

by substituting formula (12) for formula (11), we obtain:

equation (13) is expressed in the Z domain as:

wherein X, Y and V are respectivelyv_kZ transformation of (1).

This can be obtained from equation (14):

from equations (14) and (15), equation (12) is represented in the Z domain as:

wherein Z represents Z_kZ transformation of (1).

From formula (16), we obtain:

defining:

from the formulae (15), (17) and (18) to give

Obviously, z is_kIs an input to a Kalman filter, andis the output of the Kalman filter. Equation (19) thus shows the relationship between the steady state Kalman filter input and output.

The system transfer function is then expressed as:

from equation (20), the transfer function of the Kalman filter is the same as the closed loop system transfer function of the second-order class 2 DPLL. Thus, the Kalman filter may be equivalent to a second order class 2 DPLL.

Through experiments, Ks is found₁₁1, then equation (20) can be written approximately as:

for the DPLL to work properly, a delay z is introduced^-1The system transfer function of the DPLL is then expressed as:

wherein, K₀Is the loop gain.

The error transfer function of the DPLL is expressed as:

thus, a system transfer function and an error transfer function equivalent to the DPLL of the two-state variable Kalman filter are obtained. As can be seen from equations (22) and (23): parameter (Ks)₁₁,Ks₂₁,T,K₀) Determines the performance of the DPLL.

Fig. 2 depicts the algorithmic principles of the DPLL. As can be seen from FIG. 2, the amount of control over the hydrogen clock during each sampling interval is represented in the Z domain asThus, in the time domain, the relationship between the time difference of the hydrogen clock and the time difference of the hydrogen clock is expressed as:

in the formula (25), err (j) is the time difference cs (j) -Hm between the cesium clock and the hydrogen clock at the jth time_steered(j)。

S.2, deducing Ks in the system transfer function and the error transfer function according to the system transfer function and the error transfer function of the DPLL obtained in the step S.1₁₁And Ks₂₁Is composed of Q₂₂R, T.

In fact, when the Kalman filter is operated and enters a steady state, the Ks can be obtained₁₁And Ks₂₁The true value of (d). The idea of this stepMeaning that the Kalman filter may not be run, pass Q₂₂R, T to calculate Ks₁₁And Ks₂₁An approximation of (d). Simulation experiments show that: the error of the approximate value from the true value is small. Thus, the Ks may be substituted for the true value by an approximate value₁₁And Ks₂₁The expression of the approximate value of (A) is substituted for the expressions (23) and (24) to obtain the formula (Q)₂₂,R,T,K₀) A system transfer function and an error transfer function for the expression. Thus, the parameter (Q)₂₂,R,T,K₀) Determines the performance of the DPLL.

When the Kalman filter enters steady state, equation (7) can be written as:

equation (8) can be written as:

equation (10) can be written as:

in equations (25) to (27), subscript ij denotes an element corresponding to ith row and jth column of the matrix.

From formula (25), formula (26), formula (2), formula (27), formula 3, we obtain:

from formula (28) and formula (26), formula 2, we obtain:

due to the fact thatIs obtained from formula (29)

From formulae (25) to (27) to give

From formula (31) to yield

Due to Ks₂₁·T＜＜Ks₁₁From formula (32), we obtain:

by substituting formula (33) for formula (26), the following can be obtained:

from equation (34), we obtain:

from equations (30) and (35), it can be seen that: the Kalman gain is approximated by Q₂₂And the proportional relation between RAnd (6) determining.

Simulation experiments show that the error of the approximation is small. For example: when Q is₂₂＝1s²，R＝6×10²⁰s²Then, the approximate value Ks is calculated from the formula (30) and the formula (35)₁₁≈5.42×10^-4And Ks₂₁≈4.08×10^-11(ii) a And the Kalman filter is operated to obtain the true value Ks₁₁＝5.32×10^-4And Ks₂₁＝3.95×10^-11(ii) a It is clear that the error between the approximated and true values is less than 3%. Then, the calculated approximate values of the equations (30) and (35) may be used to approximate instead of the true values.

Let Q₂₂＝1s²Equation (30) and equation (35) are substituted for equation (22) and equation (23) to obtain a system transfer function and an error transfer function, which are respectively expressed as:

and

as can be seen from equations (36) and (37): when Q is₂₂＝1s²When, the parameters (R, T, K)₀) Determines the performance of the DPLL. Substituting equations (36) and (37) for equation (1) yields the relation between DPLL input and output:

s.3, determining parameters (R, T, K) according to the system transfer function and the error transfer function obtained in the step S.2₀) And optimizing the frequency stability of the DPLL output.

The frequency stability can be represented in the time domain by the Allan variance or bySSB phase noise is represented in the frequency domain. The parameters are easier to determine in the frequency domain than in the time domain. By transforming z to e^j2πf·TThe DPLL is equivalent to an equivalent APLL. The performance of the equivalent APLL in the frequency domain can then be analyzed.

In the frequency domain, the parameters (R, T, K) are selected₀) The purpose of the method is mainly to optimize the SSB phase noise performance of the equivalent APLL output signal, and the method comprises the following steps: by selecting the parameters (R, T, K)₀) The frequency at the intersection of the system transfer function and the error transfer function is made equal to the frequency at the intersection of the phase noise curve of the cesium clock and the hydrogen clock SSB.

First, the phase noise of the cesium clock and the hydrogen clock is expressed as:

and

wherein f is₀Is the carrier frequency, h_iFor noise figure, f is the sideband frequency, i is used to indicate the noise type.

By solving equation L_Cs(f)-L_Hm(f) The frequency at the intersection of the cesium clock phase noise curve and the hydrogen clock phase noise curve can be found as f' 0.

Secondly, changing z to e^j2πf·TThe system transfer function and the error transfer function of the equivalent APLL are obtained by substituting equations (36) and (37), which are respectively expressed as:

and

from equations (41) and (42), the amplitude-frequency responses 20log | h (f) and 20log | he (f) of the system transfer function and the error transfer function can be obtained, and the frequency at the intersection of these two curves can be obtained and is denoted as f ".

Third, for a given T, K is given first₀Set a value at which (T, K)₀) Under the condition, the value of f ' is changed by changing the value of R, so that f ' is equal to f '. At this time, the frequencies of the two intersections are equal. In this case, given T, the corresponding R can be determined. Thus, a set of parameters, denoted (R', K), is obtained that makes the frequencies of the two intersections equal₀'). When the frequencies of the two intersection points are equal, the DPLL filters short-term noise of the cesium clock and long-term noise of the hydrogen clock to the maximum extent, so that the phase noise of the output of the DPLL is optimized, and therefore, the frequency stability of the output of the DPLL can be considered to be optimized at this time.

Fourthly, for a given T, K is changed₀Using the same method, a number of sets of optimum parameters (R', K) can be obtained₀'). These parameters (R', K)₀') are all optimal parameters, all ensure that the frequency stability of the DPLL output is optimal, but there is still a subtle difference in the actual handling performance. Through simulation experiments, different groups (R', K) can be analyzed₀') ride performance under conditions.

And generating a hydrogen clock and a cesium clock through simulation. The hydrogen clock and the cesium clock mainly contain frequency white noise and frequency random walk noise. The noise coefficient of the hydrogen clock is h_(Hm)0＝6×10^-26And h_(Hm)-2＝2×10^-36Noise figure of cesium clock is h_(Cs)0＝3×10^-23And h_(Cs)-2＝3.33×10^-38. The sequence of hydrogen clock and cesium clock has 10000 samples each, with a sample interval T of 3600 s.

FIG. 2 shows a hydrogen clock and a cesium clock at a sampling frequency f₀When it is 10MHzThe theoretical SSB phase noise curve of (a). It can be seen from fig. 2 that the frequency at the intersection is about 10^-6.6Hz. Firstly, selecting K₀12, it was found that when R6 × 10²⁰At the intersection of the amplitude-frequency response 20log | H (f) and 20log | He (f) curves, the frequency is approximately equal to 10^-6.6Hz. Thus, a set of parameters (R', K) is obtained₀’)＝(6×10²⁰s²12) by the same method, other parameters similar to (3 × 10) were obtained²²s²,30),(5×10¹⁷s²And 2) and the like. FIG. 3 shows that (R', K)₀’)＝(6×10²⁰s²And 12) the amplitude-frequency response of the system transfer function and the error transfer function.

From the formula (3), the formula (41) and the formula (42), the parameters (R', K) can be calculated₀') and T3600 s, the theoretical SSB phase noise L of the equivalent APLL output_{Hm_steered}(f) In that respect Returning to the Z domain and the system transfer function shown in equation (36). In case 1, when (R', K)₀’)＝(5×10¹⁷s²2), the pole is 0.9968 +/-1 × 10^-8i; in case 2, when (R', K)₀’)＝(6×10²⁰s²12), poles are 0.9997 and 0.9938; in case 3, when (R', K)₀’)＝(3×10²²s²30), the poles are 0.9999 and 0.9940. In these 3 cases, the system is stable since the poles are all within the unit circle. In case 1, the system is an under-damped control system. In cases 2 and 3, the system is an over-damped control system, but the system is more sensitive in case 2. To sum up, the parameters (R', K) of case 2 were selected₀’)＝(6×10²⁰s²,12)。

FIG. 4 shows the current parameters (R', K)₀’)＝(6×10²⁰s²And 12) hydrogen clock, cesium clock, and post-drive hydrogen clock theoretical SSB phase noise. As can be seen from fig. 4, the SSB phase noise performance of the post-drive hydrogen clock is very good.

By the formula (38), parameters (R', K)₀’)＝(6×10²⁰s²12), and the generated hydrogen clock and cesium clock sequences, the output sequence of the DPLL, i.e., the sequence of the hydrogen clock after mounting, can be derived.

Fig. 5 shows the time difference of the hydrogen clock, cesium clock, and post-handling hydrogen clock for the first 4800 data points. Fig. 6 shows the alan deviations of hydrogen chimes, cesium chimes, and post-drive hydrogen chimes. As can be seen in fig. 5, the DPLL can effectively use the cesium clock to drive the hydrogen clock with less driving error. As can be seen from fig. 6: the frequency stability of the hydrogen bell after driving represents the short-term stability of the hydrogen bell and the long-term stability of the cesium bell, and the short-term stability and the long-term stability of the hydrogen bell are only slightly deteriorated compared with the hydrogen bell and the cesium bell, respectively.

Adding amplitude 3 × 10 to original hydrogen clock sequence^-21s^-1The frequency drift of (2). Fig. 7 shows the alan bias of hydrogen bell, cesium bell, and post-handling hydrogen bell when there is a drift in hydrogen bell. Comparing fig. 6 and 7, it can be seen that the frequency drift does not affect the frequency stability of the post-driving hydrogen clock.

To more clearly demonstrate the handling error (i.e., the deviation of the hydrogen clock after handling relative to the cesium clock), hydrogen clock and cesium clock sequences were regenerated, with the parameters of each sequence being identical to the parameters of the original hydrogen clock and the original cesium clock, for a total of 200000 sampling points. By using the same DPLL method, a sequence of post-drive hydrogen bells was obtained. Fig. 8 shows the steering error. As can be seen in fig. 8: the maximum steering error is 30ns, and the standard deviation is 8.99 ns. In fact, the handling error is caused by phase noise of the untracked hydrogen clock and cesium clock, and the standard deviation of the handling error, i.e., jitter, can be accurately calculated by control theory. It can therefore be presumed that: in applications where paper time is steered by UTC, the steering error will be correspondingly smaller since both paper time and UTC are more stable, containing less equivalent SSB phase noise components.

Adding amplitude 3 × 10 to original hydrogen clock sequence^-21s^-1The frequency drift of (2). Fig. 9 illustrates the handling error when the hydrogen clock contains a frequency drift. Comparing fig. 8 and 9, it can be seen that: when the hydrogen clock has frequency drift, a steady state tracking is causedError, amplitude of about 20 ns.

In conclusion, the present invention can effectively use the reference atomic clock (or time scale) to drive the driven atomic clock (or time scale), and the performance of driving error and the frequency stability after driving is superior. The drift of the driven atomic clock (or time scale) causes a steady state tracking error whose value can be accurately calculated and compensated by other algorithms.

In summary, although the present invention has been disclosed in terms of the preferred embodiment (hydrogen clock driven by cesium clock), it should not be construed as limiting the present invention, but rather as providing various modifications and alterations to the present invention without departing from the spirit and scope of the present invention, as defined by the appended claims.

Claims (3)

1. A digital phase-locked loop atomic clock steering method equivalent to a two-state variable Kalman filter, characterized by comprising the steps of:

s.1, deriving a transfer function of a DPLL equivalent to a two-state variable Kalman filter;

s.1.1 use DPLL, use cesium clock to drive hydrogen clock; the principle structure of the DPLL is: acquiring the time deviation of the cesium clock and the hydrogen clock after control; filtering the time deviation by using a loop filter to filter high-frequency noise; then, adjusting the time deviation of the hydrogen clock by using the filtered time deviation to obtain the time deviation of the hydrogen clock after driving, wherein: cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, symbol Steered Hm represents the output signal of the hydrogen clock being driven, and LF represents the loop filter;

in the Z domain, the transfer function of the loop filter LF of the DPLL is denoted g (Z); the output of the DPLL is then:

${\mathrm{Hm}}_{SteeredHm}\left(z\right)=\frac{G\left(z\right)}{1+G\left(z\right)}\·Cs\left(z\right)+\frac{1}{1+G\left(z\right)}\·Hm\left(z\right)---\left(1\right)$

it is clear that,anda system transfer function and an error transfer function of the DPLL respectively; wherein: symbol Cs represents the output signal of the cesium clock, symbol Hm represents the output signal of the hydrogen clock, and symbol Steered Hm represents the output signal of the hydrogen clock;

by z ═ e^j2πf·TWherein f represents frequency and T represents sampling period; equating the DPLL to a corresponding APLL; thus, equation (2) is expressed in terms of SSB phase noise, written as:

$\begin{array}{c}{L}_{SteeredHm}\left(f\right)=|\frac{G\left(e^{j2\mathrm{\π}f\·T}\right)}{1+G\left(e^{j2\mathrm{\π}f\·T}\right)}{|}^2\·L_{Cs}\left(f\right)+|\frac{1}{1+G\left(e^{j2\mathrm{\π}f\·T}\right)}{|}^2\·L_{Hm}\left(f\right)\\ =|H\left(e^{j2\mathrm{\π}f\·T}\right){|}^2\·L_{Cs}\left(f\right)+|He\left(e^{j2\mathrm{\π}f\·T}\right){|}^2\·L_{Hm}\left(f\right)\end{array}---\left(2\right)$

wherein, symbol L represents SSB phase noise, symbol G represents transfer function; the symbol Cs in the subscript represents the output signal of the cesium clock, the symbol Hm represents the output signal of the hydrogen clock, and the symbol steeled Hm represents the output signal of the hydrogen clock under control;

s.1.2, deriving expressions of a system transfer function and an error transfer function of the DPLL equivalent to the two-state variable Kalman filter according to the system transfer function and the error transfer function of the step S.1.1;

finally, the system transfer function of the DPLL is expressed as:

$He\left(z\right)=\frac{K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·(1-z^{-1})+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}{{(1-z^{-1})}^2+K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·(1-z^{-1})+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}---\left(22\right)$

wherein, K₀Is the loop gain; ks₁₁And Ks₂₁Respectively, the steady state gains of the Kalman filters; t is a sampling interval;

the error transfer function of the DPLL is expressed as:

$He\left(z\right)=\frac{{(1-z^{-1})}^2}{{(1-z^{-1})}^2+K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·(1-z^{-1})+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}---\left(23\right)$

thus, a system transfer function and an error transfer function of the DPLL equivalent to a two-state variable Kalman filter are obtained; represented by formula (22) and formula (23): parameter (Ks)₁₁,Ks₂₁,T,K₀) Determining the performance of the DPLL;

s.2, deducing Ks in the system transfer function and the error transfer function according to the system transfer function and the error transfer function of the DPLL obtained in the step S.1₁₁And Ks₂₁By process noise variance Q₂₂Observing noise variance R and sampling interval T to express approximate values;

when the Kalman filter enters a steady state, obtaining

${\mathrm{Ks}}_{21}\≈\sqrt{Q_{22}/R}---\left(30\right)$

And

${\mathrm{Ks}}_{11}\≈\sqrt{2T\·{\mathrm{Ks}}_{21}}\≈\sqrt{2T}\·\sqrt[4]{Q_{22}/R}---\left(35\right)$

from equations (30) and (35), the Kalman gain is given by Q₂₂And R;

let Q₂₂＝1s²Equation (30) and equation (35) are substituted for equation (22) and equation (23) to obtain a system transfer function and an error transfer function, which are respectively expressed as:

$H\left(z\right)=\frac{K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·z^{-1}\·(1-z^{-1})+K_0\·\sqrt{1/R}\·T\·z^{-2}}{{(1-z^{-1})}^2+K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·z^{-1}\·(1-z^{-1})+K_0\·\sqrt{1/R}\·T\·z^{-2}}---\left(36\right)$

and

$He\left(z\right)=\frac{{(1-z^{-1})}^2}{{(1-z^{-1})}^2+K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·z^{-1}\·(1-z^{-1})+K_0\·\sqrt{1/R}\·T\·z^{-2}}---\left(37\right)$

as can be seen from equations (36) and (37): when Q is₂₂＝1s²When, the parameters (R, T, K)₀) Determines the performance of the DPLL; substituting equations (36) and (37) for equation (1) yields the relation between DPLL input and output:

$\begin{array}{c}{\mathrm{Hm}}_{SteeredHm}\left(z\right)=\frac{2\·K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·\left(1-z^{-1}\right)+2\·K_0\·\sqrt{1/R}\·T\·z^{-1}}{{\left(1-z^{-1}\right)}^2+2\·K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·\left(1-z^{-1}\right)+2\·K_0\·\sqrt{1/R}\·T\·z^{-1}}\·Cs\left(z\right)\\ +\frac{{\left(1-z^{-1}\right)}^2}{{\left(1-z^{-1}\right)}^2+2\·K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·\left(1-z^{-1}\right)+2\·K_0\·\sqrt{1/R}\·T\·z^{-1}}\·Hm\left(z\right)\end{array}---\left(38\right)$

s.3, determining parameters (R, T, K) according to the system transfer function and the error transfer function obtained in the step S.2₀) Optimizing the frequency stability of the DPLL output specifically comprises the following steps:

first, the phase noise of the cesium clock and the hydrogen clock is expressed as:

$L_{Cs}\left(f\right)=10log(0.5\·\frac{{f_0}^2}{f^2}\·\underset{-2}{\overset{2}{\Σ}}{h}_{\left(Cs\right)i}\·{f_{\left(Cs\right)}}^i),---\left(39\right)$

and

$L_{Hm}\left(f\right)=10log(0.5\·\frac{{f_0}^2}{f^2}\·\underset{-2}{\overset{2}{\Σ}}{h}_{\left(Hm\right)i}\·{f_{\left(Hm\right)}}^i),---\left(40\right)$

wherein the symbol L represents SSB phase noise, f₀Is the carrier frequency, h_iF is the sideband frequency, i is used to indicate the noise type;

by solving equation L_Cs(f)-L_Hm(f) When the frequency is equal to 0, the frequency at the intersection of the cesium clock phase noise curve and the hydrogen clock phase noise curve can be obtained and is denoted as f';

secondly, changing z to e^j2πf·TThe system transfer function and the error transfer function of the equivalent APLL are obtained by substituting equations (36) and (37), which are respectively expressed as:

$H\left(f\right)=\frac{K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·e^{-j2\mathrm{\π}f\·T}\·(1-e^{-j2\mathrm{\π}f\·T})+K_0\·\sqrt{1/R}\·T\·e^{-j2\mathrm{\π}f\·2T}}{{(1-e^{-j2\mathrm{\π}f\·T})}^2+K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·e^{-j2\mathrm{\π}f\·T}\·(1-e^{-j2\mathrm{\π}f\·T})+K_0\·\sqrt{1/R}\·T\·e^{-j2\mathrm{\π}f\·2T}}---\left(41\right)$

and

$He\left(f\right)=\frac{{(1-e^{-j2\mathrm{\π}f\·T})}^2}{{(1-e^{-j2\mathrm{\π}f\·T})}^2+K_0\·\sqrt{2T}\·\sqrt[4]{1/R}\·e^{-j2\mathrm{\π}f\·T}\·(1-e^{-j2\mathrm{\π}f\·T})+K_0\·\sqrt{1/R}\·T\·e^{-j2\mathrm{\π}f\·2T}}.---\left(42\right)$

obtaining the amplitude-frequency response 20log | H (f) | and 20log | He (f) | of the system transfer function and the error transfer function according to the formulas (41) and (42), and obtaining the frequency at the intersection point of the two curves, which is denoted as f ";

third, for a given T, K is given first₀Set a value at which (T, K)₀) Changing the value of f 'by changing the value of R under the condition that f'; at this time, the frequencies of the two intersection points are equal; in this case, given T, the corresponding R can be determined(ii) a Thus, a set of parameters, denoted (R', K), is obtained that makes the frequencies of the two intersections equal₀') to a host; when the frequencies of the two intersection points are equal, the DPLL filters short-term noise of the cesium clock and long-term noise of the hydrogen clock to the maximum extent, so that the phase noise of the output of the DPLL is optimal, and the frequency stability of the output of the DPLL is optimal at the moment;

fourthly, for a given T, K is changed₀Using the same method as in step three, multiple sets of optimal parameters (R', K) can be obtained₀') of the parameters (R', K)₀') are all optimal parameters, all ensure that the frequency stability of the DPLL output is optimal.

2. A digital phase-locked loop atomic clock steering method equivalent to a two-state variable Kalman filter according to claim 1, characterized in that the derivation procedure of the system transfer function and the error transfer function equivalent to the DPLL of the two-state variable Kalman filter in step s.1.2 is as follows:

for a two-state variable system, the state equation is expressed as:

$\left\{\begin{array}{c}{x}_{k+1}=x_k+y_k\·T\\ y_{k+1}=y_k+u_k\end{array}\right.---\left(3\right)$

wherein x is_kAnd y_kIs two state variables, T is the sampling interval, u_kIs process noise;

the observation equation is expressed as:

z_k＝x_k+w_k(4)

wherein z is_kAs an observed quantity, w_kTo observe noise;

the two equations, the state equation and the observation equation, are expressed in the form of a matrix as:

$\left\{\begin{array}{c}{s}_{k+1}=\mathrm{\φ}\·s_k+J_k\\ z_k=H\·s_k+w_k\end{array}\right.---\left(5\right)$

wherein s is_k＝[x_ky_k]^T；J_k＝[0 u_k]^T；H＝[1 0]；

The variances of the process noise and the observation noise are respectively: r ═ E [ w_k ²]，Wherein symbol E represents a mathematical expectation;

the Kalman filter is described by the following 5 equations:

${\hat{s}}_{k,k-1}=\mathrm{\φ}\·{\hat{s}}_{k-1,k-1}---\left(6\right)$

P_k,k-1＝φP_k-1,k-1φ^T+Q (7)

K_k＝P_k,k-1·H^T(H·P_k,k-1·H^T+R)^-1(8)

${\hat{s}}_{k,k}={\hat{s}}_{k,k-1}+K_k\·(z_k-H\·{\hat{s}}_{k,k-1})---\left(9\right)$

P_k,k＝(I-K_k·H)·P_k,k-1(10)

wherein,is an estimate of the state variable at time k, P_k,kIs to estimate the error matrix of the received signal,is a single step prediction value, P, of the state variable at time k-1 to time k_k,k-1Is a single step prediction error matrix, K_kIs a Kalman gain matrix;

it can be demonstrated that the system defined by equation (5) is fully observable, so P_k,k-1、P_k,kAnd K_kAll converge to P_k,k、P_k,k-1And K_kRespectively, are recorded as: ps, Ps^-And Ks;

from equations (6) and (9), when the Kalman filter enters the steady state, there are:

$\left\{\begin{array}{c}{\hat{x}}_k={\hat{x}}_{k-1}+{\hat{y}}_{k-1}\·T+{\mathrm{Ks}}_{11}\·\left(z_k-{\hat{x}}_{k-1}-{\hat{y}}_{k-1}\·T\right)\\ {\hat{y}}_k={\hat{y}}_{k-1}+{\mathrm{Ks}}_{21}\·\left(z_k-{\hat{x}}_{k-1}-{\hat{y}}_{k-1}\·T\right)\end{array}\right.---\left(11\right)$

wherein,is an estimate of the state variable x at time k,is an estimate of the state variable y at time k, Ks_ijAn element representing the ith row and the jth column in the Ks matrix;

defining:

$v_k=(z_k-{\hat{x}}_{k-1}-{\hat{y}}_{k-1}\·T)---\left(12\right)$

by substituting formula (12) for formula (11), we obtain:

$\{\begin{array}{c}{\hat{x}}_k={\hat{x}}_{k-1}+{\hat{y}}_{k-1}\·T+{\mathrm{Ks}}_{11}\·v_k\\ {\hat{y}}_k={\hat{y}}_{k-1}+{\mathrm{Ks}}_{21}\·v_k\end{array}---\left(13\right)$

equation (13) is expressed in the Z domain as:

$\left\{\begin{array}{c}X=z^{-1}\·X+z^{-1}\·T\·Y+{\mathrm{Ks}}_{11}\·V\\ Y=z^{-1}\·Y+{\mathrm{Ks}}_{21}\·V\end{array}\right.---\left(14\right)$

wherein X, Y and V are respectivelyv_kZ-transformation of (a);

obtained from formula (14):

$X=\left(\frac{{\mathrm{Ks}}_{11}}{1-z^{-1}}+\frac{{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{(1-z^{-1})}^2}\right)\·V---\left(15\right)$

from equations (14) and (15), equation (12) is represented in the Z domain as:

$\begin{array}{c}V=Z-z^{-1}\·X-T\·z^{-1}\·Y=Z-X+{\mathrm{Ks}}_{11}\·V\\ =Z-\left(\frac{{\mathrm{Ks}}_{11}}{1-z^{-1}}+\frac{{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{(1-z^{-1})}^2}\right)\·V+{\mathrm{Ks}}_{11}\·V\end{array}---\left(16\right)$

wherein Z represents Z_kZ-transformation of (a);

from formula (16), we obtain:

$\left(1-{\mathrm{Ks}}_{11}+\frac{{\mathrm{Ks}}_{11}}{1-z^{-1}}+\frac{{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{(1-z^{-1})}^2}\right)V=Z---\left(17\right)$

defining:

$G^{\′}\left(z\right)=\frac{{\mathrm{Ks}}_{11}}{1-z^{-1}}+\frac{{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{\left(1-z^{-1}\right)}^2}=\frac{{\mathrm{Ks}}_{11}\·\left(1-z^{-1}\right)+{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{\left(1-z^{-1}\right)}^2}---\left(18\right)$

from the formulae (15), (17) and (18) to give

$X=\frac{G^{\′}\left(z\right)}{1-{\mathrm{Ks}}_{11}+G^{\′}\left(z\right)}\·Z=\frac{{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}{(1-{\mathrm{Ks}}_{11})\·{(1-z^{-1})}^2+{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}\·Z---\left(19\right)$

Obviously, z is_kIs an input to a Kalman filter, andis the output of the Kalman filter; thus, equation (19) indicates the relationship between the steady state Kalman filter input and output;

the system transfer function is then expressed as:

$H^{\′}\left(z\right)=\frac{G^{\′}\left(z\right)}{1-{\mathrm{Ks}}_{11}+G^{\′}\left(z\right)}=\frac{{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}{(1-{\mathrm{Ks}}_{11})\·{(1-z^{-1})}^2+{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}---\left(20\right)$

by equation (20), the transfer function of the Kalman filter is the same as the closed loop system transfer function of the second-order class 2 DPLL; thus, the Kalman filter is equivalent to a second order class 2 DPLL;

obviously, z is_kIs an input to a Kalman filter, andis the output of the Kalman filter; thus, equation (19) indicates the relationship between the steady state Kalman filter input and output;

the system transfer function is then expressed as:

$H^{\′}\left(z\right)=\frac{G^{\′}\left(z\right)}{1-{\mathrm{Ks}}_{11}+G^{\′}\left(z\right)}=\frac{{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}{(1-{\mathrm{Ks}}_{11})\·{(1-z^{-1})}^2+{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}---\left(20\right)$

by equation (20), the transfer function of the Kalman filter is the same as the closed loop system transfer function of the second-order class 2 DPLL; thus, the Kalman filter is equivalent to a second order class 2 DPLL;

due to Ks₁₁1, then equation (20) is written approximately as:

$H^{\′}\left(z\right)\≈\frac{G^{\′}\left(z\right)}{1+G^{\′}\left(z\right)}=\frac{{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}{{(1-z^{-1})}^2+{\mathrm{Ks}}_{11}\·(1-z^{-1})+{\mathrm{Ks}}_{21}\·T\·z^{-1}}---\left(21\right)$

for the DPLL to work properly, a delay z is introduced^-1The system transfer function of the DPLL is then expressed as:

$\begin{array}{c}H\left(z\right)=\frac{K_0\·z^{-1}\·G^{\′}\left(z\right)}{1+K_0\·z^{-1}\·G^{\′}\left(z\right)}\\ =\frac{K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·\left(1-z^{-1}\right)+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}{{\left(1-z^{-1}\right)}^2+K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·\left(1-z^{-1}\right)+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}\end{array}---\left(22\right)$

wherein, K₀Is the loop gain;

the error transfer function of the DPLL is expressed as:

$\begin{array}{c}He\left(z\right)=\frac{1}{1+K_0\·z^{-1}\·G^{\′}\left(z\right)}\\ =\frac{{\left(1-z^{-1}\right)}^2}{{\left(1-z^{-1}\right)}^2+K_0\·{\mathrm{Ks}}_{11}\·z^{-1}\·\left(1-z^{-1}\right)+K_0\·{\mathrm{Ks}}_{21}\·T\·z^{-2}}\end{array}---\left(23\right)$

thus, a system transfer function and an error transfer function equivalent to the DPLL of the two-state variable Kalman filter are obtained.

3. The digital phase-locked loop atomic clock steering method equivalent to a two-state variable Kalman filter according to claim 2, characterized in that the derivation process of equations (30) and (35) in s.2 is as follows:

when the Kalman filter enters steady state, equation (7) is written as:

$\left\{\begin{array}{c}{\mathrm{Ps}}_{11}^{-}={\mathrm{Ps}}_{11}+2\·{\mathrm{Ps}}_{12}\·T+{\mathrm{Ps}}_{22}\·T^2\\ {\mathrm{Ps}}_{12}^{-}={\mathrm{Ps}}_{12}+{\mathrm{Ps}}_{22}\·T\\ {\mathrm{Ps}}_{22}^{-}={\mathrm{Ps}}_{22}+Q_{22}\end{array}\right.---\left(25\right)$

equation (8) is written as:

$\left\{\begin{array}{c}{\mathrm{Ks}}_{11}=\frac{{\mathrm{Ps}}_{11}^{-}}{{\mathrm{Ps}}_{11}^{-}+R}\\ {\mathrm{Ks}}_{12}=\frac{{\mathrm{Ps}}_{21}^{-}}{{\mathrm{Ps}}_{21}^{-}+R}\end{array}\right.---\left(26\right)$

equation (10) is written as:

$\left\{\begin{array}{c}{\mathrm{Ps}}_{11}=\frac{R\·{\mathrm{Ps}}_{11}^{-}}{{\mathrm{Ps}}_{11}^{-}+R}\\ {\mathrm{Ps}}_{12}=\frac{R\·{\mathrm{Ps}}_{12}^{-}}{{\mathrm{Ps}}_{11}^{-}+R}\\ {\mathrm{Ps}}_{22}=\frac{-{\left({\mathrm{Ps}}_{12}^{-}\right)}^2}{{\mathrm{Ps}}_{11}^{-}+R}+{\mathrm{Ps}}_{22}^{-}\end{array}\right.---\left(27\right)$

in equations (25) to (27), subscript ij denotes an element corresponding to ith row and jth column of the matrix;

from formula (25), formula (26), formula (2), formula (27), formula 3, we obtain:

${\mathrm{Ks}}_{21}\·{\mathrm{Ps}}_{12}^{-}=Q_{22}---\left(28\right)$

from formula (28) and formula (26), formula 2, we obtain:

${\mathrm{Ks}}_{21}\·{\mathrm{Ks}}_{21}=\frac{{\mathrm{Ks}}_{21}\·{\mathrm{Ps}}_{21}^{-}}{{\mathrm{Ps}}_{11}^{-}+R}=\frac{Q_{22}}{{\mathrm{Ps}}_{11}^{-}+R}---\left(29\right)$

due to the fact thatIs obtained from formula (29)

${\mathrm{Ks}}_{21}\≈\sqrt{Q_{22}/R}---\left(30\right)$

From formulae (25) to (27) to give

$\begin{array}{c}{\mathrm{Ps}}_{11}^{-}={\mathrm{Ps}}_{11}+2\·{\mathrm{Ps}}_{12}\·T+{\mathrm{Ps}}_{22}\·T^2\\ =\left(1-{\mathrm{Ks}}_{11}\right)\·{\mathrm{Ps}}_{11}^{-}+2\·{\mathrm{Ks}}_{21}\·R\·T+{\mathrm{Ks}}_{21}\·T\·{\mathrm{Ps}}_{11}^{-}\end{array}---\left(31\right)$

From formula (31) to yield

${\mathrm{Ps}}_{11}^{-}=\frac{2\·{\mathrm{Ks}}_{21}\·T}{{\mathrm{Ks}}_{11}-{\mathrm{Ks}}_{21}\·T}\·R---\left(32\right)$

Due to Ks₂₁·T＜＜Ks₁₁From formula (32), we obtain:

${\mathrm{Ps}}_{11}^{-}\≈\frac{2\·{\mathrm{Ks}}_{21}\·T}{{\mathrm{Ks}}_{11}}\·R---\left(33\right)$

by substituting formula (33) for formula (26), the following can be obtained:

${\mathrm{Ks}}_{11}=\frac{{\mathrm{Ps}}_{11}^{-}}{{\mathrm{Ps}}_{11}^{-}+R}\≈\frac{{\mathrm{Ps}}_{11}^{-}}{R}\≈\frac{2\·{\mathrm{Ks}}_{21}\·T}{{\mathrm{Ks}}_{11}}---\left(34\right)$

from equation (34), we obtain:

${\mathrm{Ks}}_{11}\≈\sqrt{2T\·{\mathrm{Ks}}_{21}}\≈\sqrt{2T}\·\sqrt[4]{Q_{22}/R}---\left(35\right).$