CN105577175B - A kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method - Google Patents

A kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method Download PDF

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CN105577175B
CN105577175B CN201510956571.7A CN201510956571A CN105577175B CN 105577175 B CN105577175 B CN 105577175B CN 201510956571 A CN201510956571 A CN 201510956571A CN 105577175 B CN105577175 B CN 105577175B
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CN105577175A (en
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朱祥维
伍贻威
欧钢
孙广富
龚航
刘文祥
刘增军
伍微
王勇
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National University of Defense Technology
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03L—AUTOMATIC CONTROL, STARTING, SYNCHRONISATION OR STABILISATION OF GENERATORS OF ELECTRONIC OSCILLATIONS OR PULSES
    • H03L7/00—Automatic control of frequency or phase; Synchronisation
    • H03L7/06—Automatic control of frequency or phase; Synchronisation using a reference signal applied to a frequency- or phase-locked loop
    • H03L7/08—Details of the phase-locked loop
    • H03L7/085—Details of the phase-locked loop concerning mainly the frequency- or phase-detection arrangement including the filtering or amplification of its output signal
    • H03L7/093—Details of the phase-locked loop concerning mainly the frequency- or phase-detection arrangement including the filtering or amplification of its output signal using special filtering or amplification characteristics in the loop
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03L—AUTOMATIC CONTROL, STARTING, SYNCHRONISATION OR STABILISATION OF GENERATORS OF ELECTRONIC OSCILLATIONS OR PULSES
    • H03L7/00—Automatic control of frequency or phase; Synchronisation
    • H03L7/26—Automatic control of frequency or phase; Synchronisation using energy levels of molecules, atoms, or subatomic particles as a frequency reference

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  • Spectroscopy & Molecular Physics (AREA)
  • Medicines Containing Antibodies Or Antigens For Use As Internal Diagnostic Agents (AREA)
  • Position Fixing By Use Of Radio Waves (AREA)

Abstract

A kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method, the DPLL for being equivalent to Kalman filter by one and adding delayer, for controlling atomic clock；Specifically, the closed-loop system transmission function of DPLL and closed-loop error transmission function are derived first, are given it and are realized structure；Then obtain each for being controlled the adjustment amount of atomic clock, and being given the parameter selection method for making the frequency stability of DPLL output signals optimal.On this basis, cascaded up using two such DPLL and atomic clock progress two level is controlled.The algorithm controls algorithm compared to conventional atom clock, and parameter selection is easier, output signal can be made synchronous with the holding of first order reference input, and ensure that frequency stability is optimal.

Description

A kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method
Technical field
The present invention relates to temporal frequency, field of signal processing, specifically devise a kind of atomic clock and control method.
Background technology
Atomic clock is controlled technology and is played an important role in punctual laboratory and satellite navigation system.Atomic clock is carried out There are two the main purposes controlled：First, it is synchronous with referencing atom clock time to make to be controlled atomic clock, reduce them as far as possible it Between time deviation；Second is that promote the long-term stability for being controlled atomic clock.
Typical atomic clock, which controls method, includes two major classes：Method is controlled in open loop and closed loop controls method.The open loop side of controlling The core of method is the rational prediction algorithm of design.Closed loop, which controls method, mainly to be included：Linear-Quadratic Problem (LGQ) control method is opened Close (Bang-Bang) control method and digital phase-locked loop (DPLL) method.But the parameter that these closed loops control method is not allow It easily chooses, will generally be directed to each concrete application, after being compared by a large amount of emulation experiments to quantity of parameters, select One group of optimized parameter.In addition, for DPLL methods, if parameter selection is improper, it can not only ensure to control performance, can also cause Control system is unstable.
The content of the invention
In view of the defects existing in the prior art, the object of the present invention is to provide the numbers that a kind of Kalman filter adds delayer Letter lock phase annular atom clock controls method.
The principle of the present invention is：The DPLL for being equivalent to Kalman filter by one and adding delayer, for atomic clock It is controlled.The present invention has intactly derived the closed-loop system transmission function of DPLL and closed-loop error transmission function, gives it It realizes structure and each makes the frequency stabilities of DPLL output signals for being controlled the adjustment amount of atomic clock, and being given Optimal parameter selection method.On this basis, cascaded up using two such DPLL and atomic clock progress two level is controlled. Theory analysis and emulation experiment all show：The algorithm controls algorithm compared to conventional atom clock, and parameter selection is easier, can make defeated It is synchronous with the holding of first order reference input to go out signal, and ensures that frequency stability is optimal.
The technical scheme is that：
A kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method, it is characterised in that including following Step：
S.1 the transmission function for being equivalent to the DPLL that Kalman filter adds delayer is derived；
S.1.1 the observational equation and state equation of Kalman filter are provided first；
State equation is expressed as：
Wherein, xkAnd ykFor two state variables, T is the sampling interval, ukFor process noise.
Observational equation is expressed as：
zk=xk+wk (2)
Wherein, zkFor observed quantity, wkFor observation noise.
The two equations are expressed as with the form of matrix：
Wherein, sk=[xk yk]T；Jk=[0 uk]T；The side of H=[1 0], process noise and observation noise Difference is respectively：R=E [wk 2],Wherein, Q22As ukVariance.
S.1.2 the observational equation and state equation of the Kalman filter S.1.1 provided according to step, derive in Z domains Kalman filter output and input between relation, derive to provide on this basis and be equivalent to Kalman filter and add delayer DPLL transmission function；
Kalman filter can be described with following 5 steps：
Pk,k-1=φ Pk-1,k-1φT+Q (5)
Kk=Pk,k-1·HT(H·Pk,k-1·HT+R)-1 (6)
Pk,k=(I-Kk·H)·Pk,k-1 (8)
Each symbol is meant that it is well established in the art, no longer describing its meaning herein in above-mentioned 5 equations.
Wherein, KkIt is Kalman gain matrixs, Pk,kIt is evaluated error matrix, Pk,k-1It is prediction error matrix.
The system that formula (3) defines is completely observable, therefore Pk,k, Pk,k-1And KkAll restrain；Pk,k、Pk,k-1And Kk's Steady-state value is denoted as respectively：Ps、Ps-And Ks.
By formula (4) and formula (7), when Kalman filter enters stable state, have：
Wherein, subscript ij represents KkSteady-state value Ks matrixes in the i-th row jth row element.
Definition：
Formula (10) is substituted into formula (9), is obtained：
Formula (11) is expressed as in Z domains：
Wherein X, Y, V are respectivelyTransform
It can be obtained by formula (12)：
By formula (12) and formula (13), formula (10) is expressed as in Z domains：
Wherein, Z represents zkTransform.
It is obtained by formula (14)：
Definition：
By formula (13), formula (15) and formula (16), obtain
Obviously, zkIt is the input of Kalman filter as observed quantity；AndAs the estimate of state variable, it is The output of Kalman filter.Then, formula (17) is given in Z domains between the input and output of steady-state Kalman filter Relation.
In order to work normally DPLL, a delayer z is added in the loop-1.In this way, the open cycle system of DPLL transfers letter Number is expressed as：
Closed-loop system transmission function is expressed as：
Closed-loop error transmission function is expressed as：
Wherein, Ks11And Ks21The respectively steady-state gain of Kalman filter, T are the sampling interval.
Formula (18), formula (19) and formula (20), which respectively completely give this and be equivalent to Kalman filter, adds delayer Open cycle system transmission function, closed-loop system transmission function and the closed-loop error transmission function of DPLL.
In the case that in the sampling interval, T is determined it can be seen from formula (19) and formula (20), steady state Kalman gain Ks11With Ks21The performance of DPLL is determined completely.Further, steady state Kalman gain Ks11And Ks21Completely by process-noise variance Q22 It is determined with observation noise variance R.It, can be with fixation procedure noise variance Q in practical application22=1s2It is constant, such observation noise side Poor R just determines the performance of DPLL completely.
The DPLL is compared to the advantage of common 2 class DPLL maximums of second order：The parameter of common DPLL has multiple, selection conjunction The parameter of reason is not easy to, and further needs exist for the stability problem of consideration system；The DPLL of the present invention is equivalent to Kalman filter Device adds delayer, and parameter only has 1, i.e. observation noise variance R, therefore parameter is chosen relatively easily；Further, since formula (3) is fixed The system of justice is completely observable, so Kalman filter is stable.Since the DPLL is that Kalman filter adds and prolongs Slow device structure, numerical simulation show that DPLL is also stability after adding in delayer.
S.2 the transmission function of the DPLL S.1 provided according to step provides the realization structure of DPLL and each adjustment amount；
With reference to step S.1 in the obtained open cycle system transmission function of DPLL, the reality of DPLL as shown in Figure 1 can be obtained Existing structure；Wherein, Cs represents caesium clock, and Hm represents hydrogen clock, and Steered Hm represent the hydrogen clock after controlling.
It is obtained by Fig. 1：
Hmsteered(z)=G (z) (Cs (z)-Hmsteered(z))+Hm(z) (21)
Wherein, Cs represents the output signal of caesium clock, and symbol Hm represents the output signal of hydrogen clock, and symbol Hm_steered is represented Controlled the output signal of hydrogen clock.
It is obtained by formula (21)：
Formula (18) is substituted into formula (21), obtaining the adjustment amount every time for hydrogen clock in Z domains is
Error is controlled, i.e., caesium clock and control the deviation Cs-Hm between rear hydrogen clocksteeredIt is denoted as Err.Then, in time domain In, the relation for controlling the time difference and the time difference of free oscillation hydrogen clock of rear hydrogen clock is expressed as：
Wherein, Err (j) for jth moment caesium clock and is controlled the time difference Cs (j)-Hm_steered (j) between hydrogen clock.
Section 2 is in the time domain every time for the adjustment amount of hydrogen clock on the right of equal sign；Adjustment for hydrogen clock is typically profit It is realized with phase microstepper.The signal that hydrogen clock exports after phase microstepper is the signal of hydrogen clock after controlling.
S.3 the transmission function of the DPLL S.1 obtained according to step determines the frequency stability of parameter optimization DPLL outputs, Specific method is fixation procedure noise variance Q22=1s2It is constant, adjustment observation noise variance R.
S.3.1, the phase noise of caesium clock and hydrogen clock is expressed as：
With
Wherein f0For carrier frequency, hiFor noise coefficient, f is sideband frequency, and i is used to indicate noise type.
By solving equation LCs(f)-LHmIt (f)=0, can be bent in the hope of caesium clock phase noise curve and hydrogen clock phase noise The frequency of line point of intersection, is denoted as f '.
S.3.2, z=e is changed using approximationj2πf·T, formula (19) and formula (20) are substituted into, is obtained：
With
Formula (26) and formula (27) are the closed-loop system transmission function of approximate APLL and closed-loop error transmission function.Obviously, Closed-loop system transmission function shown in formula (26) is equivalent to a low-pass filter, and the closed-loop error shown in formula (27) transfers letter Number is equivalent to a high-pass filter, so their amplitude-frequency responses intersect at a point.The frequency of intersection point is denoted as f ".
S.3.3, fixation procedure noise variance Q22=1s2It is constant, the value of adjustment observation noise variance R, operation Kalman filters Ripple device obtains steady state Kalman gain Ks11And Ks21Value.APLL closed-loop systems shown in observation type (26) and formula (27) transfer letter The intersection frequency f " of number and closed-loop error transmission function.As f "=f ', corresponding R is denoted as R ', is optimized parameter.
S.3.4, R ' and corresponding steady state Kalman gain Ks are taken11And Ks21, by formula (28), it has been calculated and has controlled rear hydrogen The output power single sideband phase noise of clock；By formula (23), obtain in the time domain every time for the adjustment amount of hydrogen clock；Hydrogen clock is adjusted, Hydrogen clock after being controlled；Then the Allan deviations for controlling rear hydrogen clock are calculated.
By formula (22), the output power single sideband phase noise for controlling rear hydrogen clock is：
S.1, S.2 and S.3 parameter selection side that the transmission function and step for the DPLL that the present invention is obtained according to step obtain Method has obtained a kind of two-stage DPLL and has controlled algorithm, and specific method is cascaded up using two such DPLL, first order DPLL Input of the output as second level DPLL, atomic clock is controlled.The parameter selection method and step of each DPLL S.2 phase Together.Two-stage controls the schematic diagram of algorithm as shown in Fig. 2, wherein G1(z) and G2(z) it is respectively that the open cycle system of two DPLL is transferred Function.
The system time that present invention could apply to design phase-locked oscilaltor He establish satellite navigation system (GNSS).This hair Bright principle and step in above application is all identical.
Beneficial effects of the present invention：
1. can effectively control atomic clock, it is very good to control performance (controlling error, frequency stability)；
2. the parameter selection method of the present invention is relatively simple；
3. two-stage control can comprehensively utilize two-stage input frequency stability performance, make the short-term of output signal, mid-term, Long-term stability is all optimized.
Description of the drawings
Fig. 1 illustrates the realization structure chart of the present invention；
Fig. 2 illustrates the principle that two-stage controls algorithm；
Fig. 3 diagram illustrating hydrogen clocks and the time difference of caesium clock and Allan deviations, wherein Fig. 3 a diagram illustrating hydrogen clocks and caesium clock The time difference, Fig. 3 b illustrate the Allan deviations of hydrogen clock and caesium clock；
Fig. 4 illustrates the output power single sideband phase noise of hydrogen clock and caesium clock；
Fig. 5 illustrates the amplitude-frequency response of DPLL closed-loop systems transmission function and closed-loop error transmission function；
Fig. 6 illustrates hydrogen clock, caesium clock, the output power single sideband phase noise for controlling rear hydrogen clock；
Fig. 7 illustrates hydrogen clock, caesium clock, the Allan deviations for controlling rear hydrogen clock；
Fig. 8 illustrates hydrogen clock, caesium clock, the time difference for controlling rear hydrogen clock；
Fig. 9 illustrates NCO, hydrogen clock, the time difference of caesium clock and Allan deviations, and wherein Fig. 9 a illustrate NCO, hydrogen clock, caesium The time difference of clock, Fig. 9 b illustrate NCO, hydrogen clock, the Allan deviations of caesium clock；
Figure 10 illustrates NCO, hydrogen clock, the output power single sideband phase noise of caesium clock；
Figure 11 illustrates caesium clock, hydrogen clock, NCO and controls the time difference of rear NCO and Allan deviations, wherein Figure 11 a diagrams Illustrate caesium clock, hydrogen clock, NCO and the time difference for controlling rear NCO, Figure 11 b diagrams say caesium clock, hydrogen clock, NCO and control rear NCO's Allan deviations；
Specific embodiment
Below in conjunction with drawings and examples, the present invention is described further.
The present invention comprises the following steps.
S.1 the transmission function for being equivalent to the DPLL that Kalman filter adds delayer is derived；
S.1.1 the observational equation and state equation of Kalman filter are provided first；
For the system of a two-state variable, state equation is expressed as：
Wherein, xkAnd ykFor two state variables, T is the sampling interval, ukFor process noise.
Observational equation is expressed as：
zk=xk+wk (2)
Wherein, zkFor observed quantity, wkFor observation noise.
The two equations are expressed as with the form of matrix：
Wherein, sk=[xk yk]T；Jk=[0 uk]T；The side of H=[1 0], process noise and observation noise Difference is respectively：R=E [wk 2],Wherein, Q22As ukVariance.
S.1.2 the observational equation and state equation of Kalman filter are combined, the Kalman filter in Z domains is derived and inputs Relation between output provides the transmission function for being equivalent to the DPLL that Kalman filter adds delayer on this basis.
Kalman filter can be described with following 5 steps：
Pk,k-1=φ Pk-1,k-1φT+Q (5)
Kk=Pk,k-1·HT(H·Pk,k-1·HT+R)-1 (6)
Pk,k=(I-Kk·H)·Pk,k-1 (8)
Each symbol is meant that it is well established in the art, no longer describing its meaning herein in above-mentioned 5 equations.Its In, KkIt is Kalman gain matrixs, Pk,kIt is evaluated error matrix, Pk,k-1It is prediction error matrix.
It is completely observable, therefore P that can prove system that formula (3) definesk,k, Pk,k-1And KkIs all restrained Pk,k、 Pk,k-1And KkSteady-state value be denoted as respectively：Ps、Ps-And Ks.
By formula (4) and formula (7), when Kalman filter enters stable state, have：
Wherein, subscript ij represents KkSteady-state value Ks matrixes in the i-th row jth row element.
Definition：
Formula (10) is substituted into formula (9), is obtained：
Formula (11) is expressed as in Z domains：
Wherein X, Y, V are respectivelyTransform
It can be obtained by formula (12)：
By formula (12) and formula (13), formula (10) is expressed as in Z domains：
Wherein, Z represents zkTransform.
It is obtained by formula (14)：
Definition：
By formula (13), formula (15) and formula (16), obtain
Obviously, zkIt is the input of Kalman filter as observed quantity；AndAs the estimate of state variable, it is The output of Kalman filter.Then, formula (17) is given in Z domains between the input and output of steady-state Kalman filter Relation.
In order to work normally DPLL, a delayer z is added in the loop-1.By formula (16) and formula (17), the DPLL's Open cycle system transmission function is expressed as：
Closed-loop system transmission function is expressed as：
Closed-loop error transmission function is expressed as：
Formula (18), formula (19) and formula (20), which respectively completely give this and be equivalent to Kalman filter, adds delayer Open cycle system transmission function, closed-loop system transmission function and the closed-loop error transmission function of DPLL.
In the case that in the sampling interval, T is determined it can be seen from formula (19) and formula (20), steady state Kalman gain Ks11With Ks21The performance of DPLL is determined completely.Further, steady state Kalman gain Ks11And Ks21Completely by process-noise variance Q22 It is determined with observation noise variance R.It, can be with fixation procedure noise variance Q in practical application22=1s2It is constant, such observation noise side Poor R just determines the performance of DPLL completely.
The DPLL is compared to the advantage of common 2 class DPLL maximums of second order：The parameter of common DPLL has multiple, selection conjunction The parameter of reason is not easy to, and further needs exist for the stability problem of consideration system；The DPLL of this paper is equivalent to Kalman filter Add delayer, parameter only has 1, i.e. observation noise variance R, therefore parameter is chosen relatively easily；Further, since formula (3) defines System be completely it is observable, so Kalman filter is stable.Since the DPLL is that Kalman filter adds delay Device structure, numerical simulation show that DPLL is also stability after adding in delayer.
In the case that in the sampling interval, T is determined it can be seen from formula (19) and formula (20), steady state Kalman gain Ks11With Ks21The performance of DPLL is determined completely.Further, steady state Kalman gain Ks11And Ks21Completely by process-noise variance Q22 It is determined with observation noise variance R.It, can be with fixation procedure noise variance Q in practical application22=1s2It is constant, such observation noise side Poor R just determines the performance of DPLL completely.
S.2 the transmission function of the DPLL S.1 provided according to step provides the realization structure of DPLL and each adjustment amount；
By the open cycle system transmission function of DPLL, the realization structure of DPLL can be obtained.Fig. 1 using with caesium clock control hydrogen clock as Example describes the realization structure chart of DPLL.In Fig. 1, Cs represents caesium clock, and Hm represents hydrogen clock, and Steered Hm represent the hydrogen after controlling Clock.
The realization structure of DPLL is described with reference to Fig. 1.
It is obtained by Fig. 1：
Hmsteered(z)=G (z) (Cs (z)-Hmsteered(z))+Hm(z) (21)
It is obtained by formula (21)：
Formula (18) is substituted into formula (21), obtaining the adjustment amount every time for hydrogen clock in Z domains is
Error is controlled, i.e., caesium clock and control the deviation Cs-Hm between rear hydrogen clocksteeredIt is denoted as Err.Then, in time domain In, the relation for controlling the time difference and the time difference of free oscillation hydrogen clock of rear hydrogen clock is expressed as：
Wherein, Cs represents the output signal of caesium clock, and symbol Hm represents the output signal of hydrogen clock, and symbol Hm_steered is represented The output signal of hydrogen clock is controlled, Err (j) is jth moment caesium clock and is controlled the time difference Cs (j)-Hm_ between hydrogen clock steered(j)。
Section 2 is in the time domain every time for the adjustment amount of hydrogen clock on the right of equal sign.Adjustment for hydrogen clock is typically profit It is realized with phase microstepper.The signal that hydrogen clock exports after phase microstepper is the signal of hydrogen clock after controlling.
S.3 the transmission function of the DPLL S.1 obtained according to step determines the frequency stability of parameter optimization DPLL outputs, Specific method is fixation procedure noise variance Q22=1s2It is constant, adjustment observation noise variance R.
First, the phase noise of caesium clock and hydrogen clock is expressed as：
With
Wherein f0For carrier frequency, hiFor noise coefficient, f is sideband frequency, and i is used to indicate noise type.
By solving equation LCs(f)-LHmIt (f)=0, can be bent in the hope of caesium clock phase noise curve and hydrogen clock phase noise The frequency of line point of intersection, is denoted as f '.
Second, change z=e using approximationj2πf·T, formula (19) and formula (20) are substituted into, is obtained：
With
Formula (26) and formula (27) are the closed-loop system transmission function of approximate APLL and closed-loop error transmission function.Obviously, Closed-loop system transmission function shown in formula (26) is equivalent to a low-pass filter, and the closed-loop error shown in formula (27) transfers letter Number is equivalent to a high-pass filter, so their amplitude-frequency responses intersect at a point.The frequency of intersection point is denoted as f ".
In order to make the frequency stability of DPLL output signals optimal, methods herein is adjustment R, makes f "=f '.It is emulating Displaying this method is made the output output power single sideband phase noise of approximate APLL optimal by experimental section.
Finally, by formula (22), the output power single sideband phase noise for controlling rear hydrogen clock is：
Below the performance that the DPLL controls algorithm is illustrated as embodiment to control hydrogen clock by the use of caesium clock.
1) according to document (Kasdin N.J., Discrete Simulation ofColored Noise and Stochastic Processes and 1/f Power Law Noise Generation[J],1995,Proceedings of the IEEE,83,5,pp:Method 802-827) generates a hydrogen clock and a caesium clock.The parameter of hydrogen clock is：h(Hm)0=1 ×10-24, h(Hm)-2=8 × 10-31.The parameter of caesium clock is：h(Cs)0=5 × 10-23, h(Cs)-2=6 × 10-32.Sampling interval T= 1s.All contain 200000 data points per table clock.Fig. 3 describes emulation hydrogen clock and the time difference for emulating caesium clock and Allan is inclined Difference.If the carrier frequency for emulating hydrogen clock and caesium clock is f0=10MHz.
2) by formula (24) and formula (25), the output power single sideband phase noise curve of hydrogen clock and caesium clock is made, as shown in Figure 4.From Fig. 4 In as can be seen that the frequency f ' of this two intersections of complex curve is about 10-3.9Hz。
3) fixation procedure noise variance Q22=1s2Constant, the value of adjustment observation noise variance R runs Kalman filter, Obtain steady state Kalman gain Ks11And Ks21Value.APLL closed-loop systems transmission function shown in observation type (26) and formula (27) and The intersection frequency f " of closed-loop error transmission function.As f "=f ', corresponding R is denoted as R ', is optimized parameter.By experiment, hair Now work as R '=2 × 10-14When, the point of intersection of the amplitude-frequency response of APLL closed-loop systems transmission function and closed-loop error transmission function Frequency f " is approximately 10-3.9Hz is approximately equal to f ', as shown in Figure 5.Then, R '=2 × 10 are taken-14。
4) R '=2 × 10 are taken-14With corresponding steady state Kalman gain Ks11And Ks21, by formula (28), it has been calculated and has driven Drive the output power single sideband phase noise of rear hydrogen clock；By formula (23), obtain in the time domain every time for the adjustment amount of hydrogen clock；Hydrogen clock is carried out Adjustment, the hydrogen clock after being controlled；Then the Allan deviations for controlling rear hydrogen clock are calculated.Fig. 6 and Fig. 7 are respectively described Hydrogen clock, caesium clock and control the output power single sideband phase noise of rear hydrogen clock and Allan deviations.After Fig. 8 describes hydrogen clock, caesium clock and controls The time difference of hydrogen clock.
Fig. 6 and Fig. 7 show：Control rear hydrogen clock stability combine hydrogen clock mid-term frequency stability and caesium clock it is long-term Frequency stability；So as to demonstrate this method the frequency stability of DPLL output signals can be made optimal.Fig. 8 shows：After controlling Hydrogen clock is synchronous with caesium clock holding, demonstrates the validity for controlling algorithm.
S.2 and S.3 parameter selection method that the transmission function and step of the DPLL S.1 obtained according to step obtains, obtains Algorithm is controlled to a kind of two-stage DPLL, specifically using following steps：
It is cascaded up using two such DPLL, the input exported as second level DPLL of first order DPLL, to original Secondary clock is controlled.G1(z) and G2(z) be respectively two DPLL open cycle system transmission function.G1(z) and G2(z) parameter is Different, their parameter selection method is required for the method S.3 described using step.
The schematic diagram that two-stage controls algorithm is as shown in Figure 2；Specifically there is two-way reference input (wherein all the way for hydrogen with one Clock is all the way caesium clock) phase-locked oscilaltor be that embodiment illustrates the principle that two-stage controls algorithm：The inside of phase-locked oscilaltor Phase comparator can obtain hydrogen clock, the deviation of caesium clock and digital controlled oscillator (NCO) between any two；The phase-locked oscilaltor uses two DPLL controls digital controlled oscillator (NCO)；First DPLL controls hydrogen clock with caesium clock, forms a paper time, i.e., and the The output of one DPLL, this is controlled as the first order；Then second DPLL controls NCO with the paper time, as the second level It controls；It theoretically analyzes, which exports signal by the short-term frequency stability of comprehensive NCO, the mid-term of hydrogen clock Frequency stability, the long-term frequency stability of caesium clock, and it is synchronous with the caesium clock retention time.It is only sharp compared to traditional phase-locked oscilaltor With reference input all the way (caesium clock or hydrogen clock), this method can integrate the frequency stability of hydrogen clock and caesium clock, have simultaneously Clear superiority.
Below to verify that two-stage DPLL controls the performance of algorithm by emulation experiment.Experimental procedure is as follows：
1) according to document (Kasdin N.J., Discrete Simulation of Colored Noise and Stochastic Processes and 1/f Power Law Noise Generation[J],1995,Proceedings of the IEEE,83,5,pp:Method 802-827) generates a NCO.The parameter of NCO is：h(NCO)0=2 × 10-25, h(NCO)-2=5 × 10-30.Sampling interval T=1s.The NCO contains 200000 data points altogether.Hydrogen clock and caesium clock are using the herein the 2nd Save the identical hydrogen clock and caesium clock of generation.Fig. 9 describes emulation NCO, hydrogen clock, the time difference of caesium clock and Allan deviations.Figure 10 is described Emulation NCO, hydrogen clock, the output power single sideband phase noise of caesium clock.
2) two-stage is controlled in algorithm, and the parameter of each DPLL chooses the method S.3 described using step.As shown in Section 2, For DPLL1, work as R '=2 × 10-14When, approximation has f "=f '.Then for DPLL1, R '=2 × 10 are taken-14.It can be with by Figure 10 See, the intersection frequency of the output power single sideband phase noise curve of hydrogen clock and NCO is about 10-2.64Hz.By experiment, discovery work as R '= 1.6×10-9When, the closed-loop system transmission function of DPLL2 and the intersection frequency of closed-loop error transmission function are approximately f "=10-2.64Hz, i.e. approximation have f "=f '.Then, for DPLL2, R '=1.6 × 10 are taken-9.Then can be calculated by formula (22) Each DPLL1 is for hydrogen clock and DPLL2 for the adjustment amount of NCO.
3) the two as shown in Figure 2 cascade DPLL of operation, adjustment amount obtained by calculation, respectively to hydrogen clock and NCO is adjusted, and finally obtains the time difference of the NCO after controlling and Allan deviations, as shown in figure 11.
It can be seen from Fig. 11 that the embodiment demonstrates the conclusion of theory analysis, i.e.,：The output signal synthesis of phase-locked oscilaltor The short-term frequency stability of NCO, the middle or short term frequency stability of hydrogen clock, the secular frequency stability of caesium clock, and with first Grade input (caesium clock) retention time is synchronous.
Algorithm is controlled according to two-stage described above, describes application of the algorithm in the GNSS system time is established.
GNSS is actually a clock synchronization system, all master stations, outer station, and the atomic clock on satellite will and be Time holding of uniting is synchronous.So it is necessary to establish a GNSS system time.In addition, the GNSS system time needs again and state Border Coordinated Universal Time(UTC) UTC keeps synchronous.
By taking Beidou satellite navigation system as an example, the system time BDT of the Big Dipper is generated by master station master clock at present, and with military The local physics realization for the international coordination universal time (being denoted as UTC) that temporal frequency center (being denoted as CMTC) maintains (is denoted as：UTC (CMTC)) BDT is controlled, makes BDT synchronous with UTC (CMTC) retention times.
There is room for improvement for this mode for generating BDT.In future, BDT in the works will be a paper as GPST The face time；Atomic clock on comprehensive master station, outer station, satellite, usage time scaling algorithm are calculated on paper BDT BDT and per table clock compared with paper time BDT deviation；So the frequency stability of BDT is compared at present with reliability by list The BDT that platform master clock generates, will be greatly improved.In addition, with the foundation of satellite two-way pumping station link, can further use The local physics realization for the international coordination universal time (being denoted as UTC) that international time service center (NTSC) maintains (is denoted as：UTC(NTSC)) With UTC (CMTC) each other as backup, BDT is controlled.
Two level proposed in this paper controls algorithm and may be directly applied to establish BDT.Two level is controlled in algorithm, first order DPLL For UTC (CMTC) or UTC (NTSC) to be used to control BDT, make BDT and UTC (CMTC) or UTC (NTSC) retention times It is synchronous；Second level DPLL is used to control master station master clock using the BDT after controlling, and obtains the physics realization of BDT, is denoted as BDT(MC)。
Since UTC (CMTC) or UTC (NTSC) is it is seen that the local physics realization of UTC, it is believed that " locking " In UTC, so as UTC, there are higher long-term frequency stability；BDT combines the ground such as more hydrogen clocks, caesium clock, rubidium clocks Atomic clock on face and star, therefore with higher secular frequency stability；Master station master clock is an active hydrogen clock, tool There is higher middle or short term frequency stability.To sum up, during algorithm is controlled using two level and establishes Big Dipper time reference, UTC (CMTC) or UTC (NTSC) is equivalent to caesium clock in Fig. 2, and BDT is equivalent to the hydrogen clock in Fig. 2, and BDT (MC) is equivalent in Fig. 2 NCO.It is final to control BDT that algorithm obtains using two level and BDT (MC) be with UTC (CMTC) or UTC (NTSC) retention times It is synchronous；BDT stablizes the secular frequency stability of comprehensive BDT itself and the long run frequency of UTC (CMTC) or UTC (NTSC) Degree；And BDT (MC) will comprehensive master station master clock middle or short term frequency stability, the secular frequency stability of BDT itself and The long-term frequency stability of UTC (CMTC) or UTC (NTSC).
In conclusion although the present invention (is controlled hydrogen clock by caesium clock, passes through caesium clock, hydrogen clock two-stage with preferred embodiment Control NCO) it is disclosed above, however, it is not to limit the invention, and any those of ordinary skill in the art are not departing from the present invention Spirit and scope in, when can make it is various change and retouch, therefore protection scope of the present invention depending on claims when defining Subject to scope.

Claims (1)

1. a kind of Kalman filter adds the digital phase-locked loop atomic clock of delayer to control method, it is characterised in that：Including following Step：
S.1 it is equivalent to Kalman filter and adds open cycle system transmission functions of the DPLL of delayer in Z domains, closed-loop system passes Delivery function and closed-loop error transmission function；
S.1.1 the observational equation and state equation of Kalman filter are provided first；
State equation is expressed as：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein, xkAnd ykFor two state variables, T is the sampling interval, ukFor process noise；
Observational equation is expressed as：
zk=xk+wk (2)
Wherein, zkFor observed quantity, wkFor observation noise；
The two equations are expressed as with the form of matrix：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>J</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>=</mo> <mi>H</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
Wherein, sk=[xk yk]T；Jk=[0 uk]T；The variance difference of H=[1 0], process noise and observation noise For：R=E [wk 2],Wherein, Q22As ukVariance；
S.1.2 the observational equation and state equation of the Kalman filter S.1.1 provided according to step, derive in Z domains Kalman filter output and input between relation, derive to provide on this basis and be equivalent to Kalman filter and add delayer DPLL transmission function；
Kalman filter can be described with following 5 steps：
<mrow> <msub> <mover> <mi>s</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>s</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Pk,k-1=φ Pk-1,k-1φT+Q (5)
Kk=Pk,k-1·HT(H·Pk,k-1·HT+R)-1 (6)
<mrow> <msub> <mover> <mi>s</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>s</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mi>k</mi> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <mi>H</mi> <mo>&amp;CenterDot;</mo> <msub> <mover> <mi>s</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Pk,k=(I-Kk·H)·Pk,k-1 (8)
Wherein, KkIt is Kalman gain matrixs, Pk,kIt is evaluated error matrix, Pk,k-1It is prediction error matrix；
The system that formula (3) defines is completely observable, therefore Pk,k, Pk,k-1And KkAll restrain；Pk,k、Pk,k-1And KkStable state Value is denoted as respectively：Ps、Ps-And Ks；
By formula (4) and formula (7), when Kalman filter enters stable state, have：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein, subscript ij represents KkSteady-state value Ks matrixes in the i-th row jth row element；
Definition：
<mrow> <msub> <mi>v</mi> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
Formula (10) is substituted into formula (9), is obtained：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>v</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>y</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>v</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Formula (11) is expressed as in Z domains：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>X</mi> <mo>=</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>X</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <mi>Y</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Y</mi> <mo>=</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>Y</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>V</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
Wherein X, Y, V are respectivelyvkTransform
It can be obtained by formula (12)：
<mrow> <mi>X</mi> <mo>=</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>V</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
By formula (12) and formula (13), formula (10) is expressed as in Z domains：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>V</mi> <mo>=</mo> <mi>Z</mi> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>X</mi> <mo>-</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mi>Y</mi> <mo>=</mo> <mi>Z</mi> <mo>-</mo> <mi>X</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>V</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>Z</mi> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>V</mi> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>V</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
Wherein, Z represents zkTransform；
It is obtained by formula (14)：
<mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>)</mo> <mi>V</mi> <mo>=</mo> <mi>Z</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Definition：
<mrow> <msup> <mi>G</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
By formula (13), formula (15) and formula (16), obtain
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>X</mi> <mo>=</mo> <mfrac> <mrow> <msup> <mi>G</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>+</mo> <msup> <mi>G</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>Z</mi> </mrow> </mtd> </mtr> </mtable> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
Obviously, zkIt is the input of Kalman filter as observed quantity；AndIt is Kalman filters as the estimate of state variable The output of ripple device；Then, formula (17) gives the relation between the input and output of steady-state Kalman filter in Z domains；
A delayer z is added in the open cycle system of DPLL-1；The open cycle system transmission function of the DPLL is expressed as：
<mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msup> <mi>G</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
Closed-loop system transmission function is expressed as：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
Closed-loop error transmission function is expressed as：
<mrow> <mi>H</mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>z</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
Wherein, Ks11And Ks21The respectively steady-state gain of Kalman filter, T are the sampling interval；
Formula (18), formula (19) and formula (20) are respectively to be equivalent to Kalman filter the open cycle system of the DPLL of delayer is added to transfer Function, closed-loop system transmission function and closed-loop error transmission function；
S.2 the transmission function of the DPLL S.1 provided according to step provides the realization structure of DPLL and each adjustment amount；
Hydrogen clock is controlled using caesium clock, with reference to step S.1 in DPLL open cycle system transmission function, obtain the realization knot of DPLL Structure；
Hmsteered(z)=G (z) (Cs (z)-Hmsteered(z))+Hm(z) (21)
Wherein, Cs represents the output signal of caesium clock, and symbol Hm represents the output signal of hydrogen clock, symbol HmsteeredHydrogen is controlled in representative The output signal of clock；
It is obtained by formula (21)：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>Hm</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>e</mi> <mi>e</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>C</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>C</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>H</mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
Formula (18) is substituted into formula (21), obtaining the adjustment amount every time for hydrogen clock in Z domains is
Error is controlled, i.e., caesium clock and control the deviation Cs-Hm between rear hydrogen clocksteeredIt is denoted as Err；Then, in the time domain, drive The relation for driving the time difference and the time difference of free oscillation hydrogen clock of rear hydrogen clock is expressed as：
<mrow> <mi>H</mi> <mi>m</mi> <mo>_</mo> <mi>s</mi> <mi>t</mi> <mi>e</mi> <mi>e</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>H</mi> <mi>m</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>i</mi> </munderover> <mrow> <mo>(</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mo>(</mo> <mi>j</mi> <mo>)</mo> <mo>+</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>j</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>&amp;CenterDot;</mo> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>
Wherein, Err (j) for jth moment caesium clock and is controlled the time difference Cs (j)-Hm_steered (j) between hydrogen clock；
Section 2 is in the time domain every time for the adjustment amount of hydrogen clock on the right of equal sign；
S.3 the transmission function of the DPLL S.1 obtained according to step determines the frequency stability of parameter optimization DPLL outputs, specifically Method is fixation procedure noise variance Q22=1s2It is constant, adjustment observation noise variance R；
S.3.1, the phase noise of caesium clock and hydrogen clock is expressed as：
<mrow> <msub> <mi>L</mi> <mrow> <mi>C</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>10</mn> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <mfrac> <mrow> <msup> <msub> <mi>f</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mo>-</mo> <mn>2</mn> </mrow> <mn>2</mn> </munderover> <msub> <mi>h</mi> <mrow> <mo>(</mo> <mi>C</mi> <mi>s</mi> <mo>)</mo> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msup> <msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>C</mi> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
With
<mrow> <msub> <mi>L</mi> <mrow> <mi>H</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>10</mn> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>&amp;CenterDot;</mo> <mfrac> <mrow> <msup> <msub> <mi>f</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> </mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mo>-</mo> <mn>2</mn> </mrow> <mn>2</mn> </munderover> <msub> <mi>h</mi> <mrow> <mo>(</mo> <mi>H</mi> <mi>m</mi> <mo>)</mo> <mi>i</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msup> <msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>H</mi> <mi>m</mi> <mo>)</mo> </mrow> </msub> <mi>i</mi> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>
Wherein f0For carrier frequency, hiFor noise coefficient, f is sideband frequency, and i is used to indicate noise type；
By solving equation LCs(f)-LHmIt (f)=0, can be in the hope of caesium clock phase noise curve and hydrogen clock phase noise intersections of complex curve The frequency at place, is denoted as f '；
S.3.2, z=e is changed using approximationj2πf·T, formula (19) and formula (20) are substituted into, is obtained：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mn>2</mn> <mi>T</mi> </mrow> </msup> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>.</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mn>2</mn> <mi>T</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
With
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>H</mi> <mi>e</mi> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Ks</mi> <mn>21</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mo>/</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>Ks</mi> <mn>11</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mn>2</mn> <mi>T</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
Closed-loop system transmission function shown in formula (26) is equivalent to a low-pass filter, and the closed-loop error shown in formula (27) passes Delivery function is equivalent to a high-pass filter, so their amplitude-frequency responses intersect at a point；The frequency of intersection point is denoted as f "；
S.3.3, fixation procedure noise variance Q22=1s2Constant, the value of adjustment observation noise variance R runs Kalman filter, Obtain steady state Kalman gain Ks11And Ks21Value；APLL closed-loop systems transmission function shown in observation type (26) and formula (27) and The intersection frequency f " of closed-loop error transmission function；As f "=f ', corresponding R is denoted as R ', is optimized parameter；
S.3.4, R ' and corresponding steady state Kalman gain Ks are taken11And Ks21, by formula (28), it has been calculated and has controlled rear hydrogen clock Output power single sideband phase noise；By formula (23), obtain in the time domain every time for the adjustment amount of hydrogen clock；Hydrogen clock is adjusted, is obtained Hydrogen clock after controlling；Then the Allan deviations for controlling rear hydrogen clock are calculated；
By formula (22), the output power single sideband phase noise for controlling rear hydrogen clock is：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>L</mi> <mrow> <mi>H</mi> <mi>m</mi> <mo>_</mo> <mi>s</mi> <mi>t</mi> <mi>e</mi> <mi>e</mi> <mi>r</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <msub> <mi>L</mi> <mrow> <mi>C</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>+</mo> <mo>|</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>G</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <msub> <mi>L</mi> <mrow> <mi>H</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <msub> <mi>L</mi> <mrow> <mi>C</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>+</mo> <mo>|</mo> <mi>H</mi> <mi>e</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mi>f</mi> <mo>&amp;CenterDot;</mo> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <msub> <mi>L</mi> <mrow> <mi>H</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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