CN115437026B - A method and system for formulating a frequency planning scheme for a space-based gravitational wave detector - Google Patents

Abstract

The present invention provides a method and system for formulating a space-based gravitational wave detector frequency planning scheme. The space-based gravitational wave detector includes a satellite formation composed of three satellites, and each satellite is respectively loaded with two laser interference optical platforms; The method described includes: determining the optimization objective function and variable constraint conditions; using the multi-objective optimization algorithm to solve the optimization objective function, and constructing the constraint conditions and objective functions of six different main satellite offset frequency phase-locking schemes; The scheme is solved, and the scheme with the longest duration in the 6 schemes is selected as the final selected scheme; if the solution of the multi-objective optimization algorithm has a feasible solution, then the calculation ends, otherwise the multi-objective optimization algorithm is replaced to recalculate; the advantage of the present invention is : The idea of step-by-step optimization is adopted to solve the complex problem of frequency planning step by step and form a closed loop, which facilitates the solution of complex full-frequency planning problems and improves the solution efficiency.

Description

A method and system for formulating frequency planning scheme for space-based gravitational wave detectors

Technical Field

The present invention belongs to the field of computer technology, and in particular relates to a method and system for formulating a frequency planning scheme for a space-based gravitational wave detector.

Background Art

Gravitational wave detection is one of the hot issues in current physics research. Detecting gravitational waves by ultra-long-distance laser interferometry has been the main research method in recent decades. One of the key points in detecting gravitational wave signals is the reasonable planning of the full frequency band of lasers, that is, which frequency bands are used to measure scientific data, which frequency bands are used for intersatellite communication, and which frequency bands are used for intersatellite clock noise transmission. Reasonable setting of ultra-stable clock frequency, ADC sampling frequency, pilot frequency and other frequency-related items on each satellite during the mission is an urgent problem to be solved.

In current domestic and foreign research, only the beat frequency band used to measure scientific data has been limited and the corresponding offset frequency setting scheme has been provided. No reasonable setting scheme for all key frequencies in space-based gravitational wave detectors has been constructed, especially the ultra-stable clock frequency, ADC sampling frequency and pilot signal frequency. This is one of the problems that need to be solved at present. Among them, the proportion of carrier and sideband in the entire laser power has a great influence on the readout noise in the process of space-based gravitational wave detection; the setting of sideband has a strong coupling relationship with ADC sampling frequency and pilot frequency; the intersatellite beat frequency will also be affected by ADC sampling frequency and pilot frequency, so when setting these two frequencies, it is necessary to avoid the intersatellite beat frequency band. Under the influence of many constraints, it is necessary to formulate a full frequency planning scheme for space-based gravitational wave detectors.

Summary of the invention

The purpose of the present invention is to overcome the defect that it is impossible to formulate a full-frequency planning scheme for a space-based gravitational wave detector under the influence of numerous constraints.

In order to achieve the above object, the present invention proposes a method for formulating a frequency planning scheme for a space-based gravitational wave detector, wherein the space-based gravitational wave detector includes a satellite formation consisting of three satellites, each of which is equipped with two laser interferometer optical platforms; the method includes:

Determine the optimization objective function and variable constraints;

The multi-objective optimization algorithm is used to solve the optimization objective function, and the constraints and objective functions of the offset frequency phase-locking schemes of 6 different master satellites are constructed; the 6 schemes are solved in a parallel manner, and the scheme with the longest duration among the 6 schemes is selected as the final selected scheme; if the solution of the multi-objective optimization algorithm has a feasible solution, the calculation is completed and the frequency planning scheme is obtained, otherwise the multi-objective optimization algorithm is replaced and recalculated until the variable constraints are met; the calculation results include: ultra-stable clock modulation frequency, lower limit of intersatellite beat frequency, upper limit of intersatellite beat frequency, pilot signal frequency, ADC sampling frequency, total readout noise, adjustment parameters, main optical platform and maximum duration.

As an improvement of the above method, the method specifically includes:

Step 1: Determine the optimization objective function and construct variable constraints;

The optimization objective function is as follows:

为总读出噪声；f_upper为星间拍频频率的上限；f_lower为星间拍频频率的下限；in,

is the total readout noise; f _upper is the upper limit of the intersatellite beat frequency; f _lower is the lower limit of the intersatellite beat frequency;

The variable constraints are as follows:

为边频总读出噪声；

为载波总读出噪声；Where: f _mod is the ultra-stable clock modulation frequency;

is the total readout noise of the sideband;

is the total carrier read noise;

Step 2: Solving the optimization objective function using a multi-objective optimization algorithm;

Step 3: Adjust f _upper , f _lower and f _mod obtained in step 2. The strategy is as follows:

和

分别表示调整后的f_upper，f_lower和f_mod；Among them, round is the rounding method;

and

represent the adjusted f _upper , f _lower and f _mod respectively;

和

搜寻合适的f_ADC和f_p并再次更新f_mod；其中，f_ADC为ADC采样频率,f_p为导频信号频率；Step 4: According to

and

Search for suitable f _ADC and f _p and update f _mod again; where f _ADC is the ADC sampling frequency and f _p is the pilot signal frequency;

获取f_ADC的最小值

根据

获取f_ADC和f_p的最大间隔，即1<abs(f_ADC-f_p)<f_lower；在求解过程中，默认f_p<f_ADC<100MHz；Step 4-1: According to

Get the minimum value of f _ADC

according to

Get the maximum interval between f _ADC and f _p , that is, 1<abs(f _ADC -f _p )<f _lower ; in the solution process, the default is f _p <f _ADC <100MHz;

Step 4-2: Obtain all possible combinations of [f _mod ,f _p ,f _ADC ] in an exhaustive manner; assuming there are n possible combinations, store them in the following matrix:

距离

最近的一个组合得到

In the selection matrix,

distance

The most recent combination

Step 5: Use f _upper and f _lower adjusted in step 3 as the input of the upper and lower limits of the beat frequency in step 5, input the existing time-series Doppler frequency shift data, and construct the constraints and objective functions of the offset frequency phase-locking schemes of 6 different primary stars; solve the 6 schemes, and select the scheme with the longest duration among the 6 schemes as the final selected scheme;

重新计算总读出噪声，并更新调整参数m；更新调整参数m的方式如下：Step 6: Updated according to Steps 3 and 4

The total readout noise is recalculated and the adjustment parameter m is updated; the method for updating the adjustment parameter m is as follows:

后的总读出噪声的最小值，更新后的调整参数m记为m^new；That is, by updating the adjustment parameter m, we can obtain the updated

The minimum value of the total readout noise after , the updated adjustment parameter m is recorded as m ^new ;

The updated solution includes:

Wherein, J ₀ (m) and J ₁ (m) represent the 0th and 1st order Bessel functions respectively; m ^new is the updated adjustment parameter m; f _p ,f _ADC are the pilot frequency and ADC sampling frequency obtained in step 4; t is the maximum duration obtained in step 5; M is the main optical platform selected corresponding to the maximum duration t obtained in step 5; RN is the total readout noise after the adjustment parameter m is updated in step 6;

Step 7: After all the optimal solutions obtained in step 2 have been processed through steps 3 to 6, determine whether there is a feasible solution; if so, the feasible solution is the final calculation result; otherwise, replace the multi-objective optimization algorithm used in step 2 and restart step 2;

The determination of whether there is a feasible solution is to determine whether the optimal solution meets the variable constraints of step 1.

As an improvement of the above method, step 2 is specifically: using a multi-objective optimization algorithm to solve and obtain n groups of Pareto optimal solutions, and storing these solutions in an n×4 matrix, where the 4 columns are stored in the order of [f _mod , f _lower , f _upper , m].

As an improvement of the above method, the multi-objective optimization algorithm includes NSGA-II algorithm or MOEAD algorithm.

As an improvement of the above method, the three satellites are satellite_1, satellite_2 and satellite_3;

The satellite 1 includes an optical platform A and an optical platform B; the satellite 2 includes an optical platform C and an optical platform D; the satellite 3 includes an optical platform E and an optical platform F;

The six phase-locking schemes are:

Type 1: Optical platform A is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is optical platform D phase-locked to optical platform C, optical platform C phase-locked to optical platform B, optical platform B phase-locked to optical platform A, optical platform E phase-locked to optical platform F, and optical platform F phase-locked to optical platform A.

Type 2: Optical platform B is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is optical platform E phase-locked to optical platform F, optical platform F phase-locked to optical platform A, optical platform A phase-locked to optical platform B, optical platform D phase-locked to optical platform C, and optical platform C phase-locked to optical platform B.

The third type: Optical platform C is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is: optical platform F is phase-locked to optical platform E, optical platform E is phase-locked to optical platform D, optical platform D is phase-locked to optical platform C, optical platform A is phase-locked to optical platform B, and optical platform B is phase-locked to optical platform C.

Type 4: Optical platform D is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is: optical platform A is phase-locked to optical platform B, optical platform B is phase-locked to optical platform C, optical platform C is phase-locked to optical platform D, optical platform F is phase-locked to optical platform E, and optical platform E is phase-locked to optical platform D.

The fifth type: Optical platform E is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is: optical platform B is phase-locked to optical platform A, optical platform A is phase-locked to optical platform F, optical platform F is phase-locked to optical platform E, optical platform C is phase-locked to optical platform D, and optical platform D is phase-locked to optical platform E.

Type 6: Optical platform F is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is: optical platform C is phase-locked to optical platform D, optical platform D is phase-locked to optical platform E, optical platform E is phase-locked to optical platform F, optical platform B is phase-locked to optical platform A, and optical platform A is phase-locked to optical platform F.

As an improvement to the above method, the constraints of the six different primary stars in step 5 are as follows:

Type 1: Optical platform A as the main optical platform

Wherein, _f12 (t) is the time-series Doppler shift between satellite 1 and satellite 2 that varies with time t; _f13 (t) is the time-series Doppler shift between satellite 1 and satellite 3 that varies with time t; _f23 (t) is the time-series Doppler shift between satellite 2 and satellite 3 that varies with time t; △ _fAB is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform A and the laser frequency of optical platform B; △ _fCD is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform C and the laser frequency of optical platform D; △ _fEF is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform E and the laser frequency of optical platform F; △ _fAF is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform A and the laser frequency of optical platform F; △ _fBC is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform B and the laser frequency of optical platform C;

Type 2: Optical platform B as the main optical platform

Type 3: Optical platform C as the main optical platform

Wherein, △f _DE is the offset frequency, which represents the artificially set fixed difference between the laser frequency of optical platform D and the laser frequency of optical platform E;

Type 4: Optical platform D as the main optical platform

Type 5: Optical platform E as the main optical platform

Type 6: Optical platform F as the main optical platform

The objective function is

max(t)

Where t represents the duration;

The constructed objective function and constraint conditions are solved in a parallel manner for the six schemes, and the solving algorithm adopts a linear programming algorithm; the scheme with the longest duration among the six schemes is selected as the final selected scheme and saved; the saving order is [t, M], where t represents the duration and M represents the corresponding main optical platform.

The present invention also provides a frequency planning scheme formulation system for a space-based gravitational wave detector, wherein the space-based gravitational wave detector comprises a satellite formation consisting of three satellites, each satellite being respectively equipped with two laser interferometer optical platforms; the system comprises:

Initialization module, used to determine the optimization objective function and variable constraints;

The planning scheme calculation module is used to solve the optimization objective function using a multi-objective optimization algorithm, construct the constraints and objective functions of the offset frequency phase-locking schemes of 6 different master satellites; solve the 6 schemes in a parallel manner, and select the scheme with the longest duration among the 6 schemes as the final selected scheme; if the solution of the multi-objective optimization algorithm has a feasible solution, the calculation ends, otherwise the multi-objective optimization algorithm is replaced and recalculated; the calculation results include: ultra-stable clock modulation frequency, the lower limit of the inter-satellite beat frequency, the upper limit of the inter-satellite beat frequency, the pilot signal frequency, the ADC sampling frequency, the total readout noise and the main optical platform.

Compared with the prior art, the advantages of the present invention are:

1. The idea of step-by-step optimization is adopted to solve the complex problem of frequency planning step by step and form a closed loop, which facilitates the solution of complex full frequency planning problems and improves the solution efficiency.

2. In the process of step-by-step optimization, the algorithm is selected for each step according to the characteristics of that step, which is highly flexible.

3. This method can produce multiple feasible solutions. Researchers can formulate evaluation criteria based on actual conditions, and then quickly and easily select the optimal solution that conforms to reality from multiple feasible solutions.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 shows a flow chart of solving the full frequency planning scheme;

FIG2 shows a flow chart of solving the offset frequency planning scheme;

Figure 3 shows a schematic diagram of the time-series intersatellite Doppler shift;

FIG4 shows different main optical platform phase-locking schemes.

DETAILED DESCRIPTION

The technical solution of the present invention is described in detail below with reference to the accompanying drawings.

The purpose of the present invention is to provide a method for formulating a frequency planning scheme for a space-based gravitational wave detector, so as to solve the problem of formulating a planning scheme for the full frequency of a laser in each optical platform in the existing space gravitational wave detection.

The full-link laser frequency involves four items: intersatellite beat frequency f _het , ultra-stable clock modulation frequency f _mod , ADC sampling frequency f _ADC , and pilot signal frequency f _p . Among them, the intersatellite beat frequency f _het will change dynamically over time due to the influence of the intersatellite Doppler frequency shift, and the other three items remain unchanged during the mission operation.

Wherein, due to the limitation of ultra-stable clock, f _mod is usually not higher than 5 GHz; the frequencies of f _ADC and f _p are generated by f _mod through a divider.

The coupling relationship among f _het , f _mod , f _ADC and f _p is as follows:

Since the pilot signal frequency is usually slightly lower than the ADC sampling frequency, in order to avoid the aliasing signal produced by the ADC sampling the pilot signal to coincide with the intersatellite beat frequency, the following constraints are imposed:

Wherein, f _upper represents the upper limit of the inter-satellite beat frequency; f _lower represents the lower limit of the inter-satellite beat frequency.

During the entire mission cycle, the sideband frequency is equal to the ultrastable clock modulation frequency f _mod , which is used to transmit clock noise information between satellites. The proportion of the sideband frequency to the total laser power, the ultrastable clock modulation frequency f _{mod ,} and the intersatellite beat frequency f _het will all affect the total readout noise. In order to ensure that the total readout noise meets the mission index requirements, the following constraints are imposed:

表示总读出噪声，

表示载波总读出噪声，

表示边频总读出噪声，J₀(m)和J₁(m)分别表示0阶和1阶贝塞尔函数，m值的大小决定了边频频率在总功率中的占比。通常m∈[0.45,0.64]，对应边频占用总功率5％到10％之间。需注意的是，

与实际的任务指标相关，在求解过程中，可以进行适应性改变。in,

represents the total read noise,

represents the total carrier read noise,

represents the total readout noise of the sideband, J ₀ (m) and J ₁ (m) represent the 0th and 1st order Bessel functions respectively, and the value of m determines the proportion of the sideband frequency in the total power. Usually m∈[0.45,0.64], corresponding to the sideband occupying 5% to 10% of the total power. It should be noted that

Related to the actual task indicators, adaptive changes can be made during the solution process.

During the operation of the space-based gravitational wave detector, the linkage between three satellites (Satellite_1, Satellite_2, Satellite_3) is involved. Information is transmitted between the three satellites through lasers, and each satellite contains two optical platforms. Due to the intersatellite motion, the laser frequency will be affected by the Doppler frequency shift during the intersatellite transmission process. After the local satellite receives the laser sent by the remote satellite, the received laser interferes with the local laser to obtain scientific operation data information. During the interference process, the difference between the received laser frequency and the local laser frequency is called the beat frequency. In order to ensure that the beat frequency is within a reasonable range, that is, between [f _lower , f _upper ], it is necessary to add an offset frequency during the weak light phase locking process.

The Doppler frequency shift is expressed as f _ab (t), which represents the time series Doppler frequency shift between satellite a and satellite b that changes with time t.

The offset frequency is expressed as Δf _XY, which represents an artificially set fixed difference between the laser frequency of optical platform X and the laser frequency of optical platform Y.

The satellite_1 includes an optical platform A and an optical platform B; the satellite_2 includes an optical platform C and an optical platform D; and the satellite_3 includes an optical platform E and an optical platform F.

The phase-locking schemes include the following six types: optical platform A is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is optical platform D phase-locked to optical platform C, optical platform C phase-locked to optical platform B, optical platform B phase-locked to optical platform A, optical platform E phase-locked to optical platform F, and optical platform F phase-locked to optical platform A.

Optical platform B is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is as follows: optical platform E is phase-locked to optical platform F, optical platform F is phase-locked to optical platform A, optical platform A is phase-locked to optical platform B, optical platform D is phase-locked to optical platform C, and optical platform C is phase-locked to optical platform B.

Optical platform C is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is as follows: optical platform F is phase-locked to optical platform E, optical platform E is phase-locked to optical platform D, optical platform D is phase-locked to optical platform C, optical platform A is phase-locked to optical platform B, and optical platform B is phase-locked to optical platform C.

Optical platform D is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is optical platform A phase-locked to optical platform B, optical platform B phase-locked to optical platform C, optical platform C phase-locked to optical platform D, optical platform F phase-locked to optical platform E, and optical platform E phase-locked to optical platform D.

Optical platform E is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is as follows: optical platform B is phase-locked to optical platform A, optical platform A is phase-locked to optical platform F, optical platform F is phase-locked to optical platform E, optical platform C is phase-locked to optical platform D, and optical platform D is phase-locked to optical platform E.

Optical platform F is the main optical platform, and other satellites are slave optical platforms. The phase-locking order is as follows: optical platform C is phase-locked to optical platform D, optical platform D is phase-locked to optical platform E, optical platform E is phase-locked to optical platform F, optical platform B is phase-locked to optical platform A, and optical platform A is phase-locked to optical platform F.

表示光学平台X的激光和光学平台Y的激光在Z卫星干涉后的拍频。The constraints of the timing inter-satellite beat frequency corresponding to the different phase-locking schemes are as follows, where

It represents the beat frequency of the laser on optical platform X and the laser on optical platform Y after interference on Z satellite.

Optical platform A as main optical platform

in,

Optical platform B is the main optical platform

in,

Optical platform C as main optical platform

in,

Optical platform D as main optical platform

in,

Optical platform E is the main optical platform

in,

Optical platform F is the main optical platform

in,

As shown in FIG1 , the method for formulating a full frequency planning scheme includes the following ten steps.

Step 1: Confirm the optimization objective function and construct the corresponding variable constraints. The objective function is as follows:

There are four solution objectives in total. Objective 1 is to minimize the readout noise; Objective 2 is to maximize the beat frequency interval abs(f _upper -f _lower ), where abs represents the absolute value; According to formula (3), the total readout noise can be reduced by reducing f _het , f _het ∈[f _lower ,f _upper ], so objective 3 is introduced to minimize f _upper ; According to formula (2), f _ADC -f _p <f _lower , so objective 4 is introduced to maximize f _lower , thereby increasing the selection space of f _ADC and f _p .

The variable constraints are as follows:

Step 2: Select the currently available mature multi-objective programming algorithm to solve the optimization objective constructed in step 1. Alternative multi-objective optimization algorithms include NSGA-II algorithm and MOEAD algorithm. Obtain n groups of Pareto optimal solutions and store these solutions in an n×4 matrix for use in subsequent steps. The storage order of the 4 columns is [f _mod ,f _lower ,f _upper ,m]. It should be noted that [f _mod ,f _lower ,f _upper ,m] obtained here cannot be used directly and needs to be adjusted through subsequent steps. Select each group of solutions in step 2 in turn and enter steps 3 to 6 to fine-tune [f _mod ,f _lower ,f _upper ,m].

Step 3: Since f _upper , f _lower and f _mod obtained in step 2 are not integers, it is necessary to adjust f _upper , f _lower and f _mod obtained in step 2. The adjustment strategy is as follows:

和

分别表示调整后的f_upper，f_lower和f_mod。Among them, round means rounding.

and

They represent the adjusted f _upper , f _lower and f _mod respectively.

和

搜寻合适的f_ADC和f_p并再次更新f_mod。首先，根据

获取f_ADC的最小值

根据

获取f_ADC和f_p的最大间隔，即1<abs(f_ADC-f_p)<f_lower。另外，在求解过程中，默认f_p<f_ADC<100MHz。然后，采用穷举的方式获取所有可能的[f_mod,f_p,f_ADC]组合。假设共有n组可能的组合，存储为如下矩阵：Step 4: According to

and

Search for the appropriate f _ADC and f _p and update f _mod again. First, according to

Get the minimum value of f _ADC

according to

Get the maximum interval between f _ADC and f _p , that is, 1<abs(f _ADC -f _p )<f _lower . In addition, in the solution process, it is assumed that f _p <f _ADC <100MHz. Then, use an exhaustive method to obtain all possible combinations of [f _mod ,f _p ,f _ADC ]. Assume that there are n possible combinations, which are stored as the following matrix:

距离

最近的一个组合得到

假设

距离

最近，则

In the selection matrix,

distance

The most recent combination

Assumptions

distance

Recently,

Step 5: Use f _upper and f _lower adjusted in step 3 as the input of the upper and lower limits of the beat frequency in step 5, and input the existing time-series intersatellite Doppler frequency shift data. Construct the constraints and objective functions of the offset frequency setting schemes of six different primary stars. The constraints of the six different primary stars are as follows:

The first type: Optical platform A as the main optical platform

in,

The second type: Optical platform B is the main optical platform

in,

The third type: Optical platform C as the main optical platform

in,

The fourth type: optical platform D as the main optical platform

in,

The fifth type: Optical platform E as the main optical platform

in,

The sixth type: Optical platform F as the main optical platform

in,

The objective function is

max(t) (32)

Here, t represents the duration.

The constructed objective function and constraint conditions are solved in parallel for the six solutions, and the solution algorithm adopts the linear programming algorithm. The solution with the longest duration among the six solutions is selected as the final selected solution and saved. The saving order is [t, M], where t represents the duration and M represents the corresponding main optical platform.

重新计算总读出噪声，并更新调整参数m。更新调整参数m的方式如下Step 6: Updated according to Steps 3 and 4

The total read noise is recalculated and the adjustment parameter m is updated. The adjustment parameter m is updated as follows

后的总读出噪声的最小值，更新后的调整参数m记为m^new。That is, by updating the adjustment parameter m, we can obtain the updated

The minimum value of the total readout noise after , the updated adjustment parameter m is recorded as m ^new .

其中m^new表示更新后的调整参数m，f_p,f_ADC表示由步骤4获取的导频频率和ADC采样频率，t表示步骤5中获取的最长持续时间以及所对应选择的主光学平台M，RN表示步骤6中更新调整参数m后的总读出噪声。Step 7: Store the updated solutions into a new array in sequence. Each row of the array is arranged as follows:

Wherein m ^new represents the updated adjustment parameter m, f _p ,f _ADC represents the pilot frequency and ADC sampling frequency obtained in step 4 , t represents the maximum duration obtained in step 5 and the corresponding selected main optical platform M, and RN represents the total readout noise after the adjustment parameter m is updated in step 6 .

Step 8: After all the Pareto optimal solutions obtained in step 2 have been processed through steps 3 to 7, determine whether there is a feasible solution. If so, proceed to step 10; otherwise, proceed to step 9.

Determining whether there is a feasible solution is to determine whether the optimal solution meets the variable constraints in step 1.

Step 9: Determine whether to change the multi-objective optimization algorithm used in step 1. If yes, change it; otherwise, continue to use the previous optimization algorithm. After adjusting the optimization algorithm, go to step 1 and loop through steps 1 to 8 until the termination condition is met.

Step 10: Save the full frequency setting scheme.

An embodiment of the present invention is as follows:

Step 1: Confirm the optimization objective function and construct the corresponding variable constraints. The objective function is as follows:

There are four solution objectives in total. Objective 1 is to minimize the readout noise; Objective 2 is to maximize the beat frequency interval abs(f _upper -f _lower ), where abs represents the absolute value; According to formula (3), the total readout noise can be reduced by reducing f _het , f _het ∈[f _lower ,f _upper ], so objective 3 is introduced to minimize f _upper ; According to formula (2), f _ADC -f _p <f _lower , so objective 4 is introduced to maximize f _lower , thereby increasing the selection space of f _ADC and f _p .

The variable constraints are as follows:

Step 2: Select the currently available mature multi-objective programming algorithm to solve the optimization objective constructed in step 1. Alternative multi-objective optimization algorithms include NSGA-II algorithm and MOEAD algorithm. Obtain n groups of Pareto optimal solutions and store these solutions in an n×4 matrix for use in subsequent steps. The storage order of the 4 columns is [f _mod ,f _lower ,f _upper ,m]. It should be noted that [f _mod ,f _lower ,f _upper ,m] obtained here cannot be used directly and needs to be adjusted through subsequent steps. Select each group of solutions in step 2 in turn and enter steps 3 to 6 to fine-tune [f _mod ,f _lower ,f _upper ,m].

The Pareto optimal solution is a general representation of multiple groups of feasible solutions generated in the multi-objective planning process. These feasible solutions are not dominated by other solutions during the solution process.

Step 3: Since f _upper , f _lower and f _mod obtained in step 2 are not integers, it is necessary to adjust f _upper , f _lower and f _mod obtained in step 2. The adjustment strategy is as follows:

和

分别表示调整后的f_upper，f_lower和f_mod。Among them, round means rounding.

and

They represent the adjusted f _upper , f _lower and f _mod respectively.

和

搜寻合适的f_ADC和f_p并再次更新f_mod。首先，根据

获取f_ADC的最小值

根据

获取f_ADC和f_p的最大间隔，即1<abs(f_ADC-f_p)<f_lower。另外，在求解过程中，默认f_p<f_ADC<100MHz。然后，采用穷举的方式获取所有可能的[f_mod,f_p,f_ADC]组合。假设共有n组可能的组合，存储为如下矩阵：Step 4: According to

and

Search for the appropriate f _ADC and f _p and update f _mod again. First, according to

Get the minimum value of f _ADC

according to

Get the maximum interval between f _ADC and f _p , that is, 1<abs(f _ADC -f _p )<f _lower . In addition, in the solution process, it is assumed that f _p <f _ADC <100MHz. Then, use an exhaustive method to obtain all possible combinations of [f _mod ,f _p ,f _ADC ]. Assume that there are n possible combinations, which are stored as the following matrix:

距离

最近的一个组合得到

假设

距离

最近，则

In the selection matrix,

distance

The most recent combination

Assumptions

distance

Recently,

Step 5: As shown in Figures 2 and 3, the f _upper and f _lower adjusted in step 3 are used as the input of the upper and lower limits of the beat frequency in step 5, and the existing time-series inter-satellite Doppler frequency shift data is input. The constraints and objective functions of the offset frequency setting schemes of six different primary stars are constructed. The constraints of the six different primary stars are as follows:

The first type: Optical platform A is the main optical platform, as shown in Figure 4(a)

in,

The second type: Optical platform B is the main optical platform, as shown in Figure 4(b)

in,

The third type: Optical platform C is the main optical platform, as shown in Figure 4(c)

in,

The fourth type: Optical platform D is the main optical platform, as shown in Figure 4(d)

in,

The fifth type: Optical platform E is the main optical platform, as shown in Figure 4(e)

in,

The sixth type: Optical platform F is the main optical platform, as shown in Figure 4(f)

in,

The objective function is

max(t) (50)

Here, t represents the duration.

The constructed objective function and constraints are solved in parallel for the six solutions, and the solution algorithm adopts the linear programming algorithm. Among the six solutions, the solution with the longest duration is selected and saved as the final selected solution. The saving order is [t, M], where t represents the duration and M represents the corresponding main optical platform.

The parallelization method adopts a multi-core multi-threaded parallel method to improve computing efficiency.

重新计算总读出噪声，并更新调整参数m。更新调整参数m的方式如下Step 6: Updated according to Steps 3 and 4

The total read noise is recalculated and the adjustment parameter m is updated. The adjustment parameter m is updated as follows

后的总读出噪声的最小值，更新后的调整参数m记为m^new。That is, by updating the adjustment parameter m, we can obtain the updated

The minimum value of the total readout noise after , the updated adjustment parameter m is recorded as m ^new .

其中m^new表示更新后的调整参数m，f_p,f_ADC表示由步骤4获取的导频频率和ADC采样频率，t表示步骤5中获取的最长持续时间以及所对应选择的主光学平台M，RN表示步骤6中更新调整参数m后的总读出噪声。Step 7: Store the updated solution into a new array, and each row of the array is arranged as follows

Wherein m ^new represents the updated adjustment parameter m, f _p ,f _ADC represents the pilot frequency and ADC sampling frequency obtained in step 4 , t represents the maximum duration obtained in step 5 and the corresponding selected main optical platform M, and RN represents the total readout noise after the adjustment parameter m is updated in step 6 .

矩阵如下所示：After updating the adjustment parameter m in step 6, the obtained

The matrix looks like this:

Step 8: After all the Pareto optimal solutions obtained in step 2 have been processed through steps 3 to 7, determine whether there is a feasible solution. If so, proceed to step 10; otherwise, proceed to step 9.

则筛选出以下结果：Assume that the feasible solution required in step 8 is a duration of t = 2190 and the readout noise is less than

The following results are filtered out:

1200 2 25 0.56 48 50 2190 A 6.35E-12 1200 2 25 0.56 48 50 2190 A 6.35E-12

Step 9: Determine whether to change the multi-objective optimization algorithm used in step 1. If yes, change it, otherwise use the previous optimization algorithm. After adjusting the optimization algorithm, go to step 1 and loop through steps 1 to 8 until the termination condition is met.

Step 10: Save the full frequency setting scheme.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention is described in detail with reference to the embodiments, it should be understood by those skilled in the art that any modification or equivalent replacement of the technical solutions of the present invention does not depart from the spirit and scope of the technical solutions of the present invention and should be included in the scope of the claims of the present invention.

Claims (7)

1. A space-based gravitational wave detector frequency planning scheme formulation method, the space-based gravitational wave detector includes three satellites forming a satellite formation, each satellite is loaded with two laser interference optical platforms respectively; the method comprises the following steps:

determining an optimization objective function and variable constraint conditions;

solving an optimized objective function by utilizing a multi-objective optimization algorithm, and constructing constraint conditions and objective functions of offset frequency phase locking schemes of 6 different main stars; solving the 6 schemes in a parallelization mode, and selecting the scheme with the longest duration in the 6 schemes as a finally selected scheme; if the solution of the multi-objective optimization algorithm has a feasible solution, ending the calculation to obtain a frequency planning scheme, otherwise, replacing the multi-objective optimization algorithm to recalculate until the variable constraint condition is met; the calculation result comprises: the ultra-stable clock modulation frequency, the lower limit of the inter-satellite beat frequency, the upper limit of the inter-satellite beat frequency, the pilot signal frequency, the ADC sampling frequency, the total readout noise, the adjustment parameters, the main optical platform and the longest duration.

2. The method for formulating the frequency planning scheme of the space-based gravitational wave detector according to claim 1, wherein the method specifically comprises:

step 1: determining an optimization objective function and constructing a variable constraint;

the optimization objective function is as follows:

wherein ,

is the total readout noise; f (f) _upper The upper limit of the inter-satellite beat frequency; f (f) _lower Is the lower limit of the inter-satellite beat frequency;

the variable constraints are as follows:

wherein ：f_mod Modulating the frequency for an ultra-stable clock;

the noise is read out for the total side frequency;

Reading out noise for the carrier wave;

step 2: solving the optimization objective function by using a multi-objective optimization algorithm;

step 3: solving the step 2 to obtain f _upper ，f _lower and f_mod The adjustment is carried out according to the following strategy:

wherein round is a rounding method;

and

Respectively represents the adjusted f _upper ，f _lower and f_mod ；

Step 4: according to

and

Searching for the appropriate f _ADC and f_p And update f again _mod； wherein ,f_ADC For ADC sampling frequency, f _p Is the pilot signal frequency;

step 4-1: according to

Acquiring f _ADC Minimum value +.>

According to->

Acquiring f _ADC and f_p Maximum spacing of 1, i.e<abs(f _ADC -f _p )<f _lower The method comprises the steps of carrying out a first treatment on the surface of the Default f in the solution process _p <f _ADC <100MHz；

Step 4-2: obtaining all possible [ f ] in an exhaustive manner _mod ,f _p ,f _ADC ]Combining; there are n groups of possible combinations stored as a matrix:

in the selection matrix, the selection matrix is selected,

distance->

The most recent combination gets +>

Step 5: f, adjusting the step 3 _upper and f_lower Inputting the existing time sequence Doppler frequency shift data as the input of the upper and lower limits of the beat frequency in the step 5, and constructing constraint conditions and objective functions of offset frequency phase locking schemes of 6 different principal stars; solving the 6 schemes, and selecting the scheme with the longest duration from the 6 schemes as the finally selected scheme;

step 6: updated according to step 3 and step 4

Recalculating the total readout noise and updating the adjustment parameter m; the adjustment parameter m is updated as follows:

that is, the update is obtained by updating the adjustment parameter m

The minimum value of the total read noise after updating, and the adjustment parameter m after updating is recorded as m ^new ；

The updated solution includes:

wherein ,J₀(m) and J₁ (m) represents 0 th and 1 st order bessel functions, respectively; m is m ^new The updated adjustment parameter m; f (f) _p ,f _ADC The pilot frequency and the ADC sampling frequency obtained in the step 4 are obtained; t is the longest duration obtained in step 5; m is the main optical platform selected corresponding to the longest duration t obtained in the step 5; the RN is the total readout noise after updating the adjustment parameter m in the step 6;

step 7: judging whether feasible solutions exist after all the optimal solutions obtained in the step 2 are processed in the steps 3 to 6; if yes, the feasible solution is the final calculation result; otherwise, replacing the multi-objective optimization algorithm adopted in the step 2, and restarting to execute the step 2;

and judging whether a feasible solution exists or not, and judging whether the optimal solution accords with the variable constraint of the step 1 or not.

3. The method for formulating the frequency planning scheme of the space-based gravitational wave detector according to claim 2, wherein the step 2 is specifically: solving by using a multi-objective optimization algorithm to obtain n groups of pareto optimal solutions, storing the solutions into an n multiplied by 4 matrix, wherein the storage sequence of 4 columns is [ f ] _mod ,f _lower ,f _upper ,m]。

4. The method of claim 2, wherein the multi-objective optimization algorithm comprises NSGA-II algorithm or MOEAD algorithm.

5. The method for formulating the frequency planning scheme of the space-based gravitational wave detector in accordance with claim 2, wherein said three satellites are respectively satellite_1, satellite_2 and satellite_3;

wherein the satellite_1 comprises an optical platform A and an optical platform B; the satellite_2 comprises an optical platform C and an optical platform D; the satellite_3 comprises an optical platform E and an optical platform F;

the 6 phase locking schemes are as follows:

1 st: taking an optical platform A as a main optical platform, and taking other satellites as slave optical platforms, wherein the phase locking sequence is that an optical platform D is phase-locked to an optical platform C, the optical platform C is phase-locked to an optical platform B, the optical platform B is phase-locked to the optical platform A, the optical platform E is phase-locked to an optical platform F, and the optical platform F is phase-locked to the optical platform A;

2 nd: taking an optical platform B as a main optical platform, and taking other satellites as slave optical platforms, wherein the phase locking sequence is that an optical platform E is phase-locked to an optical platform F, the optical platform F is phase-locked to an optical platform A, the optical platform A is phase-locked to an optical platform B, the optical platform D is phase-locked to an optical platform C, and the optical platform C is phase-locked to the optical platform B;

3 rd: taking an optical platform C as a main optical platform, and taking other satellites as slave optical platforms, wherein the phase locking sequence is that an optical platform F is phase-locked to an optical platform E, the optical platform E is phase-locked to an optical platform D, the optical platform D is phase-locked to the optical platform C, the optical platform A is phase-locked to an optical platform B, and the optical platform B is phase-locked to the optical platform C;

4 th: taking an optical platform D as a main optical platform, and taking other satellites as slave optical platforms, wherein the phase locking sequence is that an optical platform A is phase-locked to an optical platform B, the optical platform B is phase-locked to an optical platform C, the optical platform C is phase-locked to the optical platform D, the optical platform F is phase-locked to an optical platform E, and the optical platform E is phase-locked to the optical platform D;

5 th: taking an optical platform E as a main optical platform, and taking other satellites as slave optical platforms, wherein the phase locking sequence is that an optical platform B is phase-locked to an optical platform A, the optical platform A is phase-locked to an optical platform F, the optical platform F is phase-locked to the optical platform E, the optical platform C is phase-locked to an optical platform D, and the optical platform D is phase-locked to the optical platform E;

6 th: the optical platform F is used as a main optical platform, other satellites are slave optical platforms, the phase locking sequence is that an optical platform C is phase-locked to an optical platform D, the optical platform D is phase-locked to an optical platform E, the optical platform E is phase-locked to the optical platform F, the optical platform B is phase-locked to an optical platform A, and the optical platform A is phase-locked to the optical platform F.

6. The method for planning frequency plan of space-based gravitational wave detector in accordance with claim 5, wherein the constraint conditions of six different main satellites in said step 5 are as follows:

1 st: optical platform A as main optical platform

wherein ,f₁₂ (t) is the time-series doppler shift between satellite 1 and satellite 2 over time t; f (f) ₁₃ (t) is the time-series doppler shift between satellite 1 and satellite 3 over time t; f (f) ₂₃ (t) is the time-series doppler shift between satellite 2 and satellite 3 over time t; Δf _AB The offset frequency is a fixed difference value which is manually set between the laser frequency of the optical platform A and the laser frequency of the optical platform B; Δf _CD For the offset frequency, a fixed difference between the laser frequency of the optical bench C and the laser frequency of the optical bench D is represented by an artificial setting; Δf _EF For the offset frequency, a fixed difference between the laser frequency of the optical bench E and the laser frequency of the optical bench F is represented by an artificial setting; Δf _AF For the offset frequency, a fixed difference between the laser frequency of the optical bench a and the laser frequency of the optical bench F is represented by an artificial setting; Δf _BC For the offset frequency, a fixed difference between the laser frequency of the optical bench B and the laser frequency of the optical bench C is represented by an artificial setting;

2 nd: optical platform B as main optical platform

3 rd: optical platform C as main optical platform

wherein ,△f_DE For the offset frequency, a fixed difference between the laser frequency of the optical bench D and the laser frequency of the optical bench E is represented by an artificial setting;

4 th: optical platform D as main optical platform

5 th: optical platform E as main optical platform

6 th: optical platform F is taken as main optical platform

The objective function is

max(t)

Wherein t represents a duration;

solving the six schemes by adopting a parallelization mode according to the constructed objective function and constraint conditions, wherein a linear programming algorithm is adopted in a solving algorithm; selecting the scheme with the longest duration from the 6 schemes as the finally selected scheme for storage; the preservation sequence is [ t, M ], wherein t represents the duration and M represents the corresponding main optical platform.

7. The system comprises a space-based gravitational wave detector and a frequency planning scheme, wherein the space-based gravitational wave detector comprises a satellite formation formed by three satellites, and each satellite is respectively loaded with two laser interference optical platforms; the system comprises:

the initialization module is used for determining an optimization objective function and variable constraint conditions;

the planning scheme calculation module is used for solving an optimized objective function by utilizing a multi-objective optimization algorithm and constructing constraint conditions and objective functions of offset frequency phase locking schemes of 6 different main stars; solving the 6 schemes in a parallelization mode, and selecting the scheme with the longest duration in the 6 schemes as a finally selected scheme; if the solution of the multi-objective optimization algorithm has a feasible solution, ending the calculation to obtain a frequency planning scheme, otherwise, replacing the multi-objective optimization algorithm to recalculate until the variable constraint condition is met; the calculation result comprises: the ultra-stable clock modulation frequency, the lower limit of the inter-satellite beat frequency, the upper limit of the inter-satellite beat frequency, the pilot signal frequency, the ADC sampling frequency, the total readout noise, the adjustment parameters, the main optical platform and the longest duration.

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