Satellite relative navigation adaptive filtering method based on online identification of power spectral density

Abstract

This invention discloses an adaptive filtering method for satellite relative navigation based on online power spectral density identification, belonging to the field of satellite formation flying and signal processing technology. The method includes: establishing a relative motion dynamic model of the satellite in a near-circular orbit, constructing continuous-time and discrete-time state-space equations; establishing an explicit analytical relationship between process noise covariance and power spectral density; adaptively determining the convergence timing of the filter using a whitening test criterion, and calculating the estimated value of the process noise covariance based on the maximum likelihood estimation method; constructing a least-squares optimization problem, and identifying the optimal power spectral density parameters online in real time. By establishing an analytical relationship between process noise covariance and power spectral density, this invention transforms the covariance matrix reconstruction problem into the estimation of a finite number of power spectral density parameters, significantly reducing the parameter dimensionality, decreasing dependence on prior noise statistics, and avoiding the generation of non-physical negative definite covariance matrices.

Description

Technical Field

[0001] This invention relates to the field of satellite formation flight and signal processing, and in particular to an adaptive filtering method for satellite relative navigation based on online power spectral density identification. Background Technology

[0002] Satellite relative navigation is a technique that estimates the real-time relative position and velocity of satellites by analyzing their relative motions, thereby achieving high-precision positioning. Under near-circular orbit conditions, the relative dynamics of satellites are typically modeled using linearized Clohessy-Wiltshire equations, and their state estimation is described by corresponding state-space equations, providing a theoretical basis for filter design. This technology supports advanced space missions such as Earth observation and gravitational wave detection by constructing multi-satellite cooperative configurations.

[0003] Kalman filters are widely used in spacecraft control and navigation due to their simplicity and computational efficiency, effectively supporting real-time state estimation and orbit determination tasks. However, their performance is highly sensitive to prior knowledge of noise statistics. In practical engineering, measurement noise can generally be obtained with high accuracy through ground calibration or sensor calibration, but process noise often originates from uncertain factors such as unmodeled dynamics and environmental disturbances, making it difficult to accurately characterize beforehand. Incorrect noise statistics can lead to a decline in filter performance or even divergence.

[0004] Traditional adaptive methods typically attempt to estimate the complete process noise covariance matrix online. This introduces a large number of parameters to be estimated, significantly increasing the computational burden and leading to slow convergence and even instability in practical applications. While Bayesian methods alleviate the reliance on prior knowledge to some extent by introducing prior distributions for joint parameter estimation, they fail to fully exploit the inherent parameterized noise structure of the system. Due to the lack of a physical model embedded with noise, Bayesian estimation may produce mathematically invalid or physically uninterpretable results, such as non-positive definite covariance matrices, thus affecting the practical usability and reliability of the filter. Summary of the Invention

[0005] Purpose of the invention: To address the above problems, the purpose of this invention is to provide a satellite relative navigation adaptive filtering method based on online power spectral density identification.

[0006] Technical solution: The first aspect of this invention, a satellite relative navigation adaptive filtering method based on online power spectral density identification, includes the following steps:

[0007] Step 1: Establish a dynamic model of the relative motion of the satellite in a near-circular orbit based on the Clohessy-Wiltshire equations, construct continuous-time and discrete-time state-space equations based on the model, and initialize the parameters.

[0008] Step 2: Based on the constructed discrete-time state-space equation, establish an explicit analytical relationship between the process noise covariance and the power spectral density;

[0009] Step 3: Run the standard Kalman filter using the initial process noise covariance, and synchronously record the innovative information sequence and Kalman gain generated during the filtering process; adaptively determine the convergence time of the filter using the whitening test criterion, and calculate the estimated value of the process noise covariance based on the maximum likelihood estimation method;

[0010] Step 4: Construct a least-squares optimization problem, identify the optimal power spectral density parameter in real time, dynamically reconstruct the process noise covariance matrix using the identification results, and update it to the Kalman filter to complete the state estimation and covariance update at the current moment.

[0011] Preferably, step 1 includes:

[0012] Step 101: For satellites in near-circular orbits, when the relative distance between satellites is much smaller than the orbital radius, the linearized Clohessy–Wiltshire equations are used to describe the relative motion dynamics, as expressed in:

[0013] ,

[0014] ,

[0015] ,

[0016] in, This represents the relative position component of the satellite with respect to the master satellite; Indicates the rate of change of the latitude angle of the main satellite; Indicates the process noise component; Indicates the control input component;

[0017] Step 102: Transform the Clohessy-Wiltshire equations into continuous-time state-space equations, in the following form:

[0018] ,

[0019] ,

[0020] in, These are state variables, including relative position and relative velocity; The process noise is represented by the process noise covariance matrix. and includes the power spectral density vector ; Indicates control input; These represent the system matrix and the input matrix, respectively. Indicates the initial time. Represents the state variables at the initial moment;

[0021] Step 103: Ignoring process noise and control input, discretize the continuous-time state-space equation to obtain the discrete-time state-space equation, expressed as:

[0022] ,

[0023] in, express The state variable at any given time; Represents the discrete time period State transition matrix; The sampling time interval;

[0024] Step 104, the observation equation corresponding to the discrete-time state-space equation is:

[0025] ,

[0026] in, Represents the observation vector; Represents the observation system matrix; To measure noise, its covariance is ;

[0027] Step 105, Initialize and set state estimation and error covariance .

[0028] Preferably, step 2 includes:

[0029] Step 201: Based on the properties of continuous-time Gaussian white noise, the process noise covariance matrix is... Represented as:

[0030] ,

[0031] in, Represents the time variable within the integral symbol;

[0032] Step 202: When the system matrix F is known, the state transition matrix has the following analytical form:

[0033] ,

[0034] in, ;

[0035] Step 203: Substitute the analytical form of the state transition matrix into the process noise covariance matrix. The integral formula, by integrating over each element, yields the following process noise covariance matrix. With power spectral density vector The analytical relationship between them is:

[0036] ,

[0037] The expression for each element is as follows:

[0038] ,

[0039] ,

[0040] ,

[0041] ,

[0042] ,

[0043] ,

[0044] ,

[0045] ,

[0046] ,

[0047] ,

[0048] ,

[0049] ,

[0050] ,

[0051] in, , , .

[0052] Preferably, step 3 includes:

[0053] Step 301: Based on the analytical relationship, select the initial coarse power spectral density parameters. Calculate the initial coarse process noise covariance and using the initial coarse process noise covariance Replacement process noise covariance The standard Kalman filter process is represented as follows:

[0054] ,

[0055] ,

[0056] ,

[0057] ,

[0058] ,

[0059] in, and They represent Prior and posterior state estimation at time t; and They represent The prior and posterior error covariance matrices at time t; Indicates the Kalman filter gain;

[0060] Define the new information sequence Its covariance is Record the sequence of innovative information at each moment during the filtering process. With Kalman filter gain ;

[0061] Step 302: Apply the whitening test criterion to adaptively determine whether the filter has entered the convergence state, and record the convergence time as... If at the current moment If so, return to step 301; if the current time Proceed to step 303;

[0062] Step 303: Within the framework of optimal filtering theory, the innovative information sequence conforms to a zero-mean Gaussian distribution, and the prior error covariance is assumed to be time-invariant. A maximum log-likelihood function is then constructed. , is represented as:

[0063] ,

[0064] in, Covariance The determinant, observation vector The dimension; Represents probability. Indicates the given parameters Under the conditions Probability of taking a value; Represents the innovation vector at the i-th discrete time step;

[0065] Step 304: To obtain the maximum log-likelihood function for the power spectral density The optimal solution is to let the maximum log-likelihood function be given by... The partial derivative is 0, that is... The following equation is derived:

[0066] ,

[0067] ,

[0068] ,

[0069] ,

[0070] in, The trace operator for a matrix; This represents the information vector at the i-th discrete time. Represents the information at the i-th discrete time. The transpose of ;

[0071] Step 305, because To satisfy the positive definite condition, the sum of the inner terms of the derivative result in step 304 must be a zero matrix, thus yielding:

[0072] ,

[0073] ,

[0074] Step 306, based on the equation relationship obtained in step 305, use the following... Estimation process noise covariance All elements, exist The recursive estimation formula for time is:

[0075] .

[0076] Preferably, step 4 includes:

[0077] Step 401, using the process noise covariance derived in Step 2. With power spectral density The analytical relationship between them is used to compare the covariance matrix estimated in step 3. By corresponding each element, a system of linear equations is constructed, which can be represented as:

[0078] ,

[0079] in, This represents a matrix vectorization operation, which arranges all elements of a matrix into a column vector.

[0080] Step 402, to solve for the optimal power spectral density parameters The least squares optimization problem is established as follows:

[0081] ,

[0082] Step 403, select the optimal power spectral density parameters. Substituting the analytical relation derived in step 203, we can reconstruct the process noise covariance matrix at the current moment, i.e. ,

[0083] Step 404, reconstruct the process noise covariance. The standard Kalman filter framework in step 301 is applied to perform prediction and update steps to complete the state estimation.

[0084] To perform state estimation for the next time step, return to step 306, which is the sequence of innovation information based on the new time step. With Kalman filter gain The gain is recalculated to determine the process noise covariance at the next time step. .

[0085] The electronic device of the second aspect of the present invention includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the satellite relative navigation adaptive filtering method based on online power spectral density identification.

[0086] The third aspect of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, performs the aforementioned satellite relative navigation adaptive filtering method based on online power spectral density identification.

[0087] Beneficial effects: Compared with the prior art, the significant advantages of this invention are:

[0088] 1. This invention establishes an analytical relationship between process noise covariance and power spectral density, transforming the covariance matrix reconstruction problem into the estimation of a finite number of power spectral density parameters. This significantly reduces the parameter dimensionality, decreases the dependence on prior noise statistics, and avoids the generation of non-physical negative definite covariance matrices.

[0089] 2. Based on structured noise modeling and online least squares optimization, this invention ensures filtering accuracy without introducing additional computational burden, effectively improving the stability and adaptability of satellite relative navigation state estimation, and is suitable for space missions such as high-precision Earth observation and gravitational wave detection.

[0090] 3. This invention maintains robustness even when there is statistical mismatch in process noise through a dynamic covariance reconstruction mechanism, which is significantly better than traditional adaptive methods. Numerical simulations verify that it achieves fast convergence and high-precision trajectory estimation under different power spectral density configurations. Attached Figure Description

[0091] Figure 1 This is a flowchart illustrating the overall implementation of the present invention;

[0092] Figure 2 The curves show the relationship between the root mean square error of the position average component and the inaccurate initial power spectral density for different methods.

[0093] Figure 3 This describes the convergence behavior of the covariance over time during filtering using different methods. Detailed Implementation

[0094] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and not intended to limit the scope of the invention. Furthermore, it should be noted that, for ease of description, the accompanying drawings show only the parts relevant to the embodiments of the present invention, and not all structures.

[0095] In the following description, specific details such as target system architecture and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of this application. However, those skilled in the art will understand that this application may also be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods are omitted so as not to obscure the description of this application with unnecessary detail.

[0096] It should be understood that, when used in this application specification and the appended claims, the term "comprising" indicates the presence of the described features, integrals, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integrals, steps, operations, elements, components and / or a collection thereof.

[0097] It should also be understood that the term “and / or” as used in this application specification and the appended claims means any combination of one or more of the associated listed items and all possible combinations, and includes such combinations.

[0098] Furthermore, in the description of this application and the appended claims, the terms "first," "second," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0099] References to "one embodiment" or "some embodiments" in this specification mean that one or more embodiments of this application include the target features, structures, or characteristics described in connection with that embodiment. Therefore, the phrases "in one embodiment," "in some embodiments," "in other embodiments," "in still other embodiments," etc., appearing in different parts of this specification do not necessarily refer to the same embodiment, but rather mean "one or more, but not all, embodiments," unless otherwise specifically emphasized.

[0100] To address the problem of inaccurate process noise statistics caused by unknown interference in satellite relative navigation, this embodiment proposes an adaptive filtering method for satellite relative navigation based on online power spectral density identification. This method establishes an explicit mathematical analytical relationship between process noise covariance and power spectral density, thereby achieving structured noise modeling, significantly reducing the number of estimation parameters, avoiding additional computational burden, and improving estimation accuracy and adaptability.

[0101] Combination Figure 1 As shown in this embodiment, the satellite relative navigation adaptive filtering method based on online power spectral density identification includes the following steps:

[0102] Step 1: Establish a dynamic model of the relative motion of the satellite in a near-circular orbit based on the Clohessy-Wiltshire equations. Construct continuous-time and discrete-time state-space equations based on this model and initialize the parameters.

[0103] A dynamic model of the satellite's relative motion in a near-circular orbit is established based on the linearized Clohessy-Wiltshire equations. Continuous-time and discrete-time state-space equations are then constructed to provide a system model for the subsequent filtering process. Subsequently, the state estimates and error covariance are initialized.

[0104] Further, step 1 includes:

[0105] Step 101: For satellites in near-circular orbits, when the relative distance between satellites is much smaller than the orbital radius, the linearized Clohessy–Wiltshire equations are used to describe the relative motion dynamics, as expressed in:

[0106] ,

[0107] ,

[0108] ,

[0109] in, This represents the relative position component of the satellite with respect to the master satellite; Indicates the rate of change of the latitude angle of the main satellite; Indicates the process noise component; Indicates the control input component;

[0110] Step 102: Transform the Clohessy-Wiltshire equations into continuous-time state-space equations, in the following form:

[0111] ,

[0112] ,

[0113] in, These are state variables, including relative position and relative velocity; The process noise is represented by the process noise covariance matrix. , including power spectral density vector ; Indicates control input; These represent the system matrix and the input matrix, respectively. Indicates the initial time. Represents the state variables at the initial moment;

[0114] Step 103: Ignoring process noise and control input, discretize the continuous-time state-space equation to obtain the discrete-time state-space equation, expressed as:

[0115] ,

[0116] in, express The state variable at any given time; Represents the discrete time period The state transition matrix is ​​calculated using a matrix method. The matrix exponent, i.e. ; The sampling time interval;

[0117] Step 104, the observation equation corresponding to the discrete-time state-space equation is:

[0118] ,

[0119] in, Represents the observation vector; Represents the observation system matrix; To measure noise, its covariance is ;

[0120] Step 105, Initialize and set state estimation and error covariance .

[0121] Step 2: Based on the constructed discrete-time state-space equation, establish an explicit analytical relationship between the process noise covariance and the power spectral density.

[0122] Further, step 2 includes:

[0123] Step 201: Based on the properties of continuous-time Gaussian white noise, the process noise covariance matrix is... Represented as:

[0124] ,

[0125] in, Represents the time variable within the integral symbol;

[0126] Step 202: When the system matrix F is known, the state transition matrix has the following analytical form:

[0127] ,

[0128] in, ;

[0129] Step 203: Substitute the analytical form of the state transition matrix into the process noise covariance matrix. The integral formula, by integrating over each element, yields the following process noise covariance matrix. With power spectral density vector The analytical relationship between them is:

[0130] ,

[0131] The expression for each element is as follows:

[0132] ,

[0133] ,

[0134] ,

[0135] ,

[0136] ,

[0137] ,

[0138] ,

[0139] ,

[0140] ,

[0141] ,

[0142] ,

[0143] ,

[0144] ,

[0145] in, , , .

[0146] Through the derivation of the above formula, the process noise covariance was finally established. With power spectral density The explicit analytical relationship between them optimizes the complex problem of estimating the entire covariance matrix (containing multiple unknown elements) into estimating only three power spectral density parameters. This significantly reduces the dimensionality of the parameter space.

[0147] Step 3: Run the standard Kalman filter using the initial process noise covariance, and synchronously record the innovative information sequence and Kalman gain generated during the filtering process; adaptively determine the convergence time of the filter using the whitening test criterion, and calculate the estimated value of the process noise covariance based on the maximum likelihood estimation method.

[0148] Furthermore, step 3 includes:

[0149] Step 301: Based on the analytical relationship, select the initial coarse power spectral density parameters. Calculate the initial coarse process noise covariance and using the initial coarse process noise covariance Replacement process noise covariance The standard Kalman filter process is represented as follows:

[0150] ,

[0151] ,

[0152] ,

[0153] ,

[0154] ,

[0155] in, and They represent Prior and posterior state estimation at time t; and They represent The prior and posterior error covariance matrices at time t; Indicates the Kalman filter gain;

[0156] Define the new information sequence Its covariance is Record the sequence of innovative information at each moment during the filtering process. With Kalman filter gain ;

[0157] Step 302: Apply the whitening test criterion to adaptively determine whether the filter has entered the convergence state, and record the convergence time as... If at the current moment If so, return to step 301; if the current time Proceed to step 303;

[0158] Step 303: Within the framework of optimal filtering theory, the innovative information sequence conforms to a zero-mean Gaussian distribution, and the prior error covariance is assumed to be time-invariant. A maximum log-likelihood function is then constructed. , is represented as:

[0159] ,

[0160] in, Covariance The determinant, observation vector The dimension; Represents probability. Indicates the given parameters Under the conditions Probability of taking a value; Represents the innovation vector at the i-th discrete time step;

[0161] Step 304: To obtain the maximum log-likelihood function for the power spectral density The optimal solution is to let the maximum log-likelihood function be given by... The partial derivative is 0, that is... The following equation is derived:

[0162] ,

[0163] ,

[0164] ,

[0165] ,

[0166] in, The trace operator represents the matrix operation, which calculates the sum of the elements on the main diagonal of a square matrix. This represents the information vector at the i-th discrete time. Represents the information at the i-th discrete time. The transpose of ;

[0167] Step 305, because To satisfy the positive definite condition, the sum of the inner terms of the derivative result in step 304 must be a zero matrix, thus yielding:

[0168] ,

[0169] ,

[0170] Step 306, based on the equation relationship obtained in step 305, use the following... Estimation process noise covariance All elements, exist The recursive estimation formula for time is:

[0171] .

[0172] This shows the process noise covariance. From the innovation information sequence Recursive estimation, but as Increased dimensionality, process noise covariance The significant increase in the number of unknown parameters inevitably leads to a decrease in estimation accuracy. Therefore, further optimization is needed.

[0173] In one example, the whitening test criterion is a method that uses the statistical characteristics of the error signal to automatically determine whether an adaptive filter has converged. Its basic principle is that when the adaptive filter reaches its optimal state, the filtered error signal should be approximately white noise, meaning the error sequence has almost no correlation between different time points. Therefore, by checking whether the error signal is whitened in real time, it is possible to infer whether the filter has converged. In practice, this is typically implemented using the following steps:

[0174] (1) Acquiring error signals:

[0175] During the operation of the filter, error sequences are continuously collected over a period of time, typically using a sliding window of error samples of a fixed length.

[0176] (2) Estimate autocorrelation:

[0177] Within this time window, the autocorrelation coefficients of the error signal under different delays are calculated, focusing primarily on the case where the delay is greater than zero. If the filter has converged, the absolute values ​​of these autocorrelation coefficients should be very small, close to zero.

[0178] (3) Constructing statistics:

[0179] By squared and weighted summation of all non-zero delay autocorrelation coefficients, a comprehensive index corresponding to the Q statistic in the Ljung-Box test is obtained, which can measure the degree of deviation of the error sequence from white noise.

[0180] (4) Set threshold:

[0181] Based on the selected significance level (e.g., 5%) and the number of delays used, a critical value is obtained from the chi-square distribution table. If the currently calculated statistic is less than this critical value, the error signal can be considered to have passed the whitening test, meaning the filter has converged; otherwise, it is considered not to have converged.

[0182] To adapt to real-time applications, this testing process can be continuous: each time a new set of error samples is obtained, the data within the sliding window is updated, the statistic is recalculated, and it is determined whether the test has passed consecutively multiple times. By setting a threshold for the number of consecutive passes, false positives caused by random fluctuations can be avoided, making the convergence determination more reliable. In this example, the whitening test criterion is applied to adaptively determine whether the filter has entered the convergence state, and a threshold is set... .

[0183] Step 4: Construct a least-squares optimization problem, identify the optimal power spectral density parameter in real time, dynamically reconstruct the process noise covariance matrix using the identification results, and update it to the Kalman filter to complete the state estimation and covariance update at the current moment.

[0184] Furthermore, step 4 includes:

[0185] Step 401, using the process noise covariance derived in Step 2. With power spectral density The analytical relationship between them is used to compare the covariance matrix estimated in step 3. By corresponding each element, a system of linear equations is constructed, which can be represented as:

[0186] ,

[0187] in, This represents a matrix vectorization operation, which arranges all elements of a matrix into a column vector.

[0188] Step 402, to solve for the optimal power spectral density parameters The least squares optimization problem is established as follows:

[0189] ,

[0190] The problem aims to find a way to reconstruct the covariance. With Estimated Covariance The power spectral density parameter with the smallest difference, due to covariance yes The problem is a linear function and can be solved analytically or numerically efficiently.

[0191] Step 403, select the optimal power spectral density parameters. Substituting the analytical relation derived in step 203, we can reconstruct the process noise covariance matrix at the current moment, i.e. ,

[0192] Step 404, reconstruct the process noise covariance. The standard Kalman filter framework in step 301 is applied to perform prediction and update steps to complete the state estimation.

[0193] To perform state estimation for the next time step, return to step 306, which is the sequence of innovation information based on the new time step. With Kalman filter gain The gain is recalculated to determine the process noise covariance at the next time step. .

[0194] Furthermore, this embodiment also compares five different filters through numerical simulation to verify the effectiveness of the present invention. The five filters are: standard Kalman filter (denoted as KF), Kalman filter based on process noise covariance matrix Q estimation (denoted as QeKF), and filter based on prediction error covariance matrix. Scaled Kalman filtering (denoted as PsKF) is based on the prediction error covariance matrix. The estimated Kalman filter (denoted as PeKF) and the novel method proposed in this invention (denoted as AKF) are presented. Based on three different power spectral density parameters under real-world conditions, the superiority of the proposed method is verified by analyzing the root mean square error of the output position components of each filter and their convergence over time.

[0195] Figure 2 The parameters are based on three different real power spectral density parameters, with Figure (a) corresponding to: Figure (b) corresponds to: Figure (c) corresponds to: This paper demonstrates the relationship between the root mean square error (RMSE) of the position-averaged position components using five different methods and inaccurate initial power spectral densities (ranging from 0.1 to 10). Figure 2 It is known that traditional methods are highly dependent on the statistical characteristics of noise, and their performance deteriorates significantly when noise parameters are mismatched. In contrast, the AKF method of this invention maintains superior accuracy and robustness throughout the entire testing range, demonstrating strong adaptability and significantly relaxing the accuracy requirements for prior information on noise parameters.

[0196] Figure 3 The parameters are based on three different real power spectral density parameters, with Figure (a) corresponding to: Figure (b) corresponds to: Figure (c) corresponds to: This paper demonstrates the convergence behavior of covariance over time (from 0-300s) during filtering using five different methods. Evaluation using the covariance error index shows that the estimated covariance of all filters deviates from the true value. However, the AKF method of this invention exhibits superior stability, accuracy, and fast convergence in all scenarios, highlighting its robust uncertainty quantification capabilities under model mismatch conditions.

[0197] This invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. It should be noted that when the processor executes the computer program, it corresponds to the specific steps of the method provided in this invention, possessing the corresponding functional modules and beneficial effects. Technical details not described in detail in this embodiment can be found in the method provided in this invention.

[0198] This invention also proposes a computer-readable storage medium storing a computer program. It should be noted that when the computer program is executed by a processor, it corresponds to the specific steps of the method provided in this invention, possessing the corresponding functional modules and beneficial effects for executing the method. Technical details not described in detail in this embodiment can be found in the method provided in this invention.

[0199] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A satellite relative navigation adaptive filtering method based on online power spectral density identification, characterized in that, Includes the following steps: Step 1: Establish a dynamic model of the relative motion of the satellite in a near-circular orbit based on the Clohessy-Wiltshire equations, construct continuous-time and discrete-time state-space equations based on the model, and initialize the parameters. Step 2: Based on the constructed discrete-time state-space equation, establish an explicit analytical relationship between the process noise covariance and the power spectral density; Step 3: Run the standard Kalman filter using the initial process noise covariance, and synchronously record the innovative information sequence and Kalman gain generated during the filtering process; adaptively determine the convergence time of the filter using the whitening test criterion, and calculate the estimated value of the process noise covariance based on the maximum likelihood estimation method; Step 4: Construct a least-squares optimization problem, identify the optimal power spectral density parameter in real time, dynamically reconstruct the process noise covariance matrix using the identification results, and update it to the Kalman filter to complete the state estimation and covariance update at the current moment.

2. The satellite relative navigation adaptive filtering method based on online power spectral density identification according to claim 1, characterized in that, Step 1 includes: Step 101: For satellites in near-circular orbits, when the relative distance between satellites is much smaller than the orbital radius, the linearized Clohessy–Wiltshire equations are used to describe the relative motion dynamics, as expressed in: ， ， ， in, This represents the relative position component of the satellite with respect to the master satellite; Indicates the rate of change of the latitude angle of the main satellite; Indicates the process noise component; Indicates the control input component; Step 102: Transform the Clohessy-Wiltshire equations into continuous-time state-space equations, in the following form: ， ， in, These are state variables, including relative position and relative velocity; The process noise is represented by the process noise covariance matrix. and includes the power spectral density vector ; Indicates control input; These represent the system matrix and the input matrix, respectively. Indicates the initial time. Represents the state variables at the initial moment; Step 103: Ignoring process noise and control input, discretize the continuous-time state-space equation to obtain the discrete-time state-space equation, expressed as: ， in, express The state variable at any given time; Represents the discrete time period State transition matrix; The sampling time interval; Step 104, the observation equation corresponding to the discrete-time state-space equation is: ， in, Represents the observation vector; Represents the observation system matrix; To measure noise, its covariance is ; Step 105, Initialize and set state estimation and error covariance .

3. The satellite relative navigation adaptive filtering method based on online power spectral density identification according to claim 2, characterized in that, Step 2 includes: Step 201: Based on the properties of continuous-time Gaussian white noise, the process noise covariance matrix is... Represented as: ， in, Indicates the time variable within the integral symbol; Step 202: When the system matrix F is known, the state transition matrix has the following analytical form: ， in, ; Step 203: Substitute the analytical form of the state transition matrix into the process noise covariance matrix. The integral formula, by integrating over each element, yields the following process noise covariance matrix. With power spectral density vector The analytical relationship between them is: ， The expression for each element is as follows: ， ， ， ， ， ， ， ， ， ， ， ， ， in, , , .

4. The satellite relative navigation adaptive filtering method based on online power spectral density identification according to claim 3, characterized in that, Step 3 includes: Step 301: Based on the analytical relationship, select the initial coarse power spectral density parameters. Calculate the initial coarse process noise covariance and using the initial coarse process noise covariance Replacement process noise covariance The standard Kalman filter process is represented as follows: ， ， ， ， ， in, and They represent Prior and posterior state estimation at time t; and They represent The prior and posterior error covariance matrices at time t; Indicates the Kalman filter gain; Define the sequence of innovative information Its covariance is Record the sequence of innovative information at each moment during the filtering process. With Kalman filter gain ; Step 302: Apply the whitening test criterion to adaptively determine whether the filter has entered the convergence state, and record the convergence time as... If at the current moment If so, return to step 301; if the current time Proceed to step 303; Step 303: Within the framework of optimal filtering theory, the innovative information sequence conforms to a zero-mean Gaussian distribution, and the prior error covariance is assumed to be time-invariant. A maximum log-likelihood function is then constructed. , is represented as: ， in, covariance The determinant, observation vector The dimension; Represents probability. Indicates the given parameters Under the conditions Probability of taking a value; Represents the innovation vector at the i-th discrete time step; Step 304: To obtain the maximum log-likelihood function for the power spectral density The optimal solution is to let the maximum log-likelihood function be given by... The partial derivative is 0, that is... The following equation is derived: ， ， ， ， in, The trace operator for a matrix; This represents the information vector at the i-th discrete time. Represents the information at the i-th discrete time. The transpose of ; Step 305, because To satisfy the positive definite condition, the sum of the inner terms of the derivative result in step 304 must be a zero matrix, thus yielding: ， ， Step 306, based on the equation relationship obtained in step 305, use the following... Estimation process noise covariance All elements, exist The recursive estimation formula for time is: 。 5. The satellite relative navigation adaptive filtering method based on online power spectral density identification according to claim 4, characterized in that, Step 4 includes: Step 401, using the process noise covariance derived in Step 2. With power spectral density The analytical relationship between them is used to compare the covariance matrix estimated in step 3. By corresponding each element, a system of linear equations is constructed, which can be represented as: ， in, This represents a matrix vectorization operation, which arranges all elements of a matrix into a column vector. Step 402, to solve for the optimal power spectral density parameters The least squares optimization problem is established as follows: ， Step 403, select the optimal power spectral density parameters. Substituting the analytical relation derived in step 203, we can reconstruct the process noise covariance matrix at the current moment, i.e. , Step 404, reconstruct the process noise covariance. The standard Kalman filter framework in step 301 is applied to perform prediction and update steps to complete the state estimation. To perform state estimation for the next time step, return to step 306, which is the sequence of innovation information based on the new time step. With Kalman filter gain The gain is recalculated to determine the process noise covariance at the next time step. .

6. An electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the satellite relative navigation adaptive filtering method based on online power spectral density identification.

7. A computer-readable storage medium storing a computer program that, when executed by a processor, performs the aforementioned satellite relative navigation adaptive filtering method based on online power spectral density identification.

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