A micro-thrust measurement signal filtering method based on total variation denoising

Abstract

An embodiment of the present invention provides a microthrust measurement signal filtering method based on total variation denoising. This method improves the total variation filtering performance by introducing an upper bound function with parameter a to the Majorize-Minimize (MM) solution for total variation filtering. The improved filtering method is then applied to steady-state thrust measurement signals to enhance the resolution of thrust signals measured by thrust frames. The method first selects appropriate filtering parameters for the thrust measurement signal using a GCV verification method. The parameter λ is used as the regularization term coefficient of the total variation filtering objective function, and the parameter a is used as the upper bound function parameter of the improved MM method. Finally, the improved MM method for total variation filtering is used to calculate the filtered result of the input signal. Compared to the low-pass filtering signal processing method commonly used in the thrust frame field, this method can address the problem of effective thrust excitation signals being obscured by large noise and difficult to restore, and can be used to improve the resolution of steady-state microthrust measurements.

Description

Technical Field

[0001] The present invention relates to the field of signal processing, and in particular to a micro-thrust measurement signal filtering method based on total variation denoising. Background Art

[0002] Gravitational wave detection is crucial for cosmic exploration, providing information about the most violent and extreme events in the universe. Accurate gravitational wave detection requires precise control of the position and attitude of small satellites. Therefore, microthrusters with sub-micronewton resolution are required to meet the requirements of gravitational wave detection. To accurately calibrate the tiny thrust generated by these microthrusters, the microthrust measurement framework must also achieve corresponding accuracy.

[0003] Currently, there are many frameworks for measuring thrust, and their measurement errors can be divided into two categories: systematic error and random error. Systematic error refers to errors such as zero offset and scale error. These errors can be suppressed through calibration and compensation. However, random error is caused by complex random interference, such as mechanical and electronic noise, and cannot be eliminated through calibration and compensation. Therefore, random error is the main factor limiting thrust measurement accuracy. To analyze and suppress random errors in thrust measurement, a variety of data processing methods are currently used in the field of thrust measurement. These methods can be mainly divided into the following four categories: bandwidth filtering algorithms, Kalman filtering algorithms, adaptive filtering algorithms, and sensor fusion algorithms. During steady-state measurement, the effective thrust signal generated by the thruster is approximately a square wave. However, the aforementioned linear filtering methods have difficulty reproducing the step edges of small changes, and even after filtering the steady-state portion of the step, they cannot achieve a stable state. In summary, the signal processing methods currently used in the field of micro-thrust measurement have certain shortcomings, making it difficult to restore the ideal approximate square wave thrust signal in the presence of large background noise.

[0004] In summary, the field of micro-thrust measurement urgently needs a set of filtering methods that can solve the above problems, thereby improving the thrust measurement accuracy of the thrust frame. Summary of the Invention

[0005] To address the challenge of restoring an ideal approximate square wave thrust signal from a thrust measurement signal with significant background noise, this paper discloses a micro-thrust measurement signal filtering method based on total variation denoising. This method, for the first time, proposes using total variation denoising to process thrust measurement signals and optimizes the filtering effect of total variation denoising from a solution perspective. This method preserves sharp step edges in the filtered signal, while also ensuring that the steady-state portion of the step is smooth after filtering. Even when the effective signal changes slightly, the tiny step signal masked by the significant background noise can be restored.

[0006] In order to achieve the above object, the technical solution of the present invention is as follows:

[0007] A micro-thrust measurement signal filtering method based on total variation denoising is proposed. First, the GCV verification method is used to select appropriate filtering parameters λ and a for the input signal. Then, the parameter λ is used as the regularization term coefficient of the total variation filtering objective function, and the parameter a is used as the parameter of the upper bound function when solving the filtering result using the improved MM method. Finally, the filtering result of the input signal is calculated using the improved MM method for total variation filtering.

[0008] Furthermore, the micro-thrust measurement signal filtering method based on total variation denoising is as follows:

[0009] 1) The actual measured thrust frame noise signal is superimposed on the standard step signal to simulate the signal of steady-state micro-thrust measurement.

[0010] 2) Using the GCV verification method, change the parameters λi and ai at a certain interval, calculate the filtering results corresponding to λi and ai one by one, and select the optimal filtering parameters λopt and aopt corresponding to the input signal based on the prior risk.

[0011] 3) The filter parameter λopt selected in step 2) is used as the regularization term coefficient of the total variation filter objective function, and the filter parameter aopt selected in step 2) is used as the parameter of the upper bound function of the MM method when the MM iterative method is used to solve the filtering result.

[0012] 4) Use the upper bound function designed in step 3) in the MM iterative method as an improved MM method; then use the improved MM method to iteratively find the x corresponding to the minimum value of the total variation filtering objective function, and then obtain the total variation filtering result improved in the solution method.

[0013] As an improvement of the present invention, in order to select the parameters of the total variation filtering method improved from the solution method, the GCV function to be minimized in step 2) is affected by two parameters λ and a, and the function is:

[0014]

[0015] where N, y, and I are the length of the input signal, the input signal, and the identity matrix, respectively. Aλ,a is the transformation matrix that maps the input signal y to the filtered signal xλ,a, i.e., xλ,a = Aλ,a·y. The optimal parameters λopt and aopt are those that minimize the GCV function G(λ,a).

[0016] As an improvement of the present invention, the transformation matrix Aλ,a, which maps the input signal y to the filtered signal xλ,a, is obtained by using the MM iterative algorithm with an improved upper bound function to obtain the total variation filtering result. The iterative formula for solving the filtering result is:

[0017]

[0018] It is guaranteed that the solution reaches convergence after 30 iterations, so is the iterative matrix obtained after 30 iterations (k=30) using parameters λi and ai, and its form is:

[0019]

[0020] in xk is the filtering solution after the kth iteration, and diag(·) is a diagonal matrix.

[0021] As an improvement to the present invention, to determine the optimal filtering parameters λopt and aopt that minimize the GCV function, a parameter sweep method was used to scan the GCV function values ​​corresponding to different parameters λi and ai at appropriate intervals. Based on the results of multiple experiments, an effective parameter range for filtering the microthrust measurement signal was determined. The range for scanning the parameter λi was set to [0.1, 50] with an interval of 0.1; the range for scanning the parameter ai was set to [0.1, 50] with an interval of 0.1. Predetermining an effective sweep range can save time in selecting the optimal parameters.

[0022] As an improvement to the present invention, step 3) first selects the optimal filter parameter λopt corresponding to aopt determined in step 2), and uses λopt as the regularization term coefficient of the total variation filter objective function. Then, in the process of filtering the thrust measurement signal, the objective function of the total variation filter is:

[0023]

[0024] As an improvement to the present invention, to improve the accuracy of total variation denoising, a new upper bound function for the MM solution method is designed, thereby optimizing the total variation filtering effect in the solution method. Step 3) The filter parameter aopt selected in step 2) is used as the parameter of the MM method upper bound function when the MM iterative method solves the filtering result. In this case, the upper bound function of the variation term is:

[0025]

[0026] in

[0027] Therefore, the upper bound function of the total variation filtering objective function is:

[0028]

[0029] As an improvement of the present invention, in step 4), after the filter parameters λopt and aopt are determined, the MM method with an improved upper bound function is used to iteratively solve the total variation filtering result. The MM method with formula (5) as the upper bound function is used to find the x corresponding to the minimum of the total variation denoising objective function (4), which is the filtered result. The iterative formula of the calculation process is:

[0030]

[0031] A stable convergent solution can be obtained after 30 iterations, so x30 is taken as the filtering result xoutput.

[0032] Compared with the existing effects, the beneficial effects of the present invention are as follows:

[0033] Compared to existing filtering techniques used for thrust measurement, this method for filtering steady-state thrust measurement signals preserves sharp step edges in the filtered signal and achieves smooth filtering of the steady-state portion of the step. Even when the effective signal changes very slightly, the effective small step signal, overshadowed by a large background noise, is restored. Compared to existing total variation denoising techniques, this filtering method improves the solution method, thereby enhancing the accuracy of total variation filtering. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 is a flow chart of the present invention;

[0035] Figure 2 This is a diagram showing the filtering effect of the present invention on the thrust measurement signal.

[0036] Figure 3 for Figure 2 Local methods for filtering effect maps. DETAILED DESCRIPTION

[0037] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are intended only to illustrate the present invention and are not intended to limit the scope of the present invention. It should be noted that the terms "front," "rear," "left," "right," "up," and "down" used in the following description refer to directions in the accompanying drawings, and the terms "inward" and "outward" refer to directions toward or away from the geometric center of a particular component, respectively.

[0038] Example 1

[0039] like Figure 1 As shown in FIG. 1 , a micro-thrust measurement signal filtering method based on total variation denoising in this embodiment is as follows:

[0040] Step 1: The measured thrust frame noise floor signal is superimposed on the standard step signal to simulate the steady-state micro-thrust measurement signal. The specific steps are as follows:

[0041] Step 1.1: Use a displacement sensor with a sampling frequency of 50 Hz to measure the background noise of the pendulum thrust frame in an atmosphere of 10-3 Pa. The measured background noise has a bilateral amplitude of approximately 2.5 μN and a natural frequency of 0.3 Hz.

[0042] Step 1.2: Detrend the background noise and compensate for the inherent drift of the measurement system.

[0043] Step 1.3: The measured response time is very short, and the ideal micro-thrust measurement signal is approximately square wave. Therefore, a standard staircase signal is constructed to simulate the ideal micro-thrust measurement signal. To highlight the advantages of this invention, the filtering effect is mainly tested under high signal-to-noise ratio conditions. Therefore, the ideal square wave signal is constructed with a range of 0.1-0.5μN.

[0044] Step 1.4: A 20,000-byte segment of the actual measured thrust noise floor signal is captured and superimposed on the constructed standard step signal to simulate the steady-state microthrust measurement signal. The subsequent steps 2) and 4) use two different segments of the thrust measurement signal: one for selecting filter parameters and the other for testing the filtering effect.

[0045] Step 2: Since the ideal effective signal is unknown during the actual measurement process, it is impossible to select the optimal filter parameters λ and a based on the variance of the filtered signal and the ideal signal. Therefore, the GCV verification method is used to select the optimal filter parameters λopt and aopt corresponding to the input signal based on the prior risk. The specific steps are as follows:

[0046] Step 2.1: Introduce the GCV verification formula, then the filtering result prior error corresponding to the filtering parameters λ and a is:

[0047]

[0048] Where N, y, and I are the length of the input signal, the input signal, and the identity matrix, respectively; Aλ,a is the transformation matrix that maps the input signal y to the filtered signal xλ,a, i.e., xλ,a = Aλ,a·y. From equation (10) in step 2.2.3), we can see that Aλ,a is the iterative matrix obtained by solving 30 iterations (k = 30) using the parameters λ and a, and its form is: When the GCV function G(λ,a) is minimized, the corresponding parameters λopt and aopt are the optimal parameters.

[0049] Step 2.2: Select the optimal filtering parameters λopt and aopt for the signal generated in step 1). To obtain the optimal parameters λopt and aopt that minimize the GCV function, use a parameter sweep method to scan the GCV function values ​​corresponding to different parameters λi and ai at appropriate intervals. Based on the results of multiple experiments, determine the effective parameter range for filtering the micro-thrust measurement signal. Set the range for scanning the parameter λi to [0.1, 50] with an interval of 0.1; set the range for scanning the parameter ai to [0.1, 50] with an interval of 0.1. Scan the filtering results corresponding to different parameters λi and ai within the above range and interval, and then find the optimal parameters λopt and aopt that minimize the GCV function as the filtering parameters for the thrust measurement signal.

[0050] The steps for calculating the filtering result corresponding to a pair of parameters λ and a in a single time are as follows, i.e., the improved total variation filtering method of the present invention:

[0051] Step 2.2.1: The total variation filtering process is an optimization process for non-convex functions, that is, solving the sequence x(n) corresponding to the minimization of the total variation filtering objective function. The following formula is the objective function:

[0052]

[0053] Where y(n) is the input sequence, x(n) is the filtered sequence, and λ>0 is the regularization coefficient. Written in norm form:

[0054]

[0055] in,

[0056] Step 2.2.2: Since L(x) is not differentiable, direct methods such as the Lagrange multiplier method cannot be used to find the x that minimizes the objective function. Using the improved MM iterative solution algorithm of the present invention, we can gradually approximate the x that minimizes the objective function. First, we design an upper bound function of |Δx(n)| = |x(n) - x(n-1)|:

[0057]

[0058] Where a>0 is a manually selected parameter, and xk(n) is the result of the k-th iteration. By summing both ends of the above equation and taking the norm, we can get the upper bound function of the variation term:

[0059]

[0060] in, diag(·) is a diagonal matrix. Combining equations (3) and (5), we can see that the upper bound function of the objective function equation (3) is:

[0061]

[0062] 2.2.3) The process of iteratively solving x using the MM algorithm is to iteratively solve the minimum value of the upper bound function of the objective function, thereby gradually approaching the x corresponding to the minimization of the true objective function. The iterative formula is as follows:

[0063] x k+1 =argmin x {G k (x)} (7)

[0064]

[0065] Among them, x k is the solution of k iterations, x k+1 is the result of substituting for the k+1th time. The upper bound function Gk(x) is a quadratic function of x, so the explicit solution of formula (8) is:

[0066]

[0067] When the above formula is iterated a certain number of times, |Dx k | will tend to 0, then will also tend to 0, so To solve this problem, the matrix inversion lemma is used to simplify the iterative formula to:

[0068]

[0069] The process of solving the filtered signal is the process of iterating formula (10) cyclically. After about 30 iterations, a stable convergence result can be obtained. Therefore, x30 is taken as the result xoutput after the total variation filtering.

[0070] Step 3: Use the optimal filter parameters λopt and aopt selected in step 2 in the subsequent filtering process of the thrust measurement signal.

[0071] Step 3.1: The filter parameter λopt is used as the regularization term coefficient of the total variation filter objective function. Then, in the process of filtering the thrust measurement signal, the objective function of the total variation filter is:

[0072]

[0073] Step 3.2: The filter parameter aopt is used as the parameter of the upper bound function of the MM method when the MM iterative method is used to solve the filtering result. The upper bound function of the variation term is:

[0074]

[0075] in, At this time, the objective function of the total variation filter is:

[0076]

[0077] Step 4: Use the upper bound function designed in step 3 for the MM iterative method as the improved MM method; then use the improved MM method to iteratively find the x corresponding to the minimum value of the total variation filtering objective function, and then obtain the total variation filtering result improved in the solution method, and use this method as the improved total variation filtering method.

[0078] Step 4.1: Test the filtering effect of this method using another thrust measurement signal synthesized in Step 1. The specific filtering steps are consistent with the solution methods in Steps 2.2.1-2.2.3.

[0079] Figure 2 The filtering effect of this method is shown in Figure 3 This is a partial amplification of the filtering effect. It can be seen that the improved total variation filtering method is better than the original total variation filtering method, and the filtered signal is closer to the ideal step signal.

[0080] The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above-mentioned embodiment, but also include technical solutions composed of any combination of the above technical features.

Claims

1. A micro-thrust measurement signal filtering method based on total variation denoising is characterized by: The specific processing steps are as follows: Step 1: Superimpose the actual measured thrust frame noise signal on the standard step signal to simulate the signal of steady-state micro-thrust measurement; Step 2: Use the GCV verification method to select the optimal filter parameter λ corresponding to the input signal based on the prior risk opt and a opt ; Step 3: Set the filter parameter λ selected in step 2 opt Used as the regularization term coefficient of the total variation filter objective function, the filter parameter a selected in step 2 is opt Used as the parameter of the upper bound function of the MM method when the MM iterative method is used to solve the filtering result; Step 4: Use the upper bound function designed in step 3) in the MM iterative method as an improved MM method; then use the improved MM method to iteratively find the x sequence corresponding to the minimum value of the total variation filtering objective function, thereby obtaining a total variation filtering result improved in the solution method.

2. The micro-thrust measurement signal filtering method based on total variation denoising according to claim 1, characterized in that: The GCV function to be minimized in step 2 is affected by two parameters, λ and a, and is: Where N, y, and I are the length of the input signal, the input signal, and the unit matrix respectively; A λ,a Mapping the input signal y to the filtered signal x λ,a The transformation matrix, that is, x λ,a =A λ,a· y; parameter λ corresponding to the minimum of the GCV function G(λ,a) opt and a opt is the optimal parameter.

3. The micro-thrust measurement signal filtering method based on total variation denoising according to claim 2, characterized in that: The input signal y is mapped to the filtered signal x λ,a The transformation matrix A λ,a , is obtained by the MM iterative algorithm with improved upper bound function to obtain the total variation filtering result. The iterative formula for solving the filtering result is: The parameter λ is used i and a i The iterative matrix obtained after iterative solution is in the form of: in x k To iterate and solve the filtering result after the kth time, diag( · ) is a diagonal matrix.

4. The micro-thrust measurement signal filtering method based on total variation denoising according to claim 2, characterized in that: Use parameter scanning method to obtain the parameter λ that minimizes the GCV function opt and a opt ; Considering the size of the thrust frame background noise, the scanning parameter λ i The range is [0.1,30], the interval is 0.1; the scanning parameter a i The range is [0.1,100], and the interval is 0.

1.

5. The micro-thrust measurement signal filtering method based on total variation denoising according to claim 1, characterized in that: Step 3: The filter parameter λ selected in step 2 opt After being used as the regularization term coefficient of the total variation filter objective function, the objective function of the total variation filter is: Where y represents the signal to be filtered, x represents the signal after filtering, and λ opt is the regularization parameter, Dx represents the variation term of the filtered signal; Step 3 replaces the filter parameter a selected in Step 2 with opt Used as the parameter of the upper bound function when solving the filtering result by the MM iterative method; at this time, the upper bound function of the variation term is: in Therefore, the upper bound function of the total variation filtering objective function is:

6. The micro-thrust measurement signal filtering method based on total variation denoising according to claim 1, characterized in that: In step 4, after the filtering parameters are determined, the improved MM method is used to iteratively solve the minimum value of the total variation filtering objective function. The iterative formula is: Dx represents the variation term of the filtered signal; When a stable convergent solution is obtained after iterating j times, take x j As the filtering result x output .

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