A recursive minimum p-order adaptive filtering positioning method based on M estimation

Abstract

The present invention discloses a recursive minimum p-order adaptive filtering positioning method based on M-estimation, comprising the following steps: S1. constructing a nonlinear model of an adaptive filtering positioning system, combining it with an expected response to obtain an error, and converting the nonlinear model into a pseudo-linear model; S2. calculating a standardized residual index to obtain a weighting function; S3. combining M-estimation theory with the recursive minimum p-order norm to construct an error function and a system cost function, providing an algorithm iteration process based on the error function and the system cost function, and updating the filter system using the algorithm; S4. performing the iteration process over a continuous period of time until a set maximum number of iterations is reached, obtaining the filter coefficients in the pseudo-linear model at that time, and predicting the target position based on the obtained coefficients. The present invention combines the logistic function of the M-estimation with the recursive minimum p-order (RLP) algorithm, can effectively operate in non-Gaussian environments, and has superior filtering performance and improved robustness, enabling accurate target positioning.

Description

Technical Field

[0001] The present invention relates to adaptive filtering, in particular to a recursive minimum p-order adaptive filtering positioning method based on M estimation. Background Art

[0002] Adaptive filtering evolved from noise cancellers. The core of an adaptive filtering system is the adaptive filter and its adaptive response algorithm. Adaptive filters play an important role in digital signal processing and communications, such as system identification, target location, and noise cancellation. Adaptive filters are primarily categorized as linear and nonlinear. Linear filters require minimal computation and have a simple structure, but their applications are limited. Nonlinear filters are more widely used but are computationally complex.

[0003] Currently, linear adaptive filters are more commonly used. In practical applications, target positioning systems based on adaptive filters are often contaminated by non-Gaussian noise, resulting in large errors in positioning results, which is not conducive to accurate target positioning. Summary of the Invention

[0004] The purpose of the present invention is to overcome the shortcomings of the existing technology and provide a recursive minimum p-order adaptive filtering positioning method based on M estimation, which has good convergence performance in the face of non-Gaussian noise, can achieve more accurate target positioning, has strong robustness, and introduces low computational complexity.

[0005] The object of the present invention is achieved through the following technical solution: a recursive minimum p-order adaptive filtering positioning method based on M estimation, characterized in that it includes the following steps:

[0006] S1. Constructing a nonlinear model of adaptive filtering positioning system Initialize the noise variance and combine it with the expected response Get the error Convert nonlinear models into pseudo-linear models;

[0007] S2. Select the weighting function of the M estimate, give the control constant c, and calculate the standardized residual index to obtain the weighting function δ(k);

[0008] S3. Combining the M-estimation theory and the recursive least p-order (RLP) norm, an error function and a system cost function are constructed. Based on the error function and the system cost function, an iterative process of the algorithm (denoted as the MRLP algorithm) is given, and the filter system is updated using the MRLP algorithm.

[0009] S4. In a continuous time, iterative processing is performed according to steps S1 to S3 until the set maximum number of iterations is reached, and the filter coefficients in the pseudo-linear model at this time are obtained, and the target position is predicted based on them.

[0010] The step S1 comprises:

[0011] S101. Assume that the adaptive filtering positioning model contains a target Target, whose position is P T =[x T ,y T ]; The mobile sensor Sensor at different times is positioned by the target observation angle information collected by the sensor. The position of the sensor is P(k) = [x(k), y(k)], and the actual distance between the sensor and the target is r(k);

[0012] S102. Constructing a nonlinear model of adaptive filtering positioning system The system output m(k) is the observation angle between the sensor and the target; the system input is the position parameter P(k) = [x(k), y(k)] of the sensor; the estimated target position is Operation The meaning is Initialization noise variance α∈(0,1) is a constant, and is the preset noise parameter;

[0013] S103. Determine the expected response of the system is the desired angle, which is determined by the target position and the sensor position and is expressed as

[0014] S104. Calculate the error function as For the above nonlinear model, by adjusting the estimated target position The observation angle m(k) can be adjusted so that the observation angle m(k) approaches the desired angle When the error e(k) reaches the minimum, it means that the observation angle m(k) closest to the actual angle is obtained, that is, the optimal target estimated position is obtained.

[0015] S105. Convert the nonlinear model into a pseudo-linear model:

[0016] From the error function in S101, when e(k) approaches 0, the equivalent infinitesimal theory is used to obtain the following relationship: So there is the following pseudo-linear relationship:

[0017]

[0018] According to the above pseudo-linear relationship, define the intermediate variable H(k) = [cos(m(k)), -sin(m(k))] T , abstract a pseudo-linear adaptive filtering system filter coefficient w(k-1) to update the intermediate variable, and thus obtain a new equivalent expression about e(k) according to the pseudo-linear relationship transformation

[0019] This results in a pseudo-linear adaptive filtering system model, where the expected regarded as The system output m(k) can be viewed as w(k-1) T H(k), the filter coefficient is w(k-1), and e(k) is obtained by the pseudo-linear relationship determined by H(k) and w(k).

[0020] The Huber function-based M-estimation block adaptive filter uses the L2 parameter criterion when the signal error is small and the L1 parameter criterion when the signal error is large. For outliers with small residuals, the Huber function loses its optimization effect, but the logistic function can still correct the error and has good steady-state performance. The present invention selects the logistic function with low computational complexity and good performance as the optimization weight function. Therefore, a new MRLP cost function can be reconstructed based on the robust optimization weight function δ(k).

[0021] It can be seen that the standardized residual indicator corrects the error signal using the median function, which can effectively reduce the impact of large outliers. Then, the Logistic function is used for further robust optimization. Therefore, the cost function reconstructed by the weighted vector can handle non-Gaussian noise well and obtain superior performance.

[0022] The weighting function of the M estimate is selected as the Logistic function; the weighting function δ(k) is defined using the Logistic function:

[0023]

[0024] Among them, z(k) is the standardized residual index, the control constant c is given, and the control constant c is generally a positive number between 1 and 2, |·| represents the absolute value function, and tan(·) represents the tangent function. From step S1, we can know Then the weighting function δ(k) can be obtained by calculating z(k) using the following formula.

[0025] z(k)=e(k) / s=0.6745(e(k) / med|e-med(e)|)

[0026] where e(k) is the error vector used to calculate the standardized residual index z(k), med(·) represents the median function, and s is the residual scale.

[0027] The step S3 comprises:

[0028] S301. Define the forgetting factor λ, 0<λ≤1, and construct the cost function as:

[0029]

[0030] Among them, p is a number greater than 0, representing the p-order norm of e(n), and δ(n) is the weighted function estimated by M. In this cost function, w T (n-1)H(n) represents the noisy observation value, w(n-1) is the filter coefficient value of the previous moment, and the initial value w(0) is a random value. Represents the true value without noise, so the difference between the two is the noise suffered by the system. By constructing an estimated target position w, the cost function is minimized, that is, the estimated position closest to the true target position is obtained when the noise v(n) is minimized.

[0031] S302. Calculate the gradient of the cost function as:

[0032]

[0033] Let the gradient is 0, there is

[0034]

[0035] S303. Define the following substitution, where R(k) represents the autocorrelation matrix and r(k) represents the cross-correlation matrix:

[0036]

[0037]

[0038] Q(k)=R -1 (k).

[0039] The relationship between R(k) and R(k-1), R(k) and r(k) is derived as follows:

[0040] R(k)=λR(k-1)+p||e(k)|| p-2 δ(k)H(k)H T (k),

[0041] w(k)=R -1 (k)r(k)=Q(k)r(k),

[0042] By the matrix inversion lemma (A+BCD) -1 =A -1 -A -1 B(C -1 +DA -1 B) -1 DA -1 It is concluded that

[0043]

[0044] S304. Construct auxiliary variable K(k) in MRLP algorithm:

[0045] K(k)=Q(k-1)H(k)[λ / p||e(k)|| p-2 δ(k)+H T (k)Q(k-1)H(k)] -1 ,

[0046] Therefore, Q(k) is simplified to the following formula:

[0047] Q(k)=[Q(k-1)-K(k)H T (k)Q(k-1)] / λ,

[0048] The initial value of Q(k) is set to ρ -1 I, where ρ -1 is a positive number ranging from 0 to 1, and I is the unit matrix.

[0049] So the calculation formula of the filter coefficient w(k) at time k can be obtained as follows:

[0050] w(k)=w(k-1)+K(k)e(k).

[0051] In step S4, iterative processing is performed according to steps S1 to S3 in a continuous time until a set maximum number of iterations is reached, and the filter coefficient in the pseudo-linear model at this time is obtained, which is recorded as the optimal filter coefficient w;

[0052] Then, the sensor is used to perform adaptive filtering positioning when the target is in an unknown position:

[0053] Given the sensor position P′=[x′,y′] and its observation angle m′, the adaptive filtering positioning method using the optimal filter coefficient w is as follows:

[0054] (1) First, use the observation angle m' to calculate the intermediate variable H' = [cos(m'), -sin(m')] T ; Then use the optimal filter coefficient w to multiply the intermediate variable H' to obtain wH';

[0055] (2) Filter and denoise the observation angle m′ to obtain the denoised observation angle, which is recorded as the estimated angle θ′:

[0056] According to the pseudo linear relationship sin(θ'-m') = P' T The inverse solution of H′-wH′ obtains a denoised estimated angle θ';

[0057] (3) By estimating the sensor position P′=[x′,y′] in angle and the estimated target position P T ′=[x′ T ,y′ T ] the nonlinear relationship θ'=arctan(D(P',P T ')) Get the target position P after denoising T ′=[x′ T ,y′ T ].

[0058] The present invention has the beneficial effect of introducing the logistic function from M-estimation theory into a recursive minimum p-order algorithm. This algorithm has excellent filtering performance for target positioning in non-Gaussian noise environments, while also exhibiting good robustness and convergence performance. The algorithm can also be applied to target positioning problems, achieving good target positioning performance and being less prone to divergence, thus solving the non-Gaussian noise problem in actual positioning environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 Schematic diagram of the target and sensor locations;

[0060] Figure 2 is a flow chart of the method of the present invention;

[0061] Figure 3 This is the result diagram of the simulation experiment. DETAILED DESCRIPTION

[0062] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings, but the protection scope of the present invention is not limited to the following.

[0063] In practical applications of adaptive filtering algorithms, although the RLS algorithm can achieve rapid convergence performance, the filtering performance of the system output has not yet reached the optimal level. The present invention uses the RLP algorithm to overcome this shortcoming, but at the cost of worsening the filtering performance of the adaptive filtering algorithm in a non-Gaussian noise environment with large outliers. Therefore, this application needs to further consider the problem of non-noise environments. There are roughly two methods for adaptive filtering algorithms to solve non-Gaussian noise: one is a robust adaptive algorithm based on information entropy theory, and the other is a robust adaptive algorithm based on M-estimation theory.

[0064] Information entropy theory connects information theory, nonparametric estimation, and Hilbert space reconstruction in a simple and unconventional way. Researchers have proposed that the correlation entropy loss function has significant advantages over the mean squared error loss function when dealing with non-Gaussian noise. This is because the correlation entropy expansion is a weighted sum of even-order moments of the error. Since the correlation entropy includes high-order moments of the error, while the mean squared error loss function only includes second-order moments of the error, adaptive algorithms using the information entropy criterion have superior performance in dealing with non-Gaussian noise. However, this introduces significant computational complexity, requires manual adjustment of the kernel width, and cannot yet achieve excellent filtering performance.

[0065] M-estimation-based filtering algorithms exhibit robust performance when dealing with non-Gaussian noise because they can effectively eliminate interference from outliers. Similar to median filtering, robust filtering algorithms based on M-estimation achieve good impulse resistance with low computational complexity. The idea behind M-estimation filtering is that to reduce the impact of outliers, different weights can be applied to different points, giving larger weights to points with smaller residuals and smaller weights to points with larger residuals. The weights are determined based on the size of the residuals, and a weighted least squares estimate is established based on this weight. The weight coefficients are refined through repeated iterations until they meet the required error accuracy.

[0066] Compared with the information entropy criterion, the robust filtering based on M estimation can be naturally combined with the adaptive algorithm, does not require the kernel width to be adjusted, and is computationally simpler. The purpose of the present invention is to introduce the robustness of M estimation into the recursive minimum p-order algorithm, which has excellent convergence performance in the face of non-Gaussian noise, is not easy to diverge, has strong robustness, and only introduces low computational complexity. Specifically:

[0067] like Figure 1 As shown, in the target positioning simulation, the present invention considers using sensors to locate static target points in a time-varying non-stationary environment for target positioning in a non-Gaussian noise environment;

[0068] like Figure 2 As shown, the recursive minimum p-order adaptive filtering algorithm based on M estimation includes the following steps:

[0069] S1. Constructing a nonlinear model of adaptive filtering positioning system Initialize the noise variance to give the expected response of the system And get the error Convert the nonlinear model into a pseudo-linear model to obtain the filter coefficient w(k), and use the pseudo-linear model of the adaptive filtering positioning system to perform calculations in subsequent steps;

[0070] The adaptive filtering positioning model includes a target Target, whose position is PT =[x T ,y T ]; The mobile sensor Sensor at different times is positioned by the target observation angle information collected by the sensor. The position of the sensor is P(k) = [x(k), y(k)], and the actual distance between the sensor and the target is r(k);

[0071] So we can build a nonlinear model of adaptive filtering positioning system The system output m(k) is the observation angle between the sensor and the target; the system input is the position parameter P(k) = [x(k), y(k)] of the sensor; the estimated target position is Operation The meaning is Initialization noise variance α∈(0,1) is a constant, and is the noise variance. The noise encountered by the initialization system is non-Gaussian noise. The initialization noise variance is expressed as v(k)~0.9N(0,0.1)+0.1N(0,10), which represents the non-Gaussian noise formed by 90% small variance Gaussian noise superimposed on 10% large variance Gaussian noise.

[0072] Next, we give the expected response of the system is the desired angle, which is determined by the target position and the sensor position and is expressed as So we can get the error function as For the above nonlinear model, by adjusting the estimated target position The observation angle m(k) can be adjusted so that the observation angle m(k) approaches the expected angle When the error e(k) reaches the minimum, it means that the most accurate observation angle m(k) is obtained, that is, the optimal target estimated position is obtained.

[0073] Next, the nonlinear model is converted into a pseudo-linear model, and the pseudo-linear model of the adaptive filtering positioning system is used to perform calculations in subsequent steps. From the error function in S101, it can be seen that when e(k) is small enough, the equivalent infinitesimal theory can be used to obtain the following relationship: So according to Figure 1 , there is the following pseudo-linear relationship:

[0074]

[0075] According to the above pseudo-linear relationship, define the intermediate variable H(k) = [cos(m(k)), -sin(m(k))] T, we can abstract a pseudo-linear adaptive filtering system filter coefficient w(k-1) to update the intermediate variable, and thus we can get a new equivalent expression about e(k) according to the pseudo-linear relationship transformation Therefore, we obtain a pseudo-linear adaptive filtering system model, where the expected regarded as The system output m(k) can be viewed as w(k-1) T H(k), the filter coefficient is w(k-1), and the pseudo-linear relationship e(k) determined by H(k) and w(k) is obtained. The obtained pseudo-linear model can then be used for calculations in subsequent steps.

[0076] S2. Select the logistic function as the weighting function of the M estimate; use the logistic function to define the weighting function δ(k):

[0077]

[0078] Where z(k) is the normalized residual index, and the control constant c is generally a positive number between 1 and 2. In the present invention, the control constant c=1.205 is selected. |·| represents the absolute value function, and tan(·) represents the tangent function. From step S1, we can see Then the weighting function δ(k) can be obtained by calculating z(k) using the following formula.

[0079] z(k)=e(k) / s=0.6745(e(k) / med|e-med(e)|)

[0080] where e(k) is the error vector used to calculate the standardized residual index z(k), med(·) represents the median function, and s is the residual scale.

[0081] S3. Combining the M-estimation theory and the recursive least p-order (RLP) norm, the iterative formula of the MRLP algorithm is given according to the error function and the system cost function as shown below. The MRLP algorithm is used to update the filter coefficients.

[0082]

[0083] The initialization parameters are as follows: λ = 0.972, p = 1.8, w(0) = [100*rand(1), 100*rand(1)], Q(0) = I. The true position of the target is Target position =[100*rand(1),100*rand(1)]. Here we ensure that the target position and the sensor position are different, and set the initial real position of the sensor to Sensor position=rand(2,3000)*100, the initial estimated position is w(0)=[100*rand(1),100*rand(1)].

[0084] S4. In a continuous time, iterative processing is performed according to steps S1 to S3 until the set maximum number of iterations is reached, and the filter coefficients in the pseudo-linear model at this time are obtained, and the target position is predicted based on them.

[0085] Specifically, the sensor is used to perform adaptive filtering positioning on the target when the target is at an unknown position:

[0086] Given the sensor position P′=[x′,y′] and its observation angle m′, the adaptive filtering positioning method using the optimal filter coefficient w is as follows:

[0087] (1) First, use the observation angle m' to calculate the intermediate variable H' = [cos(m'), -sin(m')] T ; Then use the optimal filter coefficient w to multiply the intermediate variable H' to obtain wH';

[0088] (2) Filter and denoise the observation angle m′ to obtain the denoised observation angle, which is recorded as the estimated angle θ′:

[0089] According to the pseudo linear relationship sin(θ'-m') = P' T The inverse solution of H′-wH′ obtains a denoised estimated angle θ';

[0090] (3) By estimating the sensor position P′=[x′,y′] in angle and the estimated target position P T ′=[x′ T ,y′ T ] the nonlinear relationship θ'=arctan(D(P',P T ')) Get the target position P after denoising T ′=[x′ T ,y′ T ].

[0091] The total number of sensor points set in this example is 3000. In the simulation of the example, the mean square error is used. to measure the performance of the algorithm.

[0092] The experimental results are the average results obtained after 500 simulations, such as Figure 3 As shown, the curve represented by the present invention is MRLP, and the results show that the filtering effect of the present algorithm in adaptive filtering target positioning is better than that of the existing algorithms.

[0093] The foregoing description shows and describes a preferred embodiment of the present invention. However, as previously stated, it should be understood that the present invention is not limited to the form disclosed herein and should not be construed as excluding other embodiments. Instead, the present invention is applicable to various other combinations, modifications, and environments and is capable of modification within the scope of the inventive concept described herein, through the teachings above, or through techniques or knowledge in the relevant art. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention are intended to be within the scope of the appended claims.

Claims

1. A recursive minimum p-order adaptive filtering positioning method based on M estimation, characterized by: The following steps are involved: S1. Constructing a nonlinear model of adaptive filtering positioning system Initialize the noise variance and combine it with the expected response Get the error Convert nonlinear models into pseudo-linear models; S2. Select the weighting function of the M estimate, give the control constant c, and calculate the standardized residual index to obtain the weighting function δ(k); The step S2 includes: The weighting function of the M estimate is selected as the Logistic function; the weighting function δ(k) is defined using the Logistic function: Among them, z(k) is the standardized residual index, the control constant c is given, and the control constant c is generally a positive number between 1 and 2, || represents the absolute value function, and tan(·) represents the tangent function. It can be seen from step S1 that Therefore, the weighting function δ(k) can be obtained by calculating z(k) as follows; z(k)=e(k) / s=0.6745(e(k) / med|e-med(e)|) Where e(k) is the error vector used to calculate the standardized residual index z(k), med(·) represents the median function, and s is the residual scale; S3. Combining the M-estimation theory and the recursive minimum p-order norm, the error function and the system cost function are constructed. Based on the error function and the system cost function, the iterative process of the algorithm is given, and the algorithm is used to update the filter system. The step S3 comprises: S301. Define the forgetting factor λ, 0<λ≤1, and construct the cost function as: Among them, p is a number greater than 0, representing the p-order norm of e(n), δ(n) is the weighted function estimated by M; in this cost function, w T (n-1)H(n) represents the noisy observation value, w(n-1) is the filter coefficient value of the previous moment, and the initial value w(0) is a random value. Represents the true value without noise, so the difference between the two is the noise suffered by the system. By constructing an estimated target position w, the cost function is minimized, that is, the estimated position closest to the true target position is obtained when the noise v(n) is minimized. S302. Calculate the gradient of the cost function as: Let the gradient is 0, there is S303. Define the following substitution, where R(k) represents the autocorrelation matrix and r(k) represents the cross-correlation matrix: Q(k)=R -1 (k). The relationship between R(k) and R(k-1), R(k) and r(k) is derived as follows: R(k)=λR(k-1)+p||e(k)|| p-2 δ(k)H(k)H T (k), w(k)=R -1 (k)r(k)=Q(k)r(k), By the matrix inversion lemma (A+BCD) -1 =A -1 -A -1 B(C -1 +DA -1 B) -1 DA -1 It is concluded that S304. Construct auxiliary variable K(k) in MRLP algorithm: K(k)=Q(k-1)H(k)[λ / p‖e(k)|| p-2 δ(k)+H T (k)Q(k-1)H(k)] -1 , Therefore, Q(k) is simplified to the following formula: Q(k)=[Q(k-1)-K(k)H T (k)Q(k-1)] / λ, The initial value of Q(k) is set to ρ -1 I, where ρ -1 is a positive number ranging from 0 to 1, and I is the unit matrix; the calculation formula for the filter coefficient w(k) at time k is: w(k)=w(k-1)+K(k)e(k); S4. In a continuous time, iterative processing is performed according to steps S1 to S3 until the set maximum number of iterations is reached, and the filter coefficients in the pseudo-linear model at this time are obtained, and the target position is predicted based on them.

2. The recursive minimum p-order adaptive filtering positioning method based on M estimation according to claim 1, characterized in that: The step S1 comprises: S101. Assume that the adaptive filtering positioning model contains a target Target, whose position is P T =[x T ,y T ]; The mobile sensor Sensor at different times is positioned by the target observation angle information collected by the sensor. The position of the sensor is P(k) = [x(k), y(k)], and the actual distance between the sensor and the target is r(k); S102. Constructing a nonlinear model of adaptive filtering positioning system The system output m(k) is the observation angle between the sensor and the target; the system input is the position parameter P(k) = [x(k), y(k)] of the sensor; the estimated target position is Operation The meaning is Initialization noise variance α∈(0,1) is a constant, and is the preset noise parameter; S103. Determine the expected response of the system is the desired angle, which is determined by the target position and the sensor position and is expressed as S104. Calculate the error function as For the above nonlinear model, by adjusting the estimated target position The observation angle m(k) can be adjusted so that the observation angle m(k) approaches the desired angle When the error e(k) reaches the minimum, it means that the observation angle m(k) closest to the actual angle is obtained, that is, the optimal target estimated position is obtained. S105. Convert the nonlinear model into a pseudo-linear model: From the error function in S101, when e(k) approaches 0, the equivalent infinitesimal theory is used to obtain the following relationship: So there is the following pseudo-linear relationship: According to the above pseudo-linear relationship, define the intermediate variable H(k) = [cos(m(k)), -sin(m(k))] T , abstract a pseudo-linear adaptive filtering system filter coefficient w(k-1) to update the intermediate variable, and thus obtain a new equivalent expression about e(k) according to the pseudo-linear relationship transformation This results in a pseudo-linear adaptive filtering system model, where the expected regarded as The system output m(k) can be viewed as w(k-1) T H(k), the filter coefficient is w(k-1), and e(k) is obtained by the pseudo-linear relationship determined by H(k) and w(k).

3. The recursive minimum p-order adaptive filtering positioning method based on M estimation according to claim 1, characterized in that: In step S4, iterative processing is performed according to steps S1 to S3 in a continuous time until a set maximum number of iterations is reached, and the filter coefficient in the pseudo-linear model at this time is obtained, which is recorded as the optimal filter coefficient w; Then, the sensor is used to perform adaptive filtering positioning when the target is in an unknown position: Given the sensor position P′=[x′,y′] and its observation angle m′, the adaptive filtering positioning method using the optimal filter coefficient w is as follows: (1) First, use the observation angle m' to calculate the intermediate variable H' = [cos(m'), -sin(m')] T ; Then use the optimal filter coefficient w to multiply the intermediate variable H' to obtain wH'; (2) Filter and denoise the observation angle m′ to obtain the denoised observation angle, which is recorded as the estimated angle θ′: According to the pseudo linear relationship sin(θ'-m') = P' T The inverse solution of H′-wH′ obtains a denoised estimated angle θ'; (3) By estimating the sensor position P′=[x′,y′] in angle and the estimated target position P′ T =[x′ T ,y′ T ]The nonlinear relationship θ'=arctan(D(P',P′ T )) Get the target position P′ after denoising T =[x′ T ,y′ T ].

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