Identification Methods for Model Systems with Error Variables

Abstract

The present invention discloses a method for identifying a model system containing error variables, which mainly solves the problem that existing system identification methods have low identification accuracy or even cannot identify unknown model systems containing error variables when both input and output signals are contaminated by impulse noise. Its implementation scheme is: initializing an adaptive filter; constructing a cost function of the adaptive filter using the maximum overall fractional-order correlation entropy; iteratively updating the weight coefficients of the adaptive filter using the fractional-order gradient method of the maximum overall fractional-order correlation entropy, so that the error between the weight coefficients of the adaptive filter and the weight coefficients of the unknown system is continuously reduced, and finally the unknown system weight coefficients are obtained, completing the identification of the model system containing error variables. The present invention can achieve good identification of unknown model systems containing error variables in weak and strong impulse noise environments, and improves the identification accuracy. It can be used for digital beamforming, channel equalization, noise elimination, and electrocardiogram signal interference elimination.

Description

Technical Field

[0001] The present invention belongs to the field of digital signal processing, and in particular relates to an identification method for a model system containing error variables, which can be used for digital beam forming, channel equalization, noise elimination and electrocardiogram signal interference elimination. Background Art

[0002] Many problems in fields such as radar, communications, speech, and biomedicine, such as digital beamforming, channel equalization, noise cancellation, and electrocardiogram (ECG) signal interference removal, can be viewed as system identification problems: determining the true or optimal weights of an unknown system. Currently, channel estimation or adaptive filtering methods are commonly used to determine the weights of an unknown system. Compared to the former, adaptive filtering methods do not require statistical information about the input signal and can update the filter weights online in real time, approximating the true or optimal weights of the unknown system. These advantages have led to their widespread application in system identification.

[0003] Currently, system identification algorithms based on standard regression models have been widely used in linear systems. However, in real-world environments, due to sampling errors, human errors, or modeling errors, the observed input data may be subject to noise. Consequently, the assumptions of the standard regression model no longer hold, and the performance of existing system identification algorithms degrades dramatically. To address this issue, researchers have proposed the Error in Variables (EIV) model to describe unknown systems where all input and output variables are subject to noise. However, since real-world noise does not conform to a Gaussian distribution but rather contains impulsive noise with spikes and outliers, this type of impulsive noise can also cause significant errors in existing system identification algorithms.

[0004] Patent document No. 202210066048.7 discloses a universal adaptive system identification algorithm, which first constructs an impulse noise model, then calculates the instantaneous estimation error, and constructs a universal adaptive algorithm to determine the weight vector update equation of the adaptive filter; runs the system identification model to determine that the system identification model operates smoothly; determines whether the weight vector of the adaptive filter has reached a steady state; if the weight vector has not reached a steady state and the number of iterations has not been reached, the weight vector is repeatedly updated iteratively; if the algorithm has not reached a steady state within the number of iterations, the step size needs to be readjusted and the weight vector of the adaptive filter is re-updated; the weight coefficients of the adaptive filter in the steady state are output, that is, the weight coefficients of the unknown system obtained by identification. Because the input and output signals used to update the weight coefficients of the adaptive filter in this algorithm are pure signals, the situation where the input and output signals are contaminated by noise in the actual process is not considered, resulting in a decrease in the performance of the algorithm in actual applications, and even the inability to obtain the weight coefficients of the unknown system.

[0005] In his paper "Maximum total correntropy adaptive filtering against heavy-tailed noises" (Signal Processing, Vol. 141, 2017, pp. 84-95), Fei Wang proposed the Maximum Total Correlation Entropy (MTC) algorithm. This algorithm can handle EIV models when both input and output signals are subject to non-Gaussian noise, particularly heavy-tailed noise, and performs well in weak impulse noise environments. However, due to its poor robustness, the system identification accuracy is low in strong impulse noise environments. Summary of the Invention

[0006] The purpose of the present invention is to address the deficiencies of the above-mentioned prior art and to propose an identification method for a model system containing error variables, so as to improve the robustness and accuracy of system identification.

[0007] To achieve the above objectives, the technical solution of the present invention is to construct a maximum overall fractional-order correlation entropy adaptive filter and use its fractional-order gradient to update the adaptive filter weight coefficients, so that the adaptive filter weight coefficients are close to the unknown system weight coefficients, and finally complete the system identification of the unknown error-containing variable model. The implementation steps include the following:

[0008] (1) Initialization parameters:

[0009] Set the order p of the maximum overall fractional correlation entropy and the order r of the gradient. The value range of p is 0 to 2, and the value range of r is 0 to (p, 1) min , where (p,1) min represents the minimum value between p and 1;

[0010] Set the step size μ, noise variance ratio β, kernel width η, and maximum number of iterations N max , where μ ranges from 0 to 2, η is not less than 0, N max Not less than 1000 times;

[0011] Set the adaptive filter for system identification, with the weight coefficient w;

[0012] (2) Calculate the signal error e(n) between the filtered signal of the adaptive filter and the output signal of the unknown system:

[0013]

[0014] Where, represents the input signal of the unknown system at time n, Represents the input signal at time n The output signal after passing through the unknown system, w(n) represents the unknown system weight coefficient at time n, and T represents the vector transpose operation;

[0015] (3) Construct the maximum overall fractional order correlation entropy cost function J based on the signal error e(n) p (w):

[0016]

[0017] Where exp(·) represents the exponential operation with the natural constant e as the base, ||·|| represents the Euclidean norm,

[0018] (4) Update the adaptive filter weight coefficients using the fractional gradient of the maximum overall fractional correlation entropy:

[0019]

[0020] Among them, w(n) represents the weight coefficient at time n, w(n+1) represents the updated weight coefficient at time n+1, and e(n) <p-r>< / p-r> =|e(n)| p-r ⊙sign[e(n)], The symbol ⊙ represents the element-by-element multiplication between vectors, and sign[·] represents the sign function;

[0021] (5) Determine whether the number of updates of the adaptive filter weight coefficients reaches the maximum number of iterations N set in step (1) max :

[0022] If the maximum number of iterations N is reached max , then the weight coefficient of the adaptive filter is output, which is the weight coefficient of the model system containing the error variable, and its identification is completed;

[0023] Otherwise, return to step (2).

[0024] Compared with the prior art, the present invention has the following advantages:

[0025] First, the present invention takes into account the situation where the input and output signals are contaminated by impulse noise, and uses the fractional-order method to construct the maximum overall fractional-order correlation entropy cost function, which overcomes the problem in the prior art that the integer-order cost function cannot reach the optimal solution in the impulse noise environment, thereby improving the practicality of the present invention.

[0026] Second, the present invention uses a fractional-order gradient method to update the weight coefficients of the adaptive filter, overcoming the problem of the prior art that the system identification performance degrades or even cannot be identified in a strong impulse noise environment, and improves the applicability and accuracy of the present invention in identifying error variable model systems in an impulse noise environment. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is an implementation flow chart of the present invention;

[0028] Figure 2 It is a structural diagram of the existing EIV model system;

[0029] Figure 3 This is a simulation comparison diagram of system identification using the present invention and the existing maximum total correlation entropy (MTC) method in a weak impulse noise environment;

[0030] Figure 4 It is a simulation comparison diagram of system identification in a strong impulse noise environment using the present invention and the existing maximum total correlation entropy MTC method respectively. DETAILED DESCRIPTION

[0031] The embodiments and effects of the present invention are further described in detail below with reference to the accompanying drawings.

[0032] The embodiment of the present invention is to identify the EIV model system containing error variables.

[0033] The error variable model system refers to a system in which all input and output variables are interfered by noise, and both input and output signals contain impulse noise.

[0034] Reference Figure 1 , the implementation steps of this example are as follows:

[0035] Step 1, initialize parameters.

[0036] The present invention constructs a maximum overall fractional-order correlation entropy adaptive filter and uses its fractional-order gradient to update the adaptive filter weight coefficients, so that the adaptive filter weight coefficients approach the unknown system weight coefficients, and ultimately achieves system identification of the unknown error-containing variable model. Therefore, it is necessary to first initialize the adaptive filter parameters, which include:

[0037] Set the order p of the maximum overall fractional correlation entropy and the order r of the gradient. The value range of p is 0 to 2, and the value range of r is 0 to (p, 1) min , where (p,1) min represents the minimum value between p and 1;

[0038] Set the step size μ, kernel width η, and maximum number of iterations N max , where μ ranges from 0 to 2, η is not less than 0, N max Not less than 1000 times;

[0039] Set up an adaptive filter for system identification and set its weight coefficient to w;

[0040] Set the noise variance ratio β, which is defined as follows:

[0041]

[0042] in, Indicates input signal The variance includes noise, Indicates output signal Including the variance of noise. In actual use, the noise variance ratio β can be obtained in the following two ways:

[0043] The first method is to manually input the signal With output signal The outliers in are eliminated and we get and Then calculate β;

[0044] The second way is: when the probability of input and output signals is equal, first eliminate and The first 5% noise values ​​with larger amplitudes are obtained respectively. and Then calculate β.

[0045] Step 2: Calculate the signal error e(n) between the filtered signal of the adaptive filter and the unknown system output signal.

[0046] (2.1) Obtain the input signal of the adaptive filter and unknown system output signals

[0047] Reference Figure 2 , the pure input signal x(n) passes through the unknown system to obtain the pure output signal d(n). In practice, d(n) will be contaminated by the noise v(n), so the unknown system output signal is The pure input signal x(n) is contaminated by the noise u(n) and becomes the input signal of the adaptive filter. And get the filtered signal through the adaptive filter; the filtered signal of the adaptive filter is the same as the unknown system output signal The difference between them is the signal error e(n).

[0048] and The calculation methods are as follows:

[0049]

[0050]

[0051] Among them, the covariance matrix of impulse noise u(n) is represents the variance, I L represents the identity matrix of dimension L; the variance of the pulse v(n) is

[0052] (2.2) According to and Calculate the error signal e(n)

[0053]

[0054] Where w(n) represents the weight coefficient of the adaptive filter at time n, represents the filtered signal of the adaptive filter at time n, and T represents the vector transpose operation.

[0055] Step 3: Construct the maximum overall fractional order correlation entropy cost function J according to the signal error e(n) p (w).

[0056] (3.1) The cost function J(w) of the existing maximum overall correlation entropy method is selected:

[0057]

[0058] (3.2) According to the characteristic that the second-order J(w) is not the optimal solution when both the input and output signals contain impulse noise, the fractional order p is used to replace the integer order in J(w), and the cost function J of the maximum overall fractional correlation entropy method is obtained. p (w):

[0059]

[0060] Where exp(·) represents the exponential operation with the natural constant e as the base, ||·|| represents the Euclidean norm,

[0061] Step 4: Update the adaptive filter weight coefficients using the fractional gradient of the maximum overall fractional correlation entropy.

[0062] Using fractional order gradient to update the adaptive filter weight coefficient will make the filter signal of the adaptive filter and unknown system output signals The signal error e(n) between them is decreasing. At this time, the error between the weight coefficient w(n) of the adaptive filter and the weight coefficient of the unknown system is also decreasing. When the number of updates reaches the maximum number of iterations N max When , the weight coefficient w(n) of the adaptive filter is the unknown system weight coefficient obtained by identification, and the specific implementation is as follows:

[0063] (4.1) Calculate the maximum overall fractional order correlation entropy cost function J p r-order fractional gradient of (w)

[0064]

[0065] Among them, e(n) (p-r> =|e(n)| p-r ⊙sign[e(n)], The symbol ⊙ represents the element-by-element multiplication between vectors, sign[·] represents the sign function, and ζ is a constant;

[0066] (4.2) Using fractional gradient Update the weight coefficients of the adaptive filter:

[0067]

[0068] Where w(n) represents the weight coefficient at time n, w(n+1) represents the updated weight coefficient at time n+1, and λ is a constant;

[0069] (4.3) Substituting the adaptive weight coefficient update method, the adaptive filter update rule is as follows:

[0070]

[0071] Among them, μ = ζ × λ is the update step size.

[0072] Step 5: Obtain the identification result of the unknown system.

[0073] Determine whether the number of updates of the adaptive filter weight coefficients reaches the maximum number of iterations N set in step 1 max :

[0074] If the maximum number of iterations N is reached max , then the weight coefficient of the adaptive filter is output, which is the weight coefficient of the model system containing the error variable, and its identification is completed;

[0075] Otherwise, return to step 2.

[0076] The effects of the present invention will be further described below in conjunction with simulation experiments.

[0077] 1. Simulation conditions:

[0078] The operating system of the simulation experiment of the present invention is Windows 10 64-bit operating system, and the compilation environment is MATLAB2020b.

[0079] Assume the weight coefficient h of the unknown system is:

[0080] h=[-0.1976,-0.0773,0.2896,-0.4551,-0.5332,0.4024,0.3649,-0.2910] T

[0081] Assume that the length of the input signal is 5000 points, the number of Monte Carlo is 500, and the maximum number of iterations N is max 5000 times.

[0082] 2. Simulation experiment content

[0083] Simulation 1. Under the above simulation conditions, a weak impulse noise with a characteristic index of 1.7, a scale parameter of 0.1, a skew parameter of 0, and a position parameter of 0 is set. The noise variance ratio β of the present invention is initialized to 1, the kernel width η is initialized to 1, the order of the maximum overall fractional correlation entropy p is initialized to 1.7, the order of the gradient r is initialized to 0.8, and the step size μ is initialized to 0.01. The step size μ of the existing maximum overall correlation entropy method MTC is set to 0.012. The present invention and the existing maximum overall correlation entropy method MTC are used to identify the error variable model system in a weak impulse noise environment, and the normalized mean square deviation is used as an indicator to judge the quality of the simulation results. The results are shown in FIG. Figure 3 As shown in the figure, the horizontal axis represents the number of iterations, in times; the vertical axis represents the normalized mean square deviation between the filtered signal of the adaptive filter and the output signal of the unknown system after each iterative update, in dB.

[0084] Depend on Figure 3 It can be seen that when the convergence speed of the present invention and the prior art is consistent, the normalized mean square deviation of the present invention is smaller, indicating that the system identification accuracy of the present invention is significantly higher than that of the prior art.

[0085] Simulation 2, under the above simulation conditions, set the characteristic index to 1.2, the scale parameter to 0.1, the skew parameter to 0, and the position parameter to 0 strong impulse noise; initialize the noise variance ratio β of the present invention to 1, the kernel width η to 1, the order of the maximum overall fractional correlation entropy p to 1.1 and the order of the gradient r to 0.7, and the step size μ to 0.01; set the step size μ of the existing maximum overall correlation entropy method MTC to 0.013; use the present invention and the existing maximum overall correlation entropy method MTC to identify the error variable model system in a strong impulse noise environment, and use the normalized mean square deviation as an indicator to judge the quality of the simulation results. The results are as follows Figure 4 As shown in the figure, the horizontal axis represents the number of iterations, in times; the vertical axis represents the normalized mean square deviation between the filtered signal of the adaptive filter and the output signal of the unknown system after each iterative update, in dB.

[0086] Depend on Figure 4It can be seen that in a strong impulse noise environment, the present invention can converge stably, while the prior art cannot converge stably, indicating that the present invention has better robustness in a strong impulse noise environment and higher system identification accuracy than the prior art.

[0087] The above simulation results show that the present invention has higher convergence accuracy and system identification accuracy than the existing technology in a weak pulse noise environment; under strong pulse noise, the present invention can converge stably and identify the weight coefficients of the unknown error-containing variable model system, while the existing technology cannot converge stably and cannot identify the weight coefficients of the unknown error-containing variable model system, which shows that the present invention has obvious advantages in the application of identification of error-containing variable model systems.

Claims

1. A method for identifying a model system containing error variables, characterized in that: By constructing a maximum overall fractional-order correlation entropy adaptive filter and using its fractional-order gradient to update the adaptive filter weight coefficients, the adaptive filter weight coefficients are made close to the unknown system weight coefficients, and finally the system identification of the unknown error variable model is completed. The implementation steps include the following: (1) Initialization parameters: Set the order p of the maximum overall fractional correlation entropy and the order r of the gradient. The value range of p is 0 to 2, and the value range of r is 0 to (p, 1) min , where (p,1) min represents the minimum value between p and 1; Set the step size μ, noise variance ratio β, kernel width η, and maximum number of iterations N max , where μ ranges from 0 to 2, η is not less than 0, N max Not less than 1000 times; Set the adaptive filter for system identification, with the weight coefficient w; (2) Calculate the signal error e(n) between the filtered signal of the adaptive filter and the output signal of the unknown system: Where, Represents the input signal of the adaptive filter at time n, represents the output signal of the unknown system at time n, w(n) represents the weight coefficient of the adaptive filter at time n, and T represents the vector transpose operation; (3) Construct the maximum overall fractional order correlation entropy cost function J based on the signal error e(n) p (w): Where exp(·) represents the exponential operation with the natural constant e as the base, ||·|| represents the Euclidean norm, (4) Update the adaptive filter weight coefficients using the fractional gradient of the maximum overall fractional correlation entropy: Among them, w(n) represents the weight coefficient at time n, w(n+1) represents the updated weight coefficient at time n+1, and e(n) <p-r＞ =|e(n)| p-r ⊙sign[e(n)], The symbol ⊙ represents the element-by-element multiplication between vectors, and sign[·] represents the sign function; (5) Determine whether the number of updates of the adaptive filter weight coefficients reaches the maximum number of iterations N set in step (1) max : If the maximum number of iterations N is reached max , then the weight coefficient of the adaptive filter is output, which is the weight coefficient of the model system containing the error variable, and its identification is completed; Otherwise, return to step (2).

2. The method according to claim 1, characterized in that The input signal of the adaptive filter in step (2) at time n is and the output signal of the unknown system They are represented as follows: Among them, x(n) represents the pure input signal, u(n) represents the impulse noise contained when obtaining x(n), and the covariance matrix of u(n) is represents the variance of the noise, I L represents the unit matrix of dimension L; d(n) represents the pure output signal of the unknown system, v(n) represents the impulse noise contained when obtaining d(n), and the variance of v(n) is 3. The method according to claim 1, characterized in that The noise variance ratio β in step (1) is calculated as follows: in, Indicates input signal The variance includes noise, Indicates output signal Includes the variance of the noise.

4. The method according to claim 1, wherein In step (3), the maximum overall fractional order correlation entropy cost function J is constructed based on the signal error e(n) p (w), which is implemented as follows: (3a) The cost function J(w) of the existing maximum overall correlation entropy method is selected: (3b) According to the characteristic that the second-order J(w) is not the optimal solution when both the input and output signals contain impulse noise, the fractional order p is used to replace the integer order in J(w), and the cost function J of the maximum overall fractional order correlation entropy method is obtained. p (w): The fractional order p ranges from 0 to 2.

5. The method according to claim 1, wherein In step (4), the adaptive filter weight coefficients are updated using the fractional gradient of the maximum overall fractional correlation entropy, which is implemented as follows: (4a) Calculate the maximum overall fractional order correlation entropy cost function J p r-order fractional gradient of (w) Where ζ is a constant; (4b) Using fractional gradient Update the weight coefficients of the adaptive filter: Among them, w(n) represents the weight coefficient at time n, w(n+1) represents the updated weight coefficient at time n+1, and λ represents the update step size; (4c) Substituting the adaptive weight coefficient update method, the adaptive filter update rule is as follows: Where μ = ζ × λ.

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