A frequency-domain spline efficient system identification method based on semi-quadratic criterion

Abstract

The application discloses a kind of efficient system identification methods of frequency domain spline based on semi-quadratic criterion, its steps mainly are: A, obtaining input signal and converting into frequency domain input B, by overlapping storage method obtains intermediate output s (k) ;C, according to invariance principle, obtain error e L (k)=h(k)‑s(k);D, based on semi-quadratic criterion, establish cost function and obtain frequency domain signal E, establish about weight variable step strategy μ w (k+1);F, by random gradient method, update weight;G, by spline mechanism, obtain system output y (k) ;H, obtain error e (k)=d (k)‑y (k) ;I, establish cost function and obtain time domain signal F (k) ;J, establish about spline control point variable step strategy μ q (k+1);K, by random gradient method, update local spline control point;L, iteration, repeat the steps of A to K, until nonlinear system identification ends.The method of the application can overcome the adverse effects of impulse interference, utilize the advantages of frequency domain and variable step strategy, have more rapid and efficient nonlinear system identification ability.

Description

Technical Field

[0001] This invention belongs to the field of digital signal processing technology, specifically relating to an efficient nonlinear system identification method based on semi-quadratic form criteria in the frequency domain splines. Background Technology

[0002] System identification is a fundamental problem in systems engineering. Accurate system modeling is beneficial for system analysis and performance improvement, leading to the development of a series of linear and nonlinear system identification methods for specific systems. Adaptive system identification methods based on digital signal processing techniques can effectively obtain system parameters, and the research on constructing adaptive filters to model the system has received widespread attention and practical applications.

[0003] Linear filters, through adaptive parameter learning, can acquire the impulse response of an unknown system, possessing accurate and effective linear system modeling capabilities. However, for real-world systems, inherent nonlinear factors cause the systems to be identified to generally exhibit nonlinear characteristics, rendering linear filters ineffective. Therefore, for nonlinear system identification scenarios, it is necessary to design accurate and efficient nonlinear adaptive filtering algorithms for nonlinear system identification.

[0004] For nonlinear system identification of the general Wiener-Hammerstein nonlinear model, the spline adaptive filter (SAF) is composed of a linear filter and a nonlinear spline interpolation cascade. Its filtering architecture is simple and easy to implement, making it an effective scheme for nonlinear system identification. To improve the convergence capability of the traditional spline adaptive filtering method, reference 1 "GuoW, Zhi Y. Nonlinear spline Versoria prioritizationoptimization adaptive filter for alpha-stable clutter[J].IEEE Transactions on Aerospace and Electronic Systems,2023,59(1):734-744." proposes a spline adaptive method based on the Versoria criterion. This method is robust to impulse interference. Before filtering, it uses invariance to perform priority finite impulse response identification of the unknown system and uses the intermediate error of the linear filter to adaptively learn the unknown parameters. However, the computational complexity of this method increases with the growth of the system parameter dimension. Reference 2, "Guo W, Zhi Y. Nonlinear spline prioritization optimization generalized hyperbolic secant adaptive filtering against alpha-stable noise[J]. Nonlinear Dynamics, 2023, 111: 14351-14363," proposes a frequency-domain spline adaptive method based on the hyperbolic secant function, which reduces computational complexity through fast Fourier transform operations. However, the nonlinear system identification performance of the spline adaptive filter still needs further improvement. Summary of the Invention

[0005] The purpose of this invention is to provide a high-efficiency nonlinear system identification method based on the semi-quadratic criterion in the frequency domain splines. This method combines the semi-quadratic criterion and the variable step size strategy and is applied to frequency domain spline adaptive filters. It has robustness against impulse interference, excellent convergence performance and high computational efficiency, and better nonlinear system identification results.

[0006] The technical solution adopted by the present invention to achieve the above objectives is as follows: prioritizing finite impulse response identification by utilizing the invariance principle of nonlinear system identification before filtering, reducing the sensitivity of the learning process to large outliers by using the semi-quadratic criterion, calculating linear convolution and linear correlation operations by using the fast Fourier transform method with overlapping storage, and improving convergence performance by using a variable step size strategy.

[0007] The steps of this method are as follows:

[0008] A. Obtain the tapped delay signal and convert it into a frequency domain input vector.

[0009] In the k-th data block, the system input signal is buffered to generate an input signal vector x(k) = [x(kM+1), x(kM+2), ..., x(kM+M)] of length M. T Furthermore, the data block length is the same as the tap delay length;

[0010] Based on the overlapping storage method, two adjacent input data blocks are transformed into 2M-dimensional frequency domain vectors through the fast Fourier transform. FFT stands for Fast Fourier Transform. Indicates an index in the frequency domain;

[0011] B. Linear section output

[0012] Based on the overlapping storage method, the frequency domain input signal vector is used. and 2M-dimensional frequency domain weight vector The linear M-dimensional output vector s(k) is obtained by performing the inverse fast Fourier transform (IFFT) operation:

[0013] Take the last M elements, where Represents the Hadamard product;

[0014] C. Acquisition of linear error signals

[0015] In the cascaded structure, the error vector e is obtained from the output of the linear filter. L (k) = h(k) - s(k), where h(k) is the output of the linear part of the unknown system, which cannot be directly observed. Based on the invariance principle of nonlinear system identification, the unknown system is identified using a finite impulse response filter to obtain an initial weight that approaches the ideal weight. Then, the filtered output of this weight is used as an estimate of h(k).

[0016] D. Establish a robust cost function and construct a frequency domain vector.

[0017] For the linear part, the cost function is established based on the semi-quadratic criterion:

[0018] Where the parameter λ > 0;

[0019] At time kM+j, the partial derivative of the cost function with respect to the error signal yields:

[0020]

[0021] Let f L (k)=[f(e L (kM+1)), f(e L (kM+2)), ..., f(e) L (kM+M))] T The zero-padded 2M-dimensional signal is then subjected to an FFT operation to obtain the frequency domain signal vector:

[0022]

[0023] E. Regarding the variable step size strategy for weights

[0024] For the linear part, the variable step size method is established as follows:

[0025] μ w (k+1)=ημ w (k+1)+γ||f L (k)|| 2 ,

[0026]

[0027] The parameters η and γ control the step size variation, and the step size parameter range is 0 < μ. min <μ max ;

[0028] F. Frequency domain weight vector update

[0029] Using the overlapping storage method, the linear correlation operation is calculated in the frequency domain to obtain an M-dimensional gradient vector:

[0030] Take the first M elements, where conj(·) represents the conjugate operation of complex numbers;

[0031] The frequency domain weight vector update rule obtained by the stochastic gradient method is as follows:

[0032]

[0033] G. Nonlinear output of spline interpolation mechanism

[0034] Based on the spline mechanism, obtain the normalized local interval abscissa vector u. j (k) and local spline control point vector

[0035]

[0036]

[0037] δ represents the uniform interval between two adjacent control points. The floor operator is defined, where N is the number of spline control points;

[0038] Construct a matrix U(k) = [u1(k), u2(k), ..., u M (k)] and The output signal vector of the nonlinear part is obtained:

[0039]

[0040] In the above formula, sum(·) represents the column vector containing the sum of each row, and C is the Catmull-Rom spline basis matrix;

[0041] H. Acquisition of the overall error signal

[0042] The overall error vector e(k) = d(k) - y(k) is obtained from the output of the nonlinear part, where d(k) is the desired signal of the unknown system;

[0043] I. Establish the robust cost function and construct the time-domain matrix

[0044] For the nonlinear part, the cost function is still established based on the semi-quadratic criterion:

[0045]

[0046] At time kM+j, the partial derivative of the cost function with respect to the error signal yields:

[0047]

[0048] Note that f(k)=[f(e(kM+1)), f(e(kM+2)),..., f(e(kM+M))] T Construct a time-domain matrix signal:

[0049] F(k)=[f(k), f(k), f(k), f(k)] T ;

[0050] J. Regarding the variable step size strategy for spline control points

[0051] Establish a variable step size method:

[0052] μ q (k+1)=ημ q(k+1)+γ||f(k)|| 2 ,

[0053]

[0054] K. Local spline control point update

[0055] The local spline control point update rule obtained by the stochastic gradient method is as follows:

[0056]

[0057] L. Iterate, repeating steps A to K until the nonlinear system identification is completed.

[0058] The beneficial effects of this invention are:

[0059] This invention applies a semi-quadratic criterion and a variable step-size strategy to a frequency-domain spline adaptive filter. Different error signals are used to optimize the linear and nonlinear components within the adaptive filter structure, improving system identification accuracy. This method exhibits robustness against impulse interference and significantly reduces computational complexity by performing linear convolution and linear correlation operations using a fast Fourier transform with overlapping storage. The variable step-size strategy further enhances convergence performance, resulting in faster and more efficient nonlinear system identification capabilities. Attached Figure Description

[0060] Figure 1 It is a block diagram of a frequency domain spline adaptive filter structure based on the semi-quadratic criterion and variable step size strategy, used to identify unknown Wiener-Hammerstein nonlinear systems;

[0061] Figure 2 The graph shows the mean square error curves of the method of the present invention, the method in reference 1, the method in reference 2, and the traditional SAF method in Experiment 1 when the parameters of the system to be identified are a set of specific weight vectors and spline control point vectors.

[0062] Figure 3 The graph shows the mean square error curves of the method of this invention, the method in reference 1, the method in reference 2, and the traditional SAF method in Experiment 2 when the system to be identified is a long impulse response system. Detailed Implementation

[0063] Example:

[0064] The efficient system identification method for frequency domain splines based on the semi-quadratic form criterion in this embodiment has the following specific steps:

[0065] A. Obtain the tapped delay signal and convert it into a frequency domain input vector.

[0066] In the k-th data block, the system input signal is buffered to generate an input signal vector x(k) = [x(kM+1), x(kM+2), ..., x(kM+M)] of length M. T Furthermore, the data block length is the same as the tap delay length;

[0067] Based on the overlapping storage method, two adjacent input data blocks are transformed into 2M-dimensional frequency domain vectors through the fast Fourier transform. FFT stands for Fast Fourier Transform. Indicates an index in the frequency domain;

[0068] B. Linear section output

[0069] Based on the overlapping storage method, the frequency domain input signal vector is used. and 2M-dimensional frequency domain weight vector The linear M-dimensional output vector s(k) is obtained by performing the inverse fast Fourier transform (IFFT) operation:

[0070] Take the last M elements, where Represents the Hadamard product;

[0071] C. Acquisition of linear error signals

[0072] In the cascaded structure, the error vector e is obtained from the output of the linear filter. L (k) = h(k) - s(k), where h(k) is the output of the linear part of the unknown system, which cannot be directly observed. Based on the invariance principle of nonlinear system identification, the unknown system is identified using a finite impulse response filter to obtain an initial weight that approaches the ideal weight. Then, the filtered output of this weight is used as an estimate of h(k).

[0073] D. Establish a robust cost function and construct a frequency domain vector.

[0074] For the linear part, the cost function is established based on the semi-quadratic criterion:

[0075] Where the parameter λ > 0;

[0076] At time kM+j, the partial derivative of the cost function with respect to the error signal yields:

[0077]

[0078] Let f L (k)=[f(e L (kM+1)), f(e L (kM+2)), ..., f(e)L (kM+M))] T The zero-padded 2M-dimensional signal is then subjected to an FFT operation to obtain the frequency domain signal vector:

[0079]

[0080] E. Regarding the variable step size strategy for weights

[0081] For the linear part, the variable step size method is established as follows:

[0082] μ w (k+1)=ημ w (k+1)+γ||f L (k)|| 2 ,

[0083]

[0084] The parameters η and γ control the step size variation, and the step size parameter range is 0 < μ. min <μ max ;

[0085] F. Frequency domain weight vector update

[0086] Using the overlapping storage method, the linear correlation operation is calculated in the frequency domain to obtain an M-dimensional gradient vector:

[0087] Take the first M elements, where conj(·) represents the conjugate operation of complex numbers;

[0088] The frequency domain weight vector update rule obtained by the stochastic gradient method is as follows:

[0089]

[0090] G. Nonlinear output of spline interpolation mechanism

[0091] Based on the spline mechanism, obtain the normalized local interval abscissa vector u. j (k) and local spline control point vector

[0092]

[0093]

[0094] in δ represents the uniform interval between two adjacent control points. This represents the floor operator, where N is the number of spline control points;

[0095] Construct a matrix U(k) = [u1(k), u2(k), ..., uM (k)] and The output signal vector of the nonlinear part is obtained:

[0096]

[0097] In the above formula, sum(.) represents a column vector containing the sum of each row, and C is the Catmull-Rom spline basis matrix;

[0098] H. Acquisition of the overall error signal

[0099] The overall error vector e(k) = d(k) - y(k) is obtained from the output of the nonlinear part, where d(k) is the desired signal of the unknown system;

[0100] I. Establish the robust cost function and construct the time-domain matrix

[0101] For the nonlinear part, the cost function is still established based on the semi-quadratic criterion:

[0102]

[0103] At time kM+j, the partial derivative of the cost function with respect to the error signal yields:

[0104]

[0105] Note that f(k)=[f(e(kM+1)), f(e(kM+2)),..., f(e(kM+M))] T Construct a time-domain matrix signal:

[0106] F(k)=[f(k), f(k), I(k), f(k)] T ;

[0107] J. Regarding the variable step size strategy for spline control points

[0108] Establish a variable step size method:

[0109] μ q (k+1)=ημ q (k+1)+γ||f(k)|| 2 ,

[0110]

[0111] K. Local spline control point update

[0112] The local spline control point update rule obtained by the stochastic gradient method is as follows:

[0113]

[0114] L. Iterate, repeating steps A to K until the nonlinear system identification is completed.

[0115] Numerical simulation experiment:

[0116] To verify the effectiveness of this invention, numerical simulation experiments were conducted and compared with the methods in existing literature 1 and literature 2, as well as the traditional SAF method.

[0117] In the numerical simulation, the nonlinear system to be identified in Experiment 1 consists of a cascaded nonlinear spline interpolation mechanism with linear components, where the weights of the linear components are [0.6, -0.4, 0.25, -0.15, 0.1]. T The 23 spline control points for the nonlinear component are set to [-2.2, -2, ..., -0.8, -0.91, -0.42, -0.01, -0.1, 0.1, -0.15, 0.58, 1.2, 1, 1.2, ..., 2, 2.2]. T The system input is a Gaussian signal sequence. The additional noise signal is Gaussian white noise with a signal-to-noise ratio of 30dB plus α-stable pulse noise interference. The characteristic parameter of the α-stable process is set to 1.2, and the dispersion parameter is set to 0.05.

[0118] In Experiment 1, the parameter values ​​of the method of the present invention are: μ max =0.02, μ min =0.002, η=0.98, γ=0.005, λ=1; The step size parameter of the method in Reference 1 is: μ w =μ q =0.01; The step size parameter of the method in Reference 2 is: μ w =μ q =0.06; The step size parameter of the traditional SAF method is: μ w =μ q =0.01.

[0119] In nonlinear system identification experiments, mean square error (MSE) is generally used as an evaluation index for the system identification performance. The faster the MSE curve converges and the lower the steady-state value, the better the performance of the adaptive filter.

[0120] Figure 2 This is a graph showing the mean square error (MSE) curves of the method of this invention, the method in Reference 1, the method in Reference 2, and the traditional SAF method in Experiment 1. From... Figure 2 It can be seen that the traditional SAF method, which uses mean square error as the cost function, is not robust to impulse interference, and the mean square error curve diverges. The method of this invention, as well as the methods in Reference 1 and Reference 2, can converge well. While maintaining the same steady-state error, the method of this invention has a faster convergence speed and better nonlinear system identification performance.

[0121] Experiment 2 verifies the computational efficiency of the method of this invention using a long impulse response system. The linear weight coefficients of the system to be identified are generated from the transfer function of the discrete system. The cascaded nonlinear part is a sinusoidal nonlinear function. The system input is a uniformly distributed signal sequence in the range [-1, 1]. The filter dimension is M = 120. The additional noise signal is still Gaussian white noise with a signal-to-noise ratio of 30dB plus α-stable impulse noise interference. The characteristic parameter of the α-stable process is set to 1.2, and the dispersion parameter is set to 0.05.

[0122] In Experiment 2, the parameter values ​​for the method of this invention are: μ max =0.0005, μ min =0.00005, η=0.98, γ=0.005, λ=1; The step size parameter of the method in Reference 1 is: μ w =μ q =0.0002; The step size parameter of the method in Reference 2 is: μ w =μ q =0.0015; The step size parameter of the traditional SAF method is: μ w =μ q =0.001.

[0123] Figure 3 This is a graph showing the mean square error (MSE) curves of the method of this invention, the method in Reference 1, the method in Reference 2, and the traditional SAF method in Experiment 2. From... Figure 3 It can be seen that the traditional SAF method fails under impulse noise interference, while the method of this invention exhibits good convergence performance compared to the methods in Reference 1 and Reference 2. With a computer configuration of a 2.9GHz Intel Core i5-9400F CPU and 24GB RAM, through 100 independent tests, the average execution time of the M iterations in the comparable methods is 0.1517ms for this invention, 0.7618ms for the method in Reference 1, and 0.1549ms for the method in Reference 2. This demonstrates that the method of this invention reduces computational complexity, converges faster, and possesses a faster and more efficient system identification capability.

Claims

1. A method for efficient system identification in the frequency domain based on a semi-quadratic form criterion, comprising the following steps: A. Obtain the tapped delay signal and convert it into a frequency domain input vector. In the k-th data block, the system input signal is buffered to generate an input signal vector x(k) = [x(kM+1), x(kM+2), ..., x(kM+M)] of length M. T Furthermore, the data block length is the same as the tap delay length; Based on the overlapping storage method, two adjacent input data blocks are transformed into 2M-dimensional frequency domain vectors through the fast Fourier transform. FFT stands for Fast Fourier Transform. Indicates an index in the frequency domain; B. Linear section output Based on the overlapping storage method, the frequency domain input signal vector is used. and 2M-dimensional frequency domain weight vector The linear M-dimensional output vector s(k) is obtained by performing the inverse fast Fourier transform (IFFT) operation: Take the last M elements, where Represents the Hadamard product; C. Acquisition of linear error signals In the cascaded structure, the error vector e is obtained from the output of the linear filter. L (k) = h(k) - s(k), where h(k) is the output of the linear part of the unknown system, which cannot be directly observed. Based on the invariance principle of nonlinear system identification, the unknown system is identified using a finite impulse response filter to obtain an initial weight that approaches the ideal weight. Then, the filtered output of this weight is used as an estimate of h(k). D. Establish a robust cost function and construct a frequency domain vector. For the linear part, the cost function is established based on the semi-quadratic criterion: Where the parameter λ > 0; At time kM+j, the partial derivative of the cost function with respect to the error signal yields: Let f L (k)=[f(e L (kM+1)),f(e L (kM+2)),...,f(e L (kM+M))] T The zero-padded 2M-dimensional signal is then subjected to an FFT operation to obtain the frequency domain signal vector: E. Regarding the variable step size strategy for weights For the linear part, a variable step size approach is established: m w (k+1)=day w (k+1)+γ||f L (k)|| 2 , The parameters η and γ control the step size variation, and the step size parameter range is 0 < μ. min <μ max ; F. Frequency domain weight vector update Using the overlapping storage method, the linear correlation operation is calculated in the frequency domain to obtain an M-dimensional gradient vector: Take the first M elements, where conj(·) represents the conjugate operation of complex numbers; The frequency domain weight vector update rule obtained by the stochastic gradient method is as follows: G. Nonlinear output of spline interpolation mechanism Based on the spline mechanism, obtain the normalized local interval abscissa vector u. j (k) and local spline control point vector in δ represents the uniform interval between two adjacent control points. This represents the floor operator, where N is the number of spline control points; Construct matrix U(k) = [u1(k), u2(k), ..., u M (k)] and The output signal vector of the nonlinear part is obtained: In the above formula, sum(·) represents the column vector containing the sum of each row, and C is the Catmull-Rom spline basis matrix; H. Acquisition of the overall error signal The overall error vector e(k) = d(k) - y(k) is obtained from the output of the nonlinear part, where d(k) is the desired signal of the unknown system; I. Establish the robust cost function and construct the time-domain matrix For the nonlinear part, the cost function is still established based on the semi-quadratic criterion: At time kM+j, the partial derivative of the cost function with respect to the error signal yields: Note that f(k)=[f(e(kM+1)),f(e(kM+2)),...,f(e(kM+M))] T Construct the time-domain matrix signal: F(k)=[f(k),f(k),f(k),f(k)] T ; J. Regarding the variable step size strategy for spline control points Establish a variable step size method: m q (k+1)=day q (k+1)+γ||f(k)|| 2 , K. Local spline control point update The local spline control point update rule obtained by the stochastic gradient method is as follows: L. Iterate, repeating steps A to K until the nonlinear system identification is completed.

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