Non-convex projection NLMS method and system for adaptive Hammerstein system prediction

Abstract

The present invention discloses a non-convex projection NLMS method and system for adaptive Hammerstein system prediction. The present invention utilizes system input and output measurement values ​​to design and train an adaptive Hammerstein filter so that the filter can better predict the output of the nonlinear system. Considering that parameter estimates obtained by most Hammerstein modeling methods are usually local optimal solutions, a non-convex projection NLMS method for adaptive Hammerstein system prediction is proposed. The method obtains system parameter estimates through two cyclic steps of adaptive filtering and Euclidean projection, and the obtained parameter estimates have higher accuracy and stronger stability.

Description

Technical Field

[0001] The invention relates to the field of nonlinear system prediction, in particular to a non-convex projection NLMS (normalized least mean square) method for adaptive Hammerstein system prediction. Background Art

[0002] Many scientific problems and engineering fields involve systems or processes that are essentially nonlinear, and often require prediction and identification of a nonlinear system with unknown dynamic behavior. Many actual processes and systems can be described by a mathematical model composed of a dynamic linear module and a static nonlinear module. Therefore, model-based system prediction methods are a relatively common method, which includes system identification and prediction methods based on the Hammerstein model. The basic idea of ​​most Hammerstein modeling methods is to establish a corresponding cost function according to certain criteria (such as maximum likelihood, minimum mean square, etc.), use the obtained measurement data to solve the optimization problem of the cost function, thereby calculating the parameters of the Hammerstein model, and using the model to fit the actual system. However, the structural characteristics of the Hammerstein model make the cost function obtained by using the maximum likelihood criterion or the minimum mean square error criterion often non-convex. Therefore, the parameter estimates obtained in most Hammerstein modeling methods are usually local optimal solutions to non-convex optimization problems. Summary of the invention

[0003] In view of the problems existing in the prior art, the present invention provides a non-convex projection NLMS method and system for adaptive Hammerstein system prediction.

[0004] The present invention comprises the following steps:

[0005] Step 1: Collect data and build an adaptive Hammerstein filter model:

[0006] Step 2: Compute the solution for the adaptive Hammerstein filter coefficients:

[0007] Step 3: Set the initial values ​​of the adaptive Hammerstein filter parameters is the zero vector.

[0008] Step 4: Calculate the solution of the objective function;

[0009] Step 5: Update parameters;

[0010] Step 6: Determine whether it converges: If the solution of the objective function converges, stop the iteration and execute step 7, otherwise re-execute step 4 until the iteration stop condition is met.

[0011] Step 7: Get the adaptive Hammerstein filter coefficients.

[0012] Furthermore, the specific method of step 1 is as follows:

[0013] For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)). The adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is:

[0014]

[0015] in and The dimension is L a Column vector of .

[0016] The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is:

[0017]

[0018] in:

[0019]

[0020]

[0021] h(i)=vec([x(i),X(i)]),

[0022]

[0023] vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector. The coefficients that make up the adaptive Hammerstein filter.

[0024] The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is:

[0025] d(i)=F(x(i))+v(i),

[0026] Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution.

[0027] Define the prediction error at time i:

[0028]

[0029] Furthermore, the specific method of step 2 is as follows:

[0030] 2-1. Using step 1, define the objective function E|h T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula:

[0031]

[0032] in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly. Solve this optimization problem through the following steps;

[0033] 2-2. First, the NLMS algorithm is used to solve the solution of the objective function and obtain the model parameter estimation, that is:

[0034]

[0035] Where μ is the step size and ε is set to a very small number (such as 0.0001) to prevent ||h(i)|| 2 When it is very small, the denominator in the formula is too small.

[0036] 2-3. Then find the solution according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows:

[0037]

[0038]

[0039] in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form;

[0040] 2-4. To calculate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula:

[0041]

[0042]

[0043] The global minimum of the Euclidean distance is calculated as in Q(i)=Θ(i)Θ T (i)

[0044] 2-5. Further prove that the solution of the objective function is not an empty set, and the projection value Still in D(i). In this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations. In order to avoid this instability, we introduce Replace Q(i), where λ max is the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as:

[0045]

[0046]

[0047]

[0048] Furthermore, the specific method of step 4 is as follows:

[0049] Using the formula obtained in step 2 Calculate the objective function solution

[0050] Furthermore, the specific method of step 5 is as follows:

[0051] Use the method described in step 2 to perform Euclidean projection and update the adaptive Hammerstein filter parameters

[0052] A non-convex projection NLMS system for adaptive Hammerstein system prediction includes an adaptive Hammerstein filter building module, an initial module, an objective function solving module, a parameter updating module and a convergence judgment module.

[0053] The adaptive Hammerstein filter building module is used to build an adaptive Hammerstein filter, and the specific method is as follows:

[0054] For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)). The adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is:

[0055]

[0056] in and The dimension is L a Column vector of .

[0057] The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is:

[0058]

[0059] in:

[0060]

[0061]

[0062] h(i)=vec([x(i),X(i)]),

[0063]

[0064] vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector. The coefficients that make up the adaptive Hammerstein filter.

[0065] The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is:

[0066] d(i)=F(x(i))+v(i),

[0067] Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution.

[0068] Define the prediction error at time i:

[0069]

[0070] The initialization module is used to initialize the parameters of the adaptive Hammerstein filter.

[0071] The objective function solving module defines the objective function E|h according to the adaptive Hammerstein filter constructed by the adaptive Hammerstein filter construction module. T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula:

[0072]

[0073] in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly. Solve this optimization problem through the following steps;

[0074] The NLMS algorithm is used to solve the objective function and obtain the model parameter estimation, namely:

[0075]

[0076] Where μ is the step size and ε is set to a very small number (such as 0.0001) to prevent ||h(i)|| 2 When it is very small, the denominator in the formula is too small.

[0077] The parameter updating module is used to update the parameters of the adaptive Hammerstein filter. The specific method is as follows:

[0078] Find the solution obtained by the objective function solving module according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows:

[0079]

[0080]

[0081] in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form;

[0082] To evaluate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula:

[0083]

[0084]

[0085] The global minimum of the Euclidean distance is calculated as in Q(i)=Θ(i)Θ T (i)

[0086] It is further proved that the solution of the objective function is not an empty set, and the projection value Still in D(i). In this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations. In order to avoid this instability, we introduce Replace Q(i), where λ maxis the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as:

[0087]

[0088]

[0089]

[0090] The convergence judgment module is used to judge whether the solution of the objective function of the adaptive Hammerstein filter model converges: if the solution of the objective function converges, the iteration is stopped and the adaptive Hammerstein filter coefficients are obtained; otherwise, the objective function is solved and the parameters are updated again through the objective function solving module and the parameter updating module until the iteration stop condition is met.

[0091] The present invention has the following beneficial effects:

[0092] 1. The present invention takes into account the algorithm complexity during application and uses the NLMS algorithm for adaptive filtering, which can reduce the computational complexity of the algorithm and make the output unstable due to the nonlinearity of the model.

[0093] 2. The present invention uses the Euclidean projection method to update the parameters of the adaptive Hammerstein filter, ensuring that the output prediction of the nonlinear dynamic process has higher accuracy and better stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0094] Figure 1 This is a specific flow chart of the non-convex projection NLMS method for adaptive Hammerstein system prediction of the present invention.

[0095] Figure 2 This is the adaptive Hammerstein filter model diagram. DETAILED DESCRIPTION

[0096] In order to facilitate those skilled in the art to understand and implement the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments described herein are only used to illustrate and explain the present invention, but are not used to limit the present invention.

[0097] In the practical application of nonlinear system prediction based on Hammerstein model, the basic idea of ​​most Hammerstein modeling methods is to establish corresponding cost functions according to certain criteria (such as maximum likelihood, minimum mean square, etc.) to calculate the parameters of the model. However, the structural characteristics of Hammerstein model make the cost function obtained by using maximum likelihood criterion or minimum mean square error criterion often non-convex, so the parameter estimation obtained by most Hammerstein modeling methods is usually a local optimal solution. The present invention proposes a non-convex projection NLMS method for adaptive Hammerstein system prediction, which first calculates the solution of an initial convex optimization problem using the normalized least mean square (NLMS) method, and then projects the solution to a non-convex feasible domain, i.e., Euclidean minimization, and gives a closed solution of the minimized value, thereby improving the accuracy of the parameter estimation value.

[0098] See Figure 1 :The present invention provides a non-convex projection NLMS method for adaptive Hammerstein system prediction, the steps are as follows:

[0099] Step 1: Collect data and build an adaptive Hammerstein filter model:

[0100] For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)). The prediction of nonlinear dynamic system F(·) is realized based on the adaptive Hammerstein filter model as shown in the attached figure. Figure 2 As shown, the adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is:

[0101]

[0102] in and The dimension is L a Column vector of .

[0103] The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is:

[0104]

[0105] in:

[0106]

[0107]

[0108] h(i)=vec([x(i),X(i)]),

[0109]

[0110] vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector. The coefficients that make up the adaptive Hammerstein filter.

[0111] The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is:

[0112] d(i)=F(x(i))+v(i),

[0113] Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution.

[0114] Define the prediction error at time i:

[0115]

[0116] The method provided by the present invention is to design and train an adaptive Hammerstein filter using known input signals and output measurements {x(i), d(i)} to predict the output of the unknown system as accurately as possible, i.e., the instantaneous square error e 2 (i) As small as possible.

[0117] Step 2: Compute the solution for the adaptive Hammerstein filter coefficients:

[0118] 2-1. Using step 1, define the objective function E|h T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula:

[0119]

[0120] in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly. Solve this optimization problem through the following steps;

[0121] 2-2. First, the NLMS algorithm is used to solve the solution of the objective function and obtain the model parameter estimation, that is:

[0122]

[0123] Where μ is the step size and ε is set to a very small number (such as 0.0001) to prevent ||h(i)|| 2 When it is very small, the denominator in the formula is too small.

[0124] 2-3. Then find the solution according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows:

[0125]

[0126]

[0127] in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form;

[0128] 2-4. To calculate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula:

[0129]

[0130]

[0131] The global minimum of the Euclidean distance is calculated as in Q(i)=Θ(i)Θ T (i)

[0132] 2-5. Further prove that the solution of the objective function is not an empty set, and the projection value Still in D(i). In this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations. In order to avoid this instability, we introduce Replace Q(i), where λ max is the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as:

[0133]

[0134]

[0135]

[0136] Step 3: Set the initial values ​​of the adaptive Hammerstein filter parameters is the zero vector.

[0137] Step 4: Calculate the solution of the objective function;

[0138] Using the formula obtained in step 2 Calculate the objective function solution

[0139] Step 5: Update parameters;

[0140] Use the method described in step 2 to perform Euclidean projection and update the adaptive Hammerstein filter parameters

[0141] Step 6: Determine whether it converges: If the solution of the objective function converges, stop the iteration and execute step 7, otherwise re-execute step 4 until the iteration stop condition is met.

[0142] Step 7: Get the adaptive Hammerstein filter coefficients.

[0143] A non-convex projection NLMS system for adaptive Hammerstein system prediction comprises an adaptive Hammerstein filter building module, an initial module, an objective function solving module, a parameter updating module and a convergence judgment module.

[0144] The adaptive Hammerstein filter building module is used to build an adaptive Hammerstein filter, and the specific method is as follows:

[0145] For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)). The prediction of nonlinear dynamic system F(·) is realized based on the adaptive Hammerstein filter model as shown in the attached figure. Figure 2 As shown, the adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is:

[0146]

[0147] in and The dimension is L a Column vector of .

[0148] The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is:

[0149]

[0150] in:

[0151]

[0152]

[0153] h(i)=vec([x(i),X(i)]),

[0154]

[0155] vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector. The coefficients that make up the adaptive Hammerstein filter.

[0156] The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is:

[0157] d(i)=F(x(i))+v(i),

[0158] Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution.

[0159] Define the prediction error at time i:

[0160]

[0161] The initialization module is used to initialize the parameters of the adaptive Hammerstein filter.

[0162] The objective function solving module defines the objective function E|h according to the adaptive Hammerstein filter constructed by the adaptive Hammerstein filter construction module. T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula:

[0163]

[0164] in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly. Solve this optimization problem through the following steps;

[0165] The NLMS algorithm is used to solve the objective function and obtain the model parameter estimation, namely:

[0166]

[0167] Where μ is the step size and ε is set to a very small number (such as 0.0001) to prevent ||h(i)|| 2 When it is very small, the denominator in the formula is too small.

[0168] The parameter updating module is used to update the parameters of the adaptive Hammerstein filter. The specific method is as follows:

[0169] Find the solution obtained by the objective function solving module according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows:

[0170]

[0171]

[0172] in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form;

[0173] To evaluate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula:

[0174]

[0175]

[0176] The global minimum of the Euclidean distance is calculated as in Q(i)=Θ(i)Θ T (i)

[0177] It is further proved that the solution of the objective function is not an empty set, and the projection value Still in D(i). In this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations. In order to avoid this instability, we introduce Replace Q(i), where λ max is the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as:

[0178]

[0179]

[0180]

[0181] The convergence judgment module is used to judge whether the solution of the objective function of the adaptive Hammerstein filter model converges: if the solution of the objective function converges, the iteration is stopped and the adaptive Hammerstein filter coefficients are obtained; otherwise, the objective function is solved and the parameters are updated again through the objective function solving module and the parameter updating module until the iteration stop condition is met.

[0182] The present invention utilizes system input and output measurement values ​​to design and train an adaptive Hammerstein filter so that it can better predict the output of a nonlinear system. Considering that the parameter estimates obtained by most Hammerstein modeling methods are usually local optimal solutions, a non-convex projection NLMS method for adaptive Hammerstein system prediction is proposed. This method obtains system parameter estimates through two cyclic steps of adaptive filtering and Euclidean projection, and the obtained parameter estimates have higher accuracy and stronger stability.

[0183] It should be understood that the above description of the preferred embodiment is relatively detailed and cannot be regarded as limiting the scope of patent protection of the present invention. Under the guidance of the present invention, ordinary technicians in this field can make substitutions or modifications without departing from the scope of protection of the claims of the present invention, which all fall within the scope of protection of the present invention. The scope of protection requested for the present invention shall be based on the attached claims.

Claims

1. A non-convex projection NLMS method for adaptive Hammerstein system prediction, characterized in that: The following steps are involved: Step 1: Collect data and build an adaptive Hammerstein filter model: Step 2: Compute the solution for the adaptive Hammerstein filter coefficients: Step 3: Set the initial values ​​of the adaptive Hammerstein filter parameters is the zero vector; Step 4: Calculate the solution of the objective function; Step 5: Update parameters; Step 6: Determine whether it converges: If the solution of the objective function converges, stop the iteration and execute step 7, otherwise re-execute step 4 until the iteration stop condition is met; Step 7: Get the adaptive Hammerstein filter coefficients; Step 1: For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)); The adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is: in and x(i)=[x(i),x 2 (i),···,x La (i)] T The dimension is L a Column vector of ; The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is: in: h(i)=vec([x(i),X(i)]), vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector; The coefficients that constitute the adaptive Hammerstein filter; The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is: d(i)=F(x(i))+v(i), Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution of Define the prediction error at time i: Step 2: 2-1. Using step 1, define the objective function E|h T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula: in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly; solve this optimization problem through the following steps; 2-2. First, the NLMS algorithm is used to solve the solution of the objective function and obtain the model parameter estimation, that is: where μ is the step size and ε is set to a very small number to prevent ||h(i)|| 2 When it is very small, the denominator is too small; 2-3. Then find the solution according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows: in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form; 2-4. To calculate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula: The global minimum of the Euclidean distance is calculated as in 2-5. Further prove that the solution of the objective function is not an empty set, and the projection value Still in D(i); in this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations; in order to avoid this instability, we introduce Replace Q(i), where λ max is the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as:

2. The non-convex projection NLMS method for adaptive Hammerstein system prediction according to claim 1, characterized in that: Step 4: Using the formula obtained in step 2 Calculate the objective function solution 3. The non-convex projection NLMS method for adaptive Hammerstein system prediction according to claim 2, characterized in that: Step 5: Use the method described in step 2 to perform Euclidean projection and update the adaptive Hammerstein filter parameters 4. A non-convex projection NLMS system for adaptive Hammerstein system prediction, characterized in that: It includes an adaptive Hammerstein filter building module, an initial module, an objective function solving module, a parameter updating module and a convergence judgment module; The adaptive Hammerstein filter building module is used to build an adaptive Hammerstein filter, and the specific method is as follows: For an unknown nonlinear dynamic system F(·), given the input x(i) at time i, the system output is F i =F(x(i)); The adaptive Hammerstein filter model includes two subsystems: memoryless polynomial and finite impulse response. The relationship between the input x(i) and output s(i) of the memoryless polynomial subsystem is: in and x(i)=[x(i),x 2 (i),···,x La (i)] T The dimension is L a Column vector of ; The relationship between the input s(i) and output y(i) of the finite impulse response subsystem is: in: h(i)=vec([x(i),X(i)]), vec(A) means straightening the columns of an m×n matrix A in order to form an mn-dimensional column vector; The coefficients that constitute the adaptive Hammerstein filter; The nonlinear system is not in an ideal environment. There is noise inside the system. The corresponding output is contaminated by Gaussian noise, so the output measurement value is: d(i)=F(x(i))+v(i), Where x(i)∈R, v(i)∈R, the noise v(i) has a mean of zero and a variance of σ 2 Gaussian distribution of Define the prediction error at time i: The initial module is used to initialize the parameters of the adaptive Hammerstein filter; The objective function solving module defines the objective function E|h according to the adaptive Hammerstein filter constructed by the adaptive Hammerstein filter construction module. T (i)θ-d(i)| 2 , and the solution of the adaptive Hammerstein filter coefficients is obtained by solving the optimization problem, which is shown in the following formula: in However, the feasible domain S of the objective function is non-convex, and the objective function is also non-convex, so it is difficult to solve the result directly; solve this optimization problem through the following steps; The NLMS algorithm is used to solve the objective function and obtain the model parameter estimation, namely: where μ is the step size and ε is set to a very small number to prevent ||h(i)|| 2 When it is very small, the denominator is too small; The parameter updating module is used to update the parameters of the adaptive Hammerstein filter. The specific method is as follows: Find the solution obtained by the objective function solving module according to the Euclidean minimization criterion The Euclidean projection on the non-convex set S is expressed as follows: in But this expression is difficult to calculate, and Nor can it be represented in a tractable closed form; To evaluate the Euclidean projection expression, ignore the constraint vec([a,ab T ])∈D(i), that is, directly find the minimum Euclidean distance through the following formula: The global minimum of the Euclidean distance is calculated as in It is further proved that the solution of the objective function is not an empty set, and the projection value Still in D(i); in this result, unless the maximum eigenvalue of Q(i) is 1, this solution may lead to unstable results after multiple iterations; in order to avoid this instability, we introduce Replace Q(i), where λ max is the maximum eigenvalue of the matrix Q(i), and the closed solution to minimize the filter parameters is calculated as: The convergence judgment module is used to judge whether the solution of the objective function of the adaptive Hammerstein filter model converges: if the solution of the objective function converges, the iteration is stopped and the adaptive Hammerstein filter coefficients are obtained; otherwise, the objective function is solved and the parameters are updated again through the objective function solving module and the parameter updating module until the iteration stop condition is met.

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