Data-driven bias correction and identification method for Hammerstein nonlinear systems with noisy input and output

Abstract

The present invention relates to a data-driven input-output noisy Hammerstein nonlinear system deviation correction identification method, which belongs to the field of control theory and control engineering nonlinear system identification. The method comprises: S1: collecting system observation input data and observation output data; S2: constructing an interpretable reconstruction model of the Hammerstein nonlinear system; S3: processing the Hammerstein nonlinear system noise and system data, and decoupling the noise effect; S4: solving the system noise variance by a two-dimensional search method; S5: solving the system parameter identification by a least squares deviation correction method. The present invention adopts system observation input and output data, based on an interpretable reconstruction model, to avoid the deviation of the parameter estimation process, improve the model accuracy, and provide consistent estimation.

Description

Technical Field

[0001] The invention belongs to the field of control theory and control engineering nonlinear system identification, and relates to a data-driven input-output noisy Hammerstein nonlinear system deviation correction identification method. Background Art

[0002] The Hammerstein nonlinear system is a modular nonlinear model composed of linear and nonlinear links in series. It has attracted people's attention because of its simple structure, easy identification, and good nonlinear characteristics. The Hammerstein nonlinear system has been widely used to describe actual industrial processes, such as blast furnace ironmaking systems, battery management systems, wireless power transmission systems, etc. Therefore, the study of modular nonlinear models, especially the Hammerstein nonlinear model, is of great significance. In the past few decades, a large number of nonlinear system identification methods have been proposed. The existing mainstream nonlinear system identification methods include over-parameter method, iteration method, least squares method, subspace method, maximum expectation estimation method, kernel method, artificial intelligence method, etc.

[0003] According to different representation methods of static nonlinear blocks, the identification methods of the Hammerstein model can be roughly divided into two categories: parametric methods and non-parametric methods. Among them, the parametric method has become a research hotspot in the control field due to its advantages such as good robustness and the ability to accurately give the model of the research object. The corresponding parametric methods such as the least squares method and its variants have been widely studied by researchers. However, most of the existing Hammerstein nonlinear system identification methods take noise-free systems or nonlinear systems with only linear modules containing noise as research objects, which is difficult to meet the precise identification requirements of EIV modular nonlinear systems under the conditions of coupling between input noise and measurement input that are widely present in actual industrial processes. The identification of EIV-Hammerstein nonlinear systems with additive white noise in both input and output needs to be further studied. Summary of the invention

[0004] In view of this, the purpose of the present invention is to provide a data-driven bias correction identification method for a Hammerstein nonlinear system with noisy inputs and outputs, so as to overcome the problem that the existing parameterized identification methods cannot effectively provide unbiased parameter identification of a Hammerstein system with noisy inputs and outputs, resulting in low model accuracy.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A data-driven bias correction and identification method for a Hammerstein nonlinear system with noisy input and output includes the following steps:

[0007] S1: Collection system input data and output data;

[0008] S2: Construct an interpretable reconstruction model of Hammerstein nonlinear system;

[0009] S3: Processing Hammerstein nonlinear system noise and system data, decoupling noise effects;

[0010] S4: Solve the system noise variance using a two-dimensional search method;

[0011] S5: Solve the system parameter identification using the least squares deviation correction method.

[0012] Furthermore, step S1 specifically includes: collecting data from the actual industrial process to obtain system input and output data Where t is the sampling time, N is the number of input or output data, u(t) is the system input data, and y(t) is the system output data.

[0013] Further, step S2 specifically includes the following steps:

[0014] S21: For actual industrial processes with nonlinear characteristics, a parameterized Hammerstein nonlinear model is used to describe them. The specific form is as follows:

[0015] u(t)=u0(t)+e1(t) (1)

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023] Among them, u0(t), y0(t) represent the noise-free system input and output data, u(t), y(t) represent the input and output data collected by the system, e1(t), e2(t) represent the measured noise at the input and output of the system, θ0, Represent parameter vector and information vector respectively, l represents the order of dynamic nonlinear block in Hammerstein nonlinear system, m and n represent the order of numerator and denominator of static linear block in Hammerstein nonlinear system respectively, u0(t-k1) represents the state of noiseless system input u0 before time k1, M0(t) represents the component of input signal in information vector, Further represents the component of the input signal in M0(t) at time (t-k1), represents the system parameters to be estimated.

[0024] S22: Processing the parameter Hammerstein nonlinear model of step S21 to obtain an interpretable reconstruction model to continue the subsequent parameter identification, the specific form is as follows:

[0025]

[0026]

[0027] M(t)=[μ1(t) μ2(t) … μ m (t)] (11)

[0028] μ t1 (t) = [u(t-k1) u 2 (t-k1) … u l (t-k1)] (12)

[0029] θ=[a b1d b2d … b n d] T (13)

[0030] a=[a1 a2 … a n ] (14)

[0031] d=[d1 d2 … d l ] (15)

[0032] Among them, ε(t) represents the remainder, θ, They represent the parameter vector and information vector of the interpretable reconstruction model respectively, u(t-k1) represents the state of the system collecting input u before time k1, and M(t) represents the information vector of the interpretable reconstruction model. The component of the input signal, Further represents the component of the input signal in M(t) at time (t-k1), represents the system parameters of the interpretable reconstruction model.

[0033] Furthermore, in step S2, the following assumptions are made about the system:

[0034] A1: The additive noise e1(t), e2(t) is Gaussian white noise with a mean of 0 and unknown variance, and is independent of the input and output data u0(t), y0(t) of the noise-free system;

[0035] A2: The first item of the system parameter vector d is 1, that is, d1=1.

[0036] Further, step S3 specifically includes the following steps:

[0037] S31: Perform error analysis on the classical least squares method to obtain the deviation correction matrix ψ, which satisfies the following conditions:

[0038]

[0039] Among them, the matrices φ and Δφ are specifically expressed as follows:

[0040]

[0041]

[0042] in:

[0043]

[0044]

[0045] S32: Perform block processing on the deviation correction matrix ψ to obtain a block matrix in the following form:

[0046]

[0047] Among them, E{·} represents the mathematical expectation, represents the Gaussian white noise matrix, ψ1, ψ2, ψ3, ψ4 represent the block matrices;

[0048] Since ψ contains

[0049]

[0050] Among them, e(t) means that the mean is 0 and the variance is σ 2 Gaussian white noise, m d represents the d-th moment of e(t).

[0051] S33: Study the specific forms of matrices ψ1, ψ2, ψ3, ψ4. The specific process is as follows:

[0052] For matrix ψ1, based on assumption A1, matrix ψ1 has the following form:

[0053]

[0054] in, Represents the variance of the noise e2(t) at the system output.

[0055] For the matrix ψ2, since the matrix M is composed of the input signal u, based on assumption A1, it can be known that the input signal u and the additive white noise e2(t) are uncorrelated, so the matrix ψ2 has the following form:

[0056]

[0057] For matrix ψ3, based on assumption A1, we know that the Gaussian white noise matrix There is no correlation with the matrix ΔM, so the matrix ψ3 has the following form:

[0058]

[0059] Based on the high-order rectangular form of white noise (21), the entry at (k2, k1j) in the matrix ψ3 has the following form:

[0060]

[0061] For the matrix ψ4, based on the high-order rectangular form of white noise (21), the entry at (qk3, k1j) in the matrix ψ4 has the following form:

[0062]

[0063] Among them, k1=1,2,…,m, k2=1,2,…,n, k3=1,2,…,m, j=1,2,…,l, q=1,2,…,l, m r Represents the r-th moment of the noise e2(t) at the input of the system.

[0064] S34: Since the expressions (25) and (26) of the deviation correction matrix ψ include the unmeasurable system noise-free input and output signals u0(t) and y0(t), the system noise is decoupled from the system measurement input and output data based on the high-order rectangular formula (21) of white noise.

[0065] Further, in step S34, decoupling system noise from system measurement input and output data specifically includes the following steps:

[0066] S341: For unmeasurable signals Construct remainder Satisfy the following form:

[0067]

[0068] Substituting formula (1) into (27), we can obtain:

[0069]

[0070] but and The deviation between Has the following form:

[0071]

[0072] S342: For unmeasurable signals Construct remainder Satisfy the following form:

[0073]

[0074] Substituting formula (1) into (27), we can obtain:

[0075]

[0076] but and The deviation between Has the following form:

[0077]

[0078] S343: For t = 1, 2, ..., N, recursively calculate the remainders (29) and (32) in the order of q1 = 1, 2, ..., l-1; for the remainders Can be based on previously recursively calculated terms Calculate and put it into formula (27) to decouple the system noise from the system measurement input and output data; for the remainder Same reason.

[0079] Further, step S4 specifically includes the following steps:

[0080] S41: For the least squares loss function v(θ,D N ) is reconstructed in the following form:

[0081]

[0082] Among them, φ0 represents the information matrix of the Hammerstein nonlinear system, and the specific form is as follows:

[0083]

[0084] Where N represents the number of system inputs and outputs collected.

[0085] Based on the assumption A1 and the high-order rectangular form of white noise (21), when N tends to infinity, the term and Converge to and 0; therefore, the least squares loss function v(θ,D N ) has the following form:

[0086]

[0087] in, represents the least squares estimate;

[0088] S42: Construct matrices Ω1 and Ω2 to satisfy the following conditions:

[0089]

[0090]

[0091] Since the term φ0 in (35) contains unknown noise-free input and output data, repeat step S3 to decouple the system noise from the system measured input and output data;

[0092] S43: Construct a loss function based on minimizing the output prediction error The specific form is as follows:

[0093]

[0094] Where Y represents the output vector of the Hammerstein nonlinear system, and its specific form is as follows:

[0095] Y=[y(1) … y(N)] T (39)

[0096] It should be noted that the parameter vector θ, matrices Ω1 and Ω2 are all related to the noise variance The relevant variables, when the correct noise variance value When brought into (38), theoretically,

[0097] S44: Define the search interval and Set the number of equally divided points M1 and M2 in the search interval respectively and Divide the search interval into equal parts; traverse the set noise variance combinations in turn And record the value of the corresponding loss function ∈(·); the variance combination that minimizes the loss function This is the estimate of the variance.

[0098] Further, step S5 specifically includes the following steps:

[0099] S51: Reconstruct formula (2), the specific form is as follows:

[0100]

[0101] in:

[0102]

[0103] Define the following matrix:

[0104] Δφ=φ0-φ(42)

[0105] E2=[e2(1) … e2(N)] T (43)

[0106] Calculate the true parameter vector θ0 and the least squares estimate The specific form of the deviation is as follows:

[0107]

[0108] Based on assumption A1, since there is no correlation between the Gaussian white noise matrix E2 and the matrix φ, when N tends to infinity, the term B E Converges to 0;

[0109] S52: Constructing the least squares bias correction identification framework:

[0110]

[0111] in, represents the estimate obtained by applying the least squares bias-corrected identification framework.

[0112] Since the term Δφ in (45) contains unknown noise-free input and output data, the matrix Ψ is constructed to satisfy the following conditions:

[0113]

[0114] Repeat step S3 to decouple system noise from system measurement input and output data;

[0115] S53: Estimate parameters of Hammerstein nonlinear systems.

[0116] Further, in step S53, estimating the Hammerstein nonlinear system parameters specifically includes the following steps:

[0117] S531: From Directly extract the estimated value of the system parameter vector a

[0118]

[0119] S531: Based on assumption A2, The estimated value of each item in the system parameter vector b is calculated in the following form:

[0120]

[0121] Where i = 1, 2, ..., n;

[0122] For each item d in the unified parameter vector d j , there are l different estimates

[0123]

[0124] In order to solve the problem of a large number of redundant estimates, The average value of d j Estimated value:

[0125]

[0126] The beneficial effects of the present invention are as follows: based on the collected input and output data, the present invention adopts an interpretable reconstruction model of a nonlinear system, processes nonlinear system noise and system data, decouples the noise effect, and avoids the estimation bias caused by the classical least squares method. The present invention adopts a bias-corrected least squares identification framework to achieve identification and solution of model parameters and noise variance, thereby improving model accuracy. Compared with the existing parameterized identification method, the present invention greatly improves the model accuracy of the Hammerstein nonlinear system with noisy input and output and reduces the variance of the estimated parameters.

[0127] Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0128] In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

[0129] Figure 1 It is a flowchart of the method of the present invention;

[0130] Figure 2 This is the WPT circuit topology diagram used in this embodiment;

[0131] Figure 3 Collect input and output data for the WTP system;

[0132] Figure 4 This is the result diagram of noise variance identification of the real WTP system.

[0133] Figure 5 This is a comparison chart of the simulation and identification results of the real WTP system. DETAILED DESCRIPTION

[0134] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and features in the embodiments can be combined with each other without conflict.

[0135] See also Figure 1 to Figure 5 The present invention provides a data-driven bias correction and identification method for Hammerstein nonlinear systems with noisy input and output, such as Figure 1 As shown, the method specifically comprises the following steps:

[0136] S1: Collection system input data and output data;

[0137] S2: Construct an interpretable reconstruction model of Hammerstein nonlinear system;

[0138] S3: Processing Hammerstein nonlinear system noise and system data, decoupling noise effects;

[0139] S4: Solve the system noise variance using a two-dimensional search method;

[0140] S5: Solve the system parameter identification using the least squares deviation correction method.

[0141] The method of the present invention is applied to a wireless power transfer (WPT) system to illustrate the effectiveness and superiority of the present invention. The WPT system studied in this embodiment consists of a frequency-controlled half-bridge inverter and a receiver. The circuit topology diagram and the physical structure diagram are shown in FIG. Figure 2 , where the specific parameters are shown in Table 1.

[0142] Table 1 Main parameters of WPT experimental device

[0143] parameter Interpretation value <![CDATA[C1]]> Transmit resonator capacitance 395.33nF <![CDATA[C2]]> Capacitance of the receiving resonant cavity 393.65nF <![CDATA[f S ]]> Drive frequency of the inverter 137.4kHz <![CDATA[L1]]> Inductance of the transmitting resonator 32.31μH <![CDATA[L2]]> Inductance of the receiving resonator 31.63μH <![CDATA[R o ]]> Load resistance 12.7Ω <![CDATA[v d ]]> DC supply voltage 32.5V

[0144] The specific implementation process is as follows:

[0145] S1: Collect data on the wireless power transmission system.

[0146] The input is the input voltage of the transmitting coil of the WPT system, and the output is the output current of the load. The sampling period is selected as T = 0.5μs. The transmitting end converts the 32.5V DC into a high-frequency PWM square wave with a frequency of 137.4kHz through the digital controller STM32F407, and injects it into the transmitting coil. The terminal voltage of the transmitting coil is measured with a voltage probe (P7300, 300MHz), and the output current of the load is measured with a current sensor (CP8050A). Considering that the noise introduced by the digital oscilloscope is very small, we approximate the input data to noise-free input data. In order to generate input measurement noise, the voltage probe is connected to a 10-meter-long wire with an internal resistance of 1.1Ω, and the wire is connected to the transmitting coil, thereby extending the signal line of the probe. This method actually injects noise into the measured signal. In order to verify the effectiveness of the method of the present invention, approximately noise-free data and noisy data with a length of 7500 were collected, such as Figure 3 shown.

[0147] S2: For wireless power transmission systems with nonlinear characteristics, a parameterized Hammerstein nonlinear model is used to describe them to facilitate subsequent identification. The original parameterized model based on noise-free data is processed to obtain an interpretable reconstructed model described by measurable noisy data. Figure 2 The noise-free data shown is not strictly noise-free data, but it can be approximated as noise-free data because the noise introduced by the digital oscilloscope is very small.

[0148] S3: Perform error analysis on the classical least squares method and construct a bias correction matrix ψ by recursively calculating in order from small to large. This bias correction matrix can effectively eliminate the estimation error introduced by the classical least squares method.

[0149] S4: Repeat step S3 to construct a loss function based on minimizing the output prediction error. Iterate over the pre-set variance pairs Search for the variance pair that minimizes the loss function, which is the estimated value of the noise variance. The estimated value of Figure 4 shown.

[0150] S5: Based on the estimated value of the noise variance obtained in step S4, the bias-corrected least squares identification framework proposed in the present invention is adopted, and the parameter identification results are shown in Table 2.

[0151] Based on the obtained model, the identification output curve of the WPT system is obtained as follows: Figure 5 The goodness of fit of the output data is also shown in Table 2. The goodness of fit is defined as:

[0152]

[0153] Among them, y(T) and Represent the actual output of the sample and the model output, respectively. is the average value of y(T). It can be seen that the method proposed in this paper achieves good results in parameter estimation of WPT system.

[0154] Table 2 WPT system identification results

[0155] parameter LS BCLS <![CDATA[a1]]> -0.494 -0.049 <![CDATA[a2]]> -0.498 -0.947 <![CDATA[b1]]> <![CDATA[-4.108×10 -5 ]]> <![CDATA[-3.579×10 -4 ]]> <![CDATA[b2]]> <![CDATA[4.509×10 -4 ]]> <![CDATA[-7.267×10 -4 ]]> <![CDATA[c1]]> 1 1 <![CDATA[c2]]> 0.069 -0.015 <![CDATA[c3]]> <![CDATA[4.509×10 -4 ]]> <![CDATA[-7.267×10 -4 ]]> fit(%) 55.66 88.91

[0156] The model accuracy of the obtained model basically coincides with the ideal value and the fitting degree is high, and a high-precision WPT model is established.

[0157] The data-driven input-output noisy Hammerstein nonlinear system deviation correction identification method proposed in the present invention is used to accurately identify the WPT system. The interpretable reconstruction model of the nonlinear system is used to process the nonlinear system noise and system data, decouple the noise effect, and avoid the estimation deviation caused by the classical least squares method. The present invention adopts a deviation correction least squares identification framework to realize the identification and solution of model parameters and noise variance, and improve the model accuracy. An accurate WPT model is obtained. It can be promoted and applied in other complex systems.

[0158] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in the scope of the claims of the present invention.

Claims

1. A data-driven bias correction and identification method for Hammerstein nonlinear systems with noisy input and output, characterized in that: The method specifically comprises the following steps: S1: Collect system input and output data, including: collect data from actual industrial processes, obtain system input and output data Where t is the sampling time, N is the number of input or output data, u(t) is the system input data, and y(t) is the system output data; S2: Construct an interpretable reconstruction model of Hammerstein nonlinear system, which includes the following steps: S21: For actual industrial processes with nonlinear characteristics, a parameterized Hammerstein nonlinear model is used to describe them. The specific form is as follows: u(t)=u0(t)+e1(t) (1) Among them, u0(t), y0(t) represent the noise-free system input and output data, u(t), y(t) represent the input and output data collected by the system, e1(t), e2(t) represent the measured noise at the input and output of the system, θ0, Represent parameter vector and information vector respectively, l represents the order of dynamic nonlinear block in Hammerstein nonlinear system, m and n represent the order of numerator and denominator of static linear block in Hammerstein nonlinear system respectively, u0(t-k1) represents the state of noiseless system input u0 before time k1, M0(t) represents the component of input signal in information vector, Further represents the component of the input signal in M0(t) at time (t-k1), represents the system parameters to be estimated; S22: Processing the parameter Hammerstein nonlinear model of step S21 to obtain an interpretable reconstruction model to continue the subsequent parameter identification, the specific form is as follows: M(t)=[μ1(t) μ2(t) … μ m (t)] (11) μ k1 (t)=[u(t-k1) u 2 (t-k1) … u l (t-k1)] (12) θ=[a b1d b2d … b n [d] T (13) <h2 style=";text-align:left;direction:ltr">a=[a1 a2 … a<h2 style=";text-align:left;direction:ltr"> n <h2 style=";text-align:left;direction:ltr"> ] (14) d=[d1 d2 … dl] (15) Among them, ε(t) represents the remainder, θ, They represent the parameter vector and information vector of the interpretable reconstruction model respectively, u(t-k1) represents the state of the system collecting input u before time k1, and M(t) represents the information vector of the interpretable reconstruction model. The component of the input signal, Further represents the component of the input signal in m(t) at time (t-k1), represents the system parameters of the interpretable reconstruction model; In step S2, the following assumptions are made about the system: A1: The additive noise e1(t), e2(t) is Gaussian white noise with a mean of 0 and unknown variance, and is independent of the input and output data u0(t), y0(t) of the noise-free system; A2: The first term of the system parameter vector d is 1, that is, d1=1; S3: Processing Hammerstein nonlinear system noise and system data to decouple the noise effect, specifically including the following steps: S31: Perform error analysis on the classical least squares method to obtain the deviation correction matrix ψ, which satisfies the following conditions: Among them, the matrices φ and Δφ are specifically expressed as follows: in: S32: Perform block processing on the deviation correction matrix ψ to obtain a block matrix in the following form: Among them, E{·} represents the mathematical expectation, represents the Gaussian white noise matrix, ψ1, ψ2, ψ3, ψ4 represent the block matrices; ψ contains Among them, e(t) means that the mean is 0 and the variance is σ 2 Gaussian white noise, m d represents the d-order moment of e(t); S33: Study the specific forms of matrices ψ1, ψ2, ψ3, ψ4. The specific process is as follows: Based on assumption A1, the matrix ψ1 has the following form: in, represents the variance of the noise e2(t) at the output of the system; Based on assumption A1, the matrix ψ2 has the following form: Based on assumption A1, the matrix ψ3 has the following form: Based on the high-order rectangular form (21) of white noise, the entry at (k2, k1j) in the matrix ψ3 has the following form: Based on the high-order rectangular form (21) of white noise, the entry at (qk3, k1j) in the matrix ψ4 has the following form: Among them, k1=1,2,…,m, k2=1,2,…,n, k3=1,2,…,m, j=1,2,…,l, q=1,2,…,l, m r represents the r-order moment of the noise e2(t) at the input of the system; S34: Based on the high-order rectangular form (21) of white noise, decoupling the system noise from the system measurement input and output data, specifically comprising the following steps: S341: For unmeasurable signals Construct remainder Satisfy the following form: Substituting equation (1) into (27), we obtain: but and The deviation between Has the following form: S342: For unmeasurable signals Construct remainder Satisfy the following form: Substituting equation (1) into (27), we obtain: but and The deviation between Has the following form: S343: For t = 1, 2, ..., N, recursively calculate the remainders (29) and (32) in the order of q1 = 1, 2, ..., l-1; for the remainders Based on the previously recursively calculated terms Calculate and put it into formula (27) to decouple the system noise from the system measurement input and output data; for the remainder Similarly; S4: Using a two-dimensional search method to solve the system noise variance, specifically including the following steps: S41: For the least squares loss function v(θ,D N ) is reconstructed in the following form: Among them, φ0 represents the information matrix of the Hammerstein nonlinear system, which is in the following form: Where N represents the number of system inputs and outputs collected; Based on the assumption A1 and the high-order rectangular form of white noise (21), when N tends to infinity, the term and Converge to and 0; therefore, the least squares loss function v(θ,D N ) has the following form: in, represents the least squares estimate; S42: Construct matrices Ω1 and Ω2 to satisfy the following conditions: Since the term φ0 in (35) contains unknown noise-free input and output data, repeat step S3 to decouple the system noise from the system measured input and output data; S43: Construct a loss function based on minimizing the output prediction error The specific form is as follows: Where Y represents the output vector of the Hammerstein nonlinear system, and its specific form is as follows: y=[y(1) … y(N)] T (39) S44: Define the search interval and Set the number of equally divided points M1 and M2 in the search interval respectively and Divide the search interval into equal parts; traverse the set noise variance combinations in turn And record the value of the corresponding loss function ∈(·); the variance combination that minimizes the loss function This is the estimated value of the variance; S5: Solving the system parameter identification using the least squares deviation correction method, specifically including the following steps: S51: Reconstruct formula (2), the specific form is as follows: in: Define the following matrix: Δφ=φ0-φ (42) <h2 style=";text-align:left;direction:ltr">E2 = [e2(1) … e2(N)]<h2 style=";text-align:left;direction:ltr"> t <h2 style=";text-align:left;direction:ltr"> (43) Calculate the true parameter vector θ0 and the least squares estimate The specific form of the deviation is as follows: Based on assumption A1, since there is no correlation between the Gaussian white noise matrix E2 and the matrix φ, when N tends to infinity, the term B E Converges to 0; S52: Constructing the least squares bias correction identification framework: in, represents the estimate obtained by applying the least squares bias-corrected identification framework; Since the term Δφ in (45) contains unknown noise-free input and output data, the matrix Ψ is constructed to satisfy the following conditions: Repeat step S3 to decouple system noise from system measurement input and output data; S53: Estimating the parameters of the Hammerstein nonlinear system, specifically comprising the following steps: S531: From Directly extract the estimated value of the system parameter vector a S531: Based on assumption A2, The estimated value of each item in the system parameter vector b is calculated in the following form: Where i = 1, 2, ..., n; For each item d in the unified parameter vector d j , there are l different estimates by The average value of d j Estimated value:

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