Method for reducing number of parameters to be estimated based on output parameter sensitivity matrix in nonlinear Wiener model identification

Abstract

The invention discloses a method for reducing the number of to-be-estimated parameters based on an output parameter sensitivity matrix in nonlinear Wiener model identification, and the method comprises the steps: determining the number of to-be-estimated parameters which can be reduced by a state space model in a linear part in a parameterized Wiener model through employing the output parameter sensitivity matrix; and fitting a nonlinear function part in the Wiener model by using a Lagrange interpolation method based on a Radau point, determining parameters needing to be estimated of the nonlinear part by using an AIC criterion, and identifying the parameters based on a stability constraint condition of the linear state space model and an output error mean square error criterion function. White noise input is applied to a nonlinear system, output of the system is measured, linear state space model parameter estimation and nonlinear part parameter estimation are carried out after all measurement data are obtained, and identification of the nonlinear system is achieved. According to the method for reducing the number of the to-be-estimated parameters through the parameter sensitivity matrix, the variance during parameter estimation is reduced, and the parameter estimation precision is improved.

Description

Technical Field

[0001] The invention belongs to the field of process system engineering, and in particular relates to a method for reducing the number of parameters to be estimated based on an output parameter sensitivity matrix in nonlinear Wiener model identification. Background Art

[0002] Advanced process control technologies such as adaptive control and model predictive control are widely used in many fields, such as oil refining, petrochemicals, papermaking, nuclear energy, etc. Their applications are all based on system models, so being able to accurately obtain system models is the key to the wide application of these technologies. However, many systems have complex internal mechanisms and are very difficult to model through mechanisms, so the system identification method that uses actual measured data to establish a mathematical model of a dynamic system has received a lot of attention. System identification was initially mainly the identification of linear systems, but most process industrial systems have nonlinear links, so with the deepening of system identification research, nonlinear system identification has become a hot topic in the field of system identification. For nonlinear systems, there are many types of models, such as Wiener models, hammerstein models, NARMAX models, and neural network models. The Wiener model has the characteristics of easy identification, low computational complexity, and the ability to better reflect the characteristics of the industrial production process, and is more suitable as a model for process control in actual production. The Wiener model consists of a linear part connected in series with a nonlinear part. The output of the linear part is the input of the nonlinear part. There are many models to choose from in the identification of the linear part, such as autoregressive model, sliding average model, autoregressive sliding average model and state space model, etc. The state space model can well identify multi-input and multi-output systems, and is a model that is conveniently used in computer control system design. At the same time, most optimal controllers can be effectively calculated in the form of state space models, so the state space model is the model structure used in most system identifications.

[0003] In the identification of Wiener models, many works use subspace identification methods to identify Wiener models containing linear state space models. Gómez and Baeyens used a two-step method that combines the subspace identification method of linear systems with the minimization of the two norms to identify Wiener models [Gómez JC, Baeyens E. Subspace-based identification algorithms for Hammerstein and Wiener models [J]. European Journal of Control, 2005, 11 (2): 127-136.]; this method has a large number of parameters that need to be estimated. In order to reduce the estimated parameters, Yang Qipei used the key term separation technique to divide the dynamic linear and static nonlinear of the Wiener model into two sub-models to estimate the unknown parameters separately. At the same time, the Markov parameters that are not coupled with the nonlinear parameters are estimated through the subspace predictor model, and a two-stage hierarchical iterative least squares method is proposed to estimate the unknown parameters of the two sub-models separately, reducing the number of estimated parameters [Yang Qipei. Research on subspace identification and predictive control methods for Wiener nonlinear systems [D]. Beijing: Beijing University of Chemical Technology, 2023.].

[0004] The methods mentioned in the above literature are applied to the identification of Wiener models and can solve the problems in the identification of Wiener models to a certain extent. The method proposed by Gómez and Baeyens has a large number of parameters to be estimated, which will increase the variance of parameter estimation. Although Yang Qipei's method reduces the number of estimated parameters, this method only applies to the case where there is an inverse function of the nonlinear function in the nonlinear part of the Wiener model and the inverse function can be expressed in the form of a polynomial, and cannot include other types of nonlinear functions.

[0005] Therefore, the existing methods for reducing the number of parameter estimates for different nonlinear function forms in Wiener model identification still have defects and need to be improved. Summary of the invention

[0006] The object of the present invention is to address the deficiencies of the prior art and provide a method for reducing the number of parameters to be estimated based on an output parameter sensitivity matrix in nonlinear Wiener model identification.

[0007] The object of the present invention is achieved by the following technical solution: A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification, comprising the following steps:

[0008] (1) Based on the order of the system to be identified, the linear part of the Wiener model is parameterized in the form of a state space model to determine the size of matrices A, B, and C; the nonlinear part is fitted using the Lagrange interpolation method based on Radau points, and the number of nonlinear parameters to be identified, n1, is determined using the AIC criterion;

[0009] (2) calculating and outputting a sensitivity matrix for each parameter according to the parameterized state space model obtained in step (1), and performing singular value decomposition on the obtained sensitivity matrix to obtain a singular value matrix and a right singular vector matrix; determining the number of parameters to be estimated and the number of parameters that can be fixed at initial values ​​through the singular value matrix; and converting the parameter vector into a parameter vector in which there is no correlation between the parameters by left-multiplying the transpose of the right singular vector matrix;

[0010] (3) According to the actual situation of the system to be identified, select white noise with appropriate variance and mean as input data, and record the output y(k) of the system at the kth moment as the output data for identification;

[0011] (4) Calculate the characteristic polynomial of the state vector matrix according to the parameterized state space model, and use the Jury criterion to calculate the stability constraint condition; under the stability constraint condition, use the input data and output data for identification obtained in step (3) and the nonlinear optimization method according to the criterion of minimizing the mean square value of the output error to obtain the parameters of the matrices A, B and C and the nonlinear part parameters z1,…,z in the state space model. n1 The value of .

[0012] Furthermore, the step (1) specifically includes the following sub-steps:

[0013] (1.1) According to the order n of the system to be identified, the linear part of the Wiener model is parameterized by the state space model. The form of the parameterized state space model is:

[0014] x(k)=A*x(k-1)+B*u(k-1);

[0015] yc(k)=C*x(k);

[0016] Among them, x(k-1) is the state vector at the k-1th moment; u(k-1) is the input vector at the k-1th moment; x(k) is the state vector at the kth moment; A, B and C are constant matrices respectively; yc(k) is the output vector of the linear part at the kth moment;

[0017] The state vector x(k-1) is defined as: x(k-1)=[x1(k-1)…x e (k-1)…x n (k-1)]T ; The state vector x(k) is defined as: x(k) = [x1(k)…x e (k)…x n (k)] T ; The input vector u(k-1) is defined as: u(k-1)=[u1(k-1)…u i (k-1)…u m (k-1)] T The output vector yc(k) of the linear part is defined as: yc(k) = [yc1(k)…yc s (k)…yc l (k)] T ; where x e (k-1) represents the state vector component of any k-1th moment, e = 1, ..., e, ..., n; x e (k) represents the state vector component at any k-th moment; u i (k-1) represents any input component at the k-1th moment, i = 1, ..., i, ..., m, m represents the total number of input components of the system to be identified; yc s (k) represents the output component of the linear part at any k-th moment, l represents the total number of l output components of the system to be identified, s=1,…,s,…,l;

[0018] The size of matrix A is determined to be n*n by the order n of the system to be identified, and the size of matrix B is determined to be n*m ​​by the order n of the system to be identified and the number m of the input components of the system to be identified. Finally, the size of matrix C is determined to be l*n by the order n of the system to be identified and the number l of the output components of the system to be identified. Then, matrix A is defined as Define the matrix B as Define the matrix C as Among them, a h,e represents any constant parameter in the matrix A, h=1,…,h,…,n; b e,i represents any constant parameter in matrix B; c s,e Represents any constant parameter in the matrix C;

[0019] (1.2) First, the number of nonlinear parameters is set to n1. The nonlinear part is fitted using the Lagrange interpolation polynomial based on Radau points. The formula of the Lagrange interpolation polynomial is as follows:

[0020]

[0021] Among them, f(·) is the nonlinear function of the nonlinear part;

[0022] The output vector f(yc(k)) of the nonlinear part at the kth moment is expressed as:

[0023]

[0024] Among them, f(yc s (k)) represents any output component in the output vector f(yc(k)) of the nonlinear part;

[0025] The number n1 of the nonlinear parameters is determined by the AIC criterion, and the number n1 of the nonlinear parameters is used to define the parameters of the nonlinear part to be identified in the identification as follows:

[0026] Furthermore, the step (2) specifically includes the following sub-steps:

[0027] (2.1) The sensitivity matrix S is defined according to the matrices A, B and C and the output vectors yc(1),…,yc(k),…,yc(N) of the linear part at N moments. The sensitivity matrix S is:

[0028] S=[S A S B S C ];

[0029] in,

[0030] Said for in, Said for in,

[0031] Said is an l×1 vector, where the sth component is x s (k), the rest of the components are 0;

[0032] (2.2) The sensitivity matrix S is subjected to singular value decomposition, and the singular value decomposition formula is as follows:

[0033] S=UΣV T ;

[0034] Among them, U is the left singular vector matrix, Σ is the singular value matrix, and V is the right singular vector matrix; the singular value matrix Σ obtained by the singular value decomposition of the sensitivity matrix is ​​an N*l row n 2 +n*m+n*l columns of matrix;

[0035] (2.3) All parameters in the state-space model x(k) = Ax(k-1) + Bu(k-1) are written as a column vector θ: θ = [a 1,1 a 1,2 …a h,e …a n,n b 1,1 b 1,2 …b e,i …b n,m c 1,1 c 1,2 …c s,e …c l,n ] T ;

[0036] Then, the column vector θ is multiplied by the transpose of the right singular vector matrix V to obtain the transformed parameter vector β, as follows:

[0037]

[0038] Among them, β g is any component of the vector β, g=1,…,g,…,n 2 +n*m+n*l;

[0039] The column vector θ can be replaced with the parameter vector β using the right singular vector matrix V, as follows: θ = Vβ;

[0040] Singular value matrix The values ​​on the diagonal of the block matrix from the 1st row to the qth row and from the 1st column to the qth column are not zero; the values ​​on the diagonal of the block matrix from the q+1th row to the nth column are not zero; 2 +n*m+n*l rows, q+1 columns to n 2 The absolute value of the values ​​on the diagonal of the block matrix with +n*m+n*l columns is less than 1×10 -10 ;

[0041] The parameters from the 1st to the qth row of the parameter vector β are used as the parameters to be estimated in the linear part of the Wiener model, and the parameters from the q+1th to the nth row are used as the parameters to be estimated in the linear part of the Wiener model. 2 The parameters of the +n*m+n*l rows are fixed at the initial values.

[0042] Furthermore, the step (3) is specifically as follows:

[0043] According to the actual situation of the system to be identified, the first white noise with appropriate variance and mean is selected as the input vector u(k-1), and the output y(k) = [y1(k)…y s (k)…y l (k)] TAs the data for identification, the y(k) is defined as y(k)=f(yc(k))+w(k), where w(k)=[w1(k)…w s (k)…w l (k)] T is the second white noise, where w s (k) represents the second white noise component at any k-th moment.

[0044] Furthermore, the step (4) specifically includes the following sub-steps:

[0045] (4.1) According to the parameterized state space model, the characteristic polynomial of the matrix A is calculated. The characteristic polynomial is in the following form:

[0046] λ n +h1λ n-1 +…+h n-1 λ+h n =0;

[0047] Among them, h1,…,h e ,…,h n are the coefficients of the characteristic polynomial. Any coefficient h e Both are determined by parameter a 1,1 ,…,a h,e ,…,a n,n The root λ of the characteristic polynomial is the pole of the system to be identified. In order to make the identified system stable, the modulus length of the root λ needs to be less than 1. The characteristic polynomial is applied to the coefficient criteria in the Jury table to obtain the constraint conditions that make the system stable. The form of the Jury table is as follows:

[0048]

[0049] The first and second rows of the table consist of 1 and coefficients h1,…,h n The coefficients starting from the third row in the table are calculated using the following formula:

[0050]

[0051] Among them, t=0,1,…,t,…,n; j=2,3,…,j,…,n-2; v=0,1,…,v,…,nj;

[0052] Calculate the 3 coefficients of the 2n-3th row r n-2 and until;

[0054] Using the coefficients in the Jury table, the conditions for system stability are obtained as follows:

[0055]

[0056] (4.2) Parameter estimation for a Winer system with measurement noise is to use nonlinear optimization methods to find an estimate of the output error criterion function that minimizes the following and

[0057]

[0058] The constraints for minimizing the output error criterion function are:

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065] In estimating the parameters of matrices A, B and C and the nonlinear part parameters z1,…, When , starting from the initial value, using nonlinear optimization method, under the premise of satisfying the given constraints, iteratively update The parameter estimates of the matrices A, B, and C whose values ​​are reduced and and the nonlinear part parameters z1,…,z n1 Estimated value of Until I can no longer find The estimated value of and at this time and is the final parameter estimate; Indicates that the current estimate is used and The outputs y(1),…,y(k),…,y(N) at N moments are used to simulate and calculate the value of the nonlinear output; J(·) represents the objective function of optimization. When estimating parameters, it is to find the estimated value that minimizes the objective function. and Indicates that the current estimate is used and Calculated estimates; and They represent the estimated parameter matrices respectively; represents the state vector estimated at the kth moment; is the output vector representing the linear part of the estimate at the kth moment, is the output component representing the linear part estimated at any k-th moment;

[0066] In the above formula, each parameter in the matrices A, B and C is expressed according to the β parameter, and the initial value is given during the optimization solution;

[0067] After the estimation is completed, the parameters of the matrices A, B and C in the state space model and the nonlinear part parameters z1,…,z n1 The value of .

[0068] The beneficial effects of the present invention are as follows: firstly, a parameterized state space model of the linear part is established according to the actual order of the system, and the nonlinear part in the Wiener model is fitted by a Lagrange interpolation method based on Radau points, which can effectively cope with situations of different nonlinear function forms; a parameter sensitivity matrix is ​​used to determine the number of parameters to be estimated that can be reduced by the state space model in the linear part of the parameterized Wiener model, which is a method for effectively reducing the number of parameters to be estimated, reducing the variance during parameter estimation, and improving the estimation accuracy of the parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 A flowchart of a method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification. DETAILED DESCRIPTION

[0070] In order to make the purpose, technical scheme and advantages of the present invention more clear, the present invention is further described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0071] Example 1

[0072] like Figure 1 As shown, the present invention provides a method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification, comprising the following steps:

[0073] (1) Based on the order of the system to be identified, the linear part of the Wiener model is parameterized in the form of a state-space model to determine the sizes of matrices A, B, and C. The nonlinear part is fitted using the Lagrange interpolation method based on Radau points, and the number of nonlinear parameters to be identified, n1, is determined using the AIC criterion.

[0074] The step (1) specifically includes the following sub-steps:

[0075] (1.1) According to the order n of the system to be identified, the linear part of the Wiener model is parameterized by the state space model. The form of the parameterized state space model is:

[0076] x(k)=A*x(k-1)+B*u(k-1);

[0077] yc(k)=C*x(k);

[0078] Among them, x(k-1) is the state vector at the k-1th moment; u(k-1) is the input vector at the k-1th moment; x(k) is the state vector at the kth moment; A, B and C are constant matrices respectively; yc(k) is the output vector of the linear part at the kth moment.

[0079] The state vector x(k-1) is defined as: x(k-1)=[x1(k-1)…x e (k-1)…x n (k-1)] T ; The state vector x(k) is defined as: x(k) = [x1(k)…x e (k)…x n (k)] T ; The input vector u(k-1) is defined as: u(k-1)=[u1(k-1)…u i (k-1)…u m (k-1)] T The output vector yc(k) of the linear part is defined as: yc(k) = [yc1(k)…yc s (k)…yc l (k)] T ; where x e (k-1) represents the state vector component of any k-1th moment, e = 1, ..., e, ..., n; x e (k) represents the state vector component at any k-th moment; u i (k-1) represents any input component at the k-1th moment, i = 1, ..., i, ..., m, m represents the total number of input components of the system to be identified; yc s(k) represents the output component of the linear part at any k-th moment, l represents the l output components of the system to be identified, s=1,…,s,…,l.

[0080] The size of matrix A is determined to be n*n by the order n of the system to be identified, and the size of matrix B is determined to be n*m ​​by the order n of the system to be identified and the number m of the input components of the system to be identified. Finally, the size of matrix C is determined to be l*n by the order n of the system to be identified and the number l of the output components of the system to be identified. Then, matrix A is defined as Define the matrix B as Define the matrix C as Among them, a h,e represents any constant parameter in the matrix A, h=1,…,h,…,n; b e,i represents any constant parameter in matrix B; c s,e Represents any constant parameter in the matrix C.

[0081] (1.2) First, the number of nonlinear parameters is set to n1. The nonlinear part is fitted using the Lagrange interpolation polynomial based on Radau points. The formula of the Lagrange interpolation polynomial is as follows:

[0082]

[0083] Where f(·) is the nonlinear function of the nonlinear part.

[0084] The output vector f(yc(k)) of the nonlinear part at the kth moment is expressed as:

[0085]

[0086] Among them, f(yc s (k)) represents any output component in the output vector f(yc(k)) of the nonlinear part.

[0087] The number n1 of the nonlinear parameters is determined by the AIC criterion, and the number n1 of the nonlinear parameters is used to define the parameters of the nonlinear part to be identified in the identification as follows:

[0088] (2) Calculate and output the sensitivity matrix of each parameter according to the parameterized state-space model obtained in step (1), and perform singular value decomposition on the obtained sensitivity matrix to obtain a singular value matrix and a right singular vector matrix; determine the number of parameters to be estimated and the number of parameters that can be fixed at initial values ​​through the singular value matrix; and convert the parameter vector into a parameter vector in which there is no correlation between the parameters by left-multiplying the parameter vector by the transpose of the right singular vector matrix.

[0089] The step (2) specifically includes the following sub-steps:

[0090] (2.1) The sensitivity matrix S is defined according to the matrices A, B and C and the output vectors yc(1),…,yc(k),…,yc(N) of the linear part at N moments. The sensitivity matrix S is:

[0091] S=[S A S B S C ];

[0092] in,

[0093] Said for in, Said for ,in,

[0094] ; is an l×1 vector, where the sth component is x s (k), and the rest of the components are 0.

[0095] (2.2) The sensitivity matrix S is subjected to singular value decomposition, and the singular value decomposition formula is as follows:

[0096] S=UΣV T ;

[0097] Among them, U is the left singular vector matrix, Σ is the singular value matrix, and V is the right singular vector matrix; the singular value matrix Σ obtained by the singular value decomposition of the sensitivity matrix is ​​an N*l row n 2 A matrix with +n*m+n*l columns.

[0098] (2.3) All parameters in the state-space model x(k) = Ax(k-1) + Bu(k-1) are written as a column vector θ: θ = [a 1,1 a 1,2 …a h,e …a n,n b 1,1 b 1,2 …b e,i …b n,m c 1,1 c 1,2 …c s,e …c l,n ] T .

[0099] Then, the column vector θ is multiplied by the transpose of the right singular vector matrix V to obtain the transformed parameter vector β, as follows:

[0100]

[0101] Among them, β g is any component of the vector β, g=1,…,g,…,n 2 +n*m+n*l.

[0102] The column vector θ can be replaced with the parameter vector β using the right singular vector matrix V, as shown in the following formula: θ=Vβ.

[0103] In the first n singular value matrix Σ 2 The values ​​of the diagonal of the block matrix with +nm+nl rows will change significantly, and the singular value matrix The values ​​on the diagonal of the block matrix from the 1st row to the qth row and from the 1st column to the qth column are not zero; the values ​​on the diagonal of the block matrix from the q+1th row to the nth column are not zero; 2 +n*m+n*l rows, q+1 columns to n 2 The absolute value of the values ​​on the diagonal of the block matrix with +n*m+n*l columns is less than 1×10 -10 .

[0104] The parameters from the 1st to the qth row of the parameter vector β are used as the parameters to be estimated in the linear part of the Wiener model, and the parameters from the q+1th to the nth row are used as the parameters to be estimated in the linear part of the Wiener model. 2 The parameters of the +n*m+n*l rows are fixed at the initial values.

[0105] (3) According to the actual situation of the system to be identified, select white noise with appropriate variance and mean as the input data, and record the system output y(k) at the kth moment as the output data for identification.

[0106] The step (3) is specifically:

[0107] According to the actual situation of the system to be identified, the first white noise with appropriate variance and mean is selected as the input vector u(k-1), and the output y(k) = [y1(k)…y s (k)…y l (k)] T As the data for identification, the y(k) is defined as y(k)=f(yc(k))+w(k), where w(k)=[w1(k)…w s (k)…w l (k)] T is the second white noise, where w s (k) represents the second white noise component at any k-th moment.

[0108] (4) Calculate the characteristic polynomial of the state vector matrix according to the parameterized state space model, and use the Jury criterion to calculate the stability constraint condition; under the stability constraint condition, use the input data and output data for identification obtained in step (3) and the nonlinear optimization method according to the criterion of minimizing the mean square value of the output error to obtain the parameters of the matrices A, B and C and the nonlinear part parameters z1,…,z in the state space model. n1 The value of .

[0109] The step (4) specifically includes the following sub-steps:

[0110] (4.1) According to the parameterized state space model, the characteristic polynomial of the matrix A is calculated. The characteristic polynomial is in the following form:

[0111] λ n +h1λ n-1 +…+h n-1 λ+h n =0;

[0112] Among them, h1,…,h e ,…,h n are the coefficients of the characteristic polynomial. Any coefficient h e Both are determined by parameter a 1,1 ,…,a h,e ,…,a n,n The root λ of the characteristic polynomial is the pole of the system to be identified. In order to make the identified system stable, the modulus length of the root λ needs to be less than 1. The characteristic polynomial is applied to the coefficient criteria in the Jury table to obtain the constraint conditions that make the system stable. The form of the Jury table is as follows:

[0113]

[0114]

[0115] The first and second rows of the table consist of 1 and coefficients h1,…,h n The coefficients starting from the third row in the table are calculated using the following formula:

[0116]

[0117] Among them, t=0,1,…,t,…,n; j=2,3,…,j,…,n-2; v=0,1,…,v,…,nj;

[0118] Calculate the 3 coefficients of the 2n-3th row r1 n-2 and until.

[0119] Using the coefficients in the Jury table, the conditions for system stability are obtained as follows:

[0120]

[0121] (4.2) Parameter estimation for a Winer system with measurement noise is to use nonlinear optimization methods to find an estimate of the output error criterion function that minimizes the following and

[0122]

[0123] The constraints for minimizing the output error criterion function are:

[0124]

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] In estimating the parameters of matrices A, B, and C and the nonlinear part parameters z1,…,z n1 When , starting from the initial value, using nonlinear optimization method, under the premise of satisfying the given constraints, iteratively update The parameter estimates of the matrices A, B, and C whose values ​​are reduced and and nonlinear part parameters Estimated value of Until I can no longer find The estimated value of and at this time and is the final parameter estimate; Indicates that the current estimate is used and The outputs y(1),…,y(k),…,y(N) at N moments are used to simulate and calculate the value of the nonlinear output; J(·) represents the objective function of optimization. When estimating parameters, it is to find the estimated value that minimizes the objective function. and Indicates that the current estimate is used and Calculated estimates; and They represent the estimated parameter matrices respectively; represents the state vector estimated at the kth moment; is the output vector representing the linear part of the estimate at the kth moment, is the output component representing the linear part of the estimate at any k-th moment.

[0131] In the above formula, each parameter in the matrices A, B and C is expressed according to the β parameter, and the initial value is given during the optimization solution.

[0132] After the estimation is completed, the parameters of the matrices A, B and C in the state space model and the nonlinear part parameters z1,…,z n1 The value of .

[0133] Example 2

[0134] The present invention provides a method for reducing the number of parameters to be estimated based on an output parameter sensitivity matrix in nonlinear Wiener model identification, comprising the following steps:

[0135] (1) According to the present embodiment, based on the order n=4 of the system to be identified, the linear part of the Wiener model is parameterized in the form of a state space model to determine the sizes of matrices A, B and C; the nonlinear part is fitted using the Lagrange interpolation method based on Radau points, and the number n1 of nonlinear parameters to be identified is determined using the AIC criterion.

[0136] The step (1) specifically includes the following sub-steps:

[0137] (1.1) According to the order n of the system to be identified, the linear part of the Wiener model is parameterized by the state space model. The form of the parameterized state space model is:

[0138]

[0139]

[0140] In this embodiment, m=1, l=1.

[0141] (1.2) First, the number of nonlinear parameters is set to n1. The nonlinear part is fitted using the Lagrange interpolation polynomial based on Radau points. The formula of the Lagrange interpolation polynomial is as follows:

[0142]

[0143] Wherein, f(·) is a nonlinear function of the nonlinear part, and f(·) has an inverse function. The output of the linear system is the inverse function of f(·). Since the hierarchical iterative method can only be used when the inverse function can be written as a polynomial, in order to facilitate comparison with the method proposed in the present invention, the inverse function is written in the form of a polynomial, as shown below:

[0144] yc(k)=f -1 (y(k))=y(k)+r1y(k) 2 +…+r n-1 y(k) n ;

[0145] In this embodiment, the order of the inverse function polynomial is 3rd order, and r1=0.08, r2=0.05.

[0146] The number n1 of the nonlinear parameters is determined by the AIC criterion, and the number n1 of the nonlinear parameters is used to define the parameters of the nonlinear part to be identified in the identification as follows: is the selected Radau point, It is the value of the nonlinear function corresponding to each Radau point, and is also the parameter of the nonlinear part that needs to be identified in the identification.

[0147] n1 is determined by the following AIC criterion formula:

[0148]

[0149] MSE is the mean square error between the true output and the model output, and 8+n1 is the total number of parameters that need to be estimated. Let n1=4, n1=5, n1=6, n1=7, n1=8, and n1=9 calculate the value of AIC, and the values ​​obtained are shown in Table 1:

[0150] Table 1 AIC values ​​under different n1

[0151] <![CDATA[n1]]> 4 5 6 7 8 9 AIC 0.0139 0.0247 0.0106 0.0105 0.0113 0.0119

[0152] It can be seen from Table 1 that when n1=7, the value of AIC is the smallest, so n1=7 is selected.

[0153] When n1=7, the Radau points in the range [0,1] are:

[0154] d1=0;

[0155] d2=0.3980985705146874e-01;

[0156] d3=0.1980134178736082e+00;

[0157] d4=0.4379748102473861e+00;

[0158] d5=0.6954642733536361e+00;

[0159] d6=0.9014649142011736e+00;

[0160] d7=0.1000000000000000e+01;

[0161] z1…z7 are the values ​​of the nonlinear function corresponding to each Radau point, and are also the parameters of the nonlinear part that needs to be identified in the identification.

[0162] (2) Calculate and output the sensitivity matrix of each parameter according to the parameterized state-space model obtained in step (1), and perform singular value decomposition on the obtained sensitivity matrix to obtain a singular value matrix and a right singular vector matrix; determine the number of parameters to be estimated and the number of parameters that can be fixed at initial values ​​through the singular value matrix; and convert the parameter vector into a parameter vector in which there is no correlation between the parameters by left-multiplying the parameter vector by the transpose of the right singular vector matrix.

[0163] The step (2) specifically includes the following sub-steps:

[0164] (2.1) The sensitivity matrix S is defined based on the matrices A, B and C and the output vectors yc(1),…,yc(k),…,yc(N) of the linear part at N moments.

[0165] (2.2) The sensitivity matrix S is subjected to singular value decomposition, and the singular value decomposition formula is as follows:

[0166] S=UΣV T ;

[0167] Among them, the singular value matrix Σ obtained by the singular value decomposition of the sensitivity matrix is ​​a matrix of N rows and 24 columns. The values ​​of the diagonal of the block matrix in the first 24 rows of the matrix will have an obvious change. The first 8 values ​​are non-zero values, and the last 16 values ​​are almost zero values.

[0168] (2.3) Write all parameters in the state-space model x(k)=Ax(k-1)+Bu(k-1) as a column vector θ.

[0169] Then, the column vector θ is multiplied by the transpose of the right singular vector matrix V to obtain the transformed parameter vector β.

[0170] The column vector θ can be replaced with the parameter vector β using the right singular vector matrix V, as follows: θ = Vβ;

[0171] Singular value matrix Σ∈R N×24The values ​​on the diagonal of the block matrix from the 1st row to the q=8th row and the 1st column to the q=8th column are not zero; the absolute values ​​of the values ​​on the diagonal of the block matrix from the q+1=9th row to the 24th row and the q+1=9th column to the 24th column are less than 1×10 -10 ;

[0172] The parameters from the 1st to the qth row of the parameter vector β are taken as the parameters that need to be estimated in the linear part of the Wiener model, and the parameters from the q+1th to the 24th row are fixed at the initial values.

[0173] (3) According to the actual situation of the system to be identified, select white noise with appropriate variance and mean as the input data, and record the system output y(k) at the kth moment as the output data for identification.

[0174] Select the first white noise with a suitable mean of 0 and variance of 1 as the input data u(k-1), record the output y(k) of the system at the kth moment as the output data for identification, and have a second white noise with a variance of 0.0001 and a mean of 0 during output measurement.

[0175] (4) Calculate the characteristic polynomial of the state vector matrix according to the parameterized state space model, and use the Jury criterion to calculate the stability constraint condition; under the stability constraint condition, use the input data and output data for identification obtained in step (3) and the nonlinear optimization method to solve the parameters of the matrices A, B and C and the nonlinear part parameters in the state space model according to the criterion of minimizing the mean square value of the output error. The value of .

[0176] (4.1) According to the parameterized state space model, the characteristic polynomial of the matrix A is calculated. The characteristic polynomial is in the following form:

[0177] λ 4 +h1λ 3 +h2λ 2 +h3λ+h4=0;

[0178] Among them, h1,…,h e ,…,h n are the coefficients of the characteristic polynomial, as follows:

[0179] h1=-a 11 -a 22 -a 33 -a 44 ;

[0180]

[0181]

[0182]

[0183] The root of the polynomial is the pole of the system. In order to make the identified system stable, the root of the polynomial needs to be within the unit circle. For this purpose, the stability constraint conditions need to be calculated using the Jury criterion, as follows:

[0184]

[0185] (4.2) Parameter estimation for a Winer system with measurement noise is to use nonlinear optimization methods to find an estimate of the output error criterion function that minimizes the following and

[0186]

[0187] The constraints for minimizing the output error criterion function are:

[0188]

[0189]

[0190]

[0191]

[0192]

[0193]

[0194] When estimating the parameters of the matrices A, B and C and the nonlinear part parameters z1,…,z7, starting from the initial values, a nonlinear optimization method is used to iterate and update continuously under the premise of satisfying the given constraints. The parameter estimates of the matrices A, B, and C whose values ​​are reduced and and the estimated values ​​of the nonlinear part parameters z1,…,z7 Until I can no longer find The estimated value of and z1,…,z7, at this time and z1,…,z7 are the final parameter estimates; among them, Indicates that the current estimate is used and The outputs y(1),…,y(k),…,y(N) at N moments are used to simulate and calculate the value of the nonlinear output; J(·) represents the objective function of optimization. When estimating parameters, it is to find the estimated value that minimizes the objective function. and z1,…,z7; Indicates that the current estimate is used and the estimated values ​​calculated from z1,…,z7; and They represent the estimated parameter matrices respectively; represents the state vector estimated at the kth moment; is the output vector representing the linear part of the estimate at the kth moment,

[0195] In the above formula, each parameter in the matrices A, B and C is expressed according to the β parameter, and the initial value is given during the optimization solution.

[0196] After the estimation is completed, the parameters of the matrices A, B and C in the state space model and the values ​​of the nonlinear part parameters z1,…,z7 are obtained.

[0197] In this embodiment, the example used in Yang Qipei's paper is used for comparison with the hierarchical iteration method proposed by him. The experimental results are shown in Tables 2 and 3.

[0198] Table 2 Comparison of linear part system pole estimation

[0199]

[0200] Table 3 Comparison of mean square error of identification system output

[0201]

[0202] According to Table 1 and Table 2, the method provided by the present invention can have a good identification effect when the nonlinear function has an inverse function and the inverse function can be written in a polynomial form. This embodiment mainly illustrates the situation that the hierarchical iterative method can handle, and the present invention can also have a good identification effect.

[0203] Example 3

[0204] The present invention provides a method for reducing the number of parameters to be estimated based on an output parameter sensitivity matrix in nonlinear Wiener model identification, comprising the following steps:

[0205] (1) According to the present embodiment, based on the order n=2 of the system to be identified, the linear part of the Wiener model is parameterized in the form of a state space model to determine the size of matrices A, B and C; the nonlinear part is fitted using the Lagrange interpolation method based on Radau points, and the number n1 of nonlinear parameters to be identified is determined using the AIC criterion. The step (1) specifically includes the following sub-steps:

[0206] (1.1) According to the differential equation of the valve system of this embodiment:

[0207]

[0208]

[0209] According to the order of the system to be identified n = 2, the linear part of the Wiener model is parameterized by the state space model. The form of the parameterized state space model is:

[0210]

[0211]

[0212] In this embodiment, m=1, l=1.

[0213] (1.2) First, the number of nonlinear parameters is set to n1. The nonlinear part is fitted using the Lagrange interpolation polynomial based on Radau points. The formula of the Lagrange interpolation polynomial is as follows:

[0214]

[0215] Where f(·) is the nonlinear function of the nonlinear part, and the formula is as follows:

[0216]

[0217] The number n1 of the nonlinear parameters is determined by the AIC criterion, and the number n1 of the nonlinear parameters is used to define the parameters of the nonlinear part to be identified in the identification as follows:

[0218] n1 is determined by the following AIC criterion formula:

[0219]

[0220] Where N is the number of data, MSE is the mean square error between the true output and the model output, and 4+n1 is the total number of parameters that need to be estimated. Let n1=3, n1=4, n1=5, n1=6, n1=7, and n1=8 calculate the AIC values, and the values ​​obtained are shown in Table 4:

[0221] Table 4 AIC values ​​under different n1

[0222] <![CDATA[n1]]> 3 4 5 6 7 8 AIC 0.0211 0.0057 0.0064 0.0070 0.0074 0.0081

[0223] It can be seen from Table 4 that when n1=4, the value of AIC is the smallest, so n1=4 is selected.

[0224] When n1=4, the Radau points in the range [0,1] are:

[0225] d1=0;

[0226] d2=0.1550510257216822e+00;

[0227] d3=0.6449489742783178e+00;

[0228] d4=0.1000000000000000e+01;

[0229] z1…z4 are the values ​​of the nonlinear function corresponding to each Radau point, and are also the parameters of the nonlinear part that needs to be identified in the identification.

[0230] (2) Calculate and output the sensitivity matrix of each parameter according to the parameterized state-space model obtained in step (1), and perform singular value decomposition on the obtained sensitivity matrix to obtain a singular value matrix and a right singular vector matrix; determine the number of parameters to be estimated and the number of parameters that can be fixed at initial values ​​through the singular value matrix; and convert the parameter vector into a parameter vector in which there is no correlation between the parameters by left-multiplying the parameter vector by the transpose of the right singular vector matrix.

[0231] The step (2) specifically includes the following sub-steps:

[0232] (2.1) Define the sensitivity matrix S based on the matrices A, B and C and the output vectors yc(1),…,yc(k),…,yc(N) of the linear part at N moments;

[0233] (2.2) The calculated sensitivity matrix S is subjected to singular value decomposition. The formula of singular value decomposition is as follows:

[0234] S=UΣV T ;

[0235] Among them, the singular value matrix obtained by the singular value decomposition of the sensitivity matrix is ​​a matrix of N rows and 8 columns. The values ​​of the diagonal of the block matrix in the first 8 rows of the matrix will have an obvious change. The first 4 values ​​are non-zero values, and the last 4 values ​​are almost zero values.

[0236] (2.3) Write all parameters in the state-space model x(k)=Ax(k-1)+Bu(k-1) as a column vector θ.

[0237] Then, the column vector θ is multiplied by the transpose of the right singular vector matrix V to obtain the transformed parameter vector β.

[0238] The column vector θ can be replaced with the parameter vector β using the right singular vector matrix V, as follows: θ = Vβ;

[0239] Singular value matrix Σ∈R N×8The values ​​on the diagonal of the block matrix from the 1st row to the q=4th row and the 1st column to the q=4th column are not zero; the absolute values ​​of the values ​​on the diagonal of the block matrix from the q+1=5th row to the 8th row and the q+1=5th column to the 8th column are less than 1×10 -10 ;

[0240] The parameters from the 1st to the qth row of the parameter vector β are taken as the parameters that need to be estimated in the linear part of the Wiener model, and the parameters from the q+1th to the 8th row are fixed at the initial values.

[0241] (3) According to the actual situation of the system to be identified, select white noise with appropriate variance and mean as the input data, and record the system output y(k) at the kth moment as the output data for identification.

[0242] Select the first white noise with a suitable mean of 0.4 and a variance of 0.5 as the input data u(k-1), record the output y(k) of the system at the kth moment as the output data for identification, and have a second white noise with a variance of 0.0001 and a mean of 0 during output measurement.

[0243] (4) Calculate the characteristic polynomial of the state vector matrix according to the parameterized state space model, and use the Jury criterion to calculate the stability constraint condition; under the stability constraint condition, use the input data and output data for identification obtained in step (3) and the nonlinear optimization method to solve the parameters of the matrices A, B and C and the nonlinear part parameters in the state space model according to the criterion of minimizing the mean square value of the output error. The value of .

[0244] The step (4) is specifically:

[0245] (4.1) According to the parameterized state space model, the characteristic polynomial of matrix A is calculated. The polynomial is in the following form:

[0246] λ 2 +h1λ+h2=0;

[0247] Where h1 and h2 are the coefficients of the characteristic polynomial, as follows:

[0248]

[0249] h2=a 11 a 22 -a 12 a 21 ;

[0250] The root of the polynomial is the pole of the system. In order to make the identified system stable, the root of the polynomial needs to be within the unit circle. For this purpose, the stability constraint conditions need to be calculated using the Jury criterion, as follows:

[0251]

[0252] (4.2) Parameter estimation for a Winer system with measurement noise is to use nonlinear optimization methods to find an estimate of the output error criterion function that minimizes the following and

[0253]

[0254] The constraints for minimizing the output error criterion function are:

[0255]

[0256]

[0257]

[0258]

[0259]

[0260]

[0261] In estimating the parameters of matrices A, B, and C and the nonlinear part parameters When , starting from the initial value, using nonlinear optimization method, under the premise of satisfying the given constraints, iteratively update The parameter estimates of the matrices A, B, and C whose values ​​are reduced and and nonlinear part parameters Estimated value of Until I can no longer find The estimated value of and at this time and is the final parameter estimate; Indicates that the current estimate is used and The outputs y(1),…,y(k),…,y(N) at N moments are used to simulate and calculate the value of the nonlinear output; J(·) represents the objective function of optimization. When estimating parameters, it is to find the estimated value that minimizes the objective function. and Indicates that the current estimate is used and Calculated estimates; and They represent the estimated parameter matrices respectively; represents the state vector estimated at the kth moment; is the output vector representing the linear part of the estimate at the kth moment,

[0262] In the above formula, each parameter in the matrices A, B and C is expressed according to the β parameter, and the initial value is given during the optimization solution.

[0263] After the estimation is completed, the parameters of matrices A, B and C in the state space model and the nonlinear part parameters are obtained The value of .

[0264] In this embodiment, the experimental results are shown in Tables 5 and 6 in comparison with the hierarchical iterative method. The hierarchical iterative method cannot solve a stable system, so Table 6 only gives the experimental results of the method proposed in the present invention.

[0265] Table 5 Comparison of linear part system pole estimation

[0266]

[0267]

[0268] Table 6 Comparison of mean square error of identification system output

[0269] A method to reduce the number of parameters to be estimated based on parameter sensitivity matrix Output mean square error <![CDATA[3.3470×10 -4 ]]>

[0270] According to Table 5 and Table 6, the method provided by the present invention can well identify the wiener valve model and has good identification accuracy. This embodiment mainly illustrates that the present invention can also achieve good identification effect in situations that the hierarchical iterative method cannot handle.

[0271] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification, characterized in that: The following steps are involved: (1) Based on the order of the system to be identified, the linear part of the Wiener model is parameterized in the form of a state space model to determine the size of matrices A, B, and C; the nonlinear part is fitted using the Lagrange interpolation method based on Radau points, and the number of nonlinear parameters to be identified, n1, is determined using the AIC criterion; (2) calculating and outputting a sensitivity matrix for each parameter according to the parameterized state space model obtained in step (1), and performing singular value decomposition on the obtained sensitivity matrix to obtain a singular value matrix and a right singular vector matrix; determining the number of parameters to be estimated and the number of parameters that can be fixed at initial values ​​through the singular value matrix; and converting the parameter vector into a parameter vector in which there is no correlation between the parameters by left-multiplying the transpose of the right singular vector matrix; (3) According to the actual situation of the system to be identified, select white noise with appropriate variance and mean as input data, and record the output y(k) of the system at the kth moment as the output data for identification; (4) Calculate the characteristic polynomial of the state vector matrix according to the parameterized state space model, and use the Jury criterion to calculate the stability constraint condition; under the stability constraint condition, use the input data and output data for identification obtained in step (3) and the nonlinear optimization method to solve the parameters of the matrices A, B and C and the nonlinear part parameters in the state space model according to the criterion of minimizing the mean square value of the output error. The value of .

2. A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification according to claim 1, characterized in that: The step (1) specifically includes the following sub-steps: (1.1) According to the order n of the system to be identified, the linear part of the Wiener model is parameterized by the state space model. The form of the parameterized state space model is: x(k)=A*x(k-1)+B*u(k-1); yc(k)=C*x(k); Among them, x(k-1) is the state vector at the k-1th moment; u(k-1) is the input vector at the k-1th moment; x(k) is the state vector at the kth moment; A, B and C are constant matrices respectively; yc(k) is the output vector of the linear part at the kth moment; The state vector x(k-1) is defined as: The state vector x(k) is defined as: The input vector u(k-1) is defined as: The output vector yc(k) of the linear part is defined as: Among them, x e (k-1) represents the state vector component of any k-1th moment, e = 1, ..., e, ..., n; x e (k) represents the state vector component at any k-th moment; u i (k-1) represents any input component at the k-1th moment, i = 1, ..., i, ..., m, m represents the total number of input components of the system to be identified; yc s (k) represents the output component of the linear part at any k-th moment, l represents the total number of l output components of the system to be identified, s=1,…,s,…,l; The size of matrix A is determined to be n*n by the order n of the system to be identified, and the size of matrix B is determined to be n*m ​​by the order n of the system to be identified and the number m of the input components of the system to be identified. Finally, the size of matrix C is determined to be l*n by the order n of the system to be identified and the number l of the output components of the system to be identified. Then, matrix A is defined as Define the matrix B as Define the matrix C as Among them, a h,e represents any constant parameter in the matrix A, h=1,…,h,…,n; b e,i represents any constant parameter in matrix B; c s,e Represents any constant parameter in the matrix C; (1.2) First, the number of nonlinear parameters is set to n1. The nonlinear part is fitted using the Lagrange interpolation polynomial based on Radau points. The formula of the Lagrange interpolation polynomial is as follows: Among them, f(·) is the nonlinear function of the nonlinear part; The output vector f(yc(k)) of the nonlinear part at the kth moment is expressed as: Among them, f(yc s (k)) represents any output component in the output vector f(yc(k)) of the nonlinear part; The number n1 of the nonlinear parameters is determined by the AIC criterion, and the number n1 of the nonlinear parameters is used to define the parameters of the nonlinear part to be identified in the identification as follows:

3. A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification according to claim 2, characterized in that: The step (2) specifically includes the following sub-steps: (2.1) The sensitivity matrix S is defined according to the matrices A, B and C and the output vectors yc(1),…,yc(k),…,yc(N) of the linear part at N moments. The sensitivity matrix S is: S=[S A S B S C ]; in, Said for in, Said for in, Said is an l×1 vector, where the sth component is x s (k), the rest of the components are 0; (2.2) The sensitivity matrix S is subjected to singular value decomposition, and the singular value decomposition formula is as follows: S=UΣV T ; Among them, U is the left singular vector matrix, Σ is the singular value matrix, and V is the right singular vector matrix; the singular value matrix Σ obtained by the singular value decomposition of the sensitivity matrix is ​​an N*l row n 2 +n*m+n*l columns of matrix; (2.3) All parameters in the state-space model x(k) = Ax(k-1) + Bu(k-1) are written as a column vector θ: θ = [a 1, 1a 1,2 …a h,e …a n,n b 1,1 b 1,2 …b e,i …b n,m c 1,1 c 1,2 …c s,e …c l,n ] T ; Then, the column vector θ is multiplied by the transpose of the right singular vector matrix V to obtain the transformed parameter vector β, as follows: Among them, β g is any component of the vector β, g=1,…,g,…,n 2 +n*m+n*l; The column vector θ can be replaced with the parameter vector β using the right singular vector matrix V, as follows: θ = Vβ; Singular value matrix The values ​​on the diagonal of the block matrix from the 1st row to the qth row and from the 1st column to the qth column are not zero; the values ​​on the diagonal of the block matrix from the q+1th row to the nth column are not zero; 2 +n*m+n*l rows, q+1 columns to n 2 The absolute value of the values ​​on the diagonal of the block matrix with +n*m+n*l columns is less than 1×10 -10 ; The parameters from the 1st to the qth row of the parameter vector β are used as the parameters to be estimated in the linear part of the Wiener model, and the parameters from the q+1th to the nth row are used as the parameters to be estimated in the linear part of the Wiener model. 2 The parameters of the +n*m+n*l rows are fixed at the initial values.

4. A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification according to claim 3, characterized in that: The step (3) is specifically: According to the actual situation of the system to be identified, select the first white noise with appropriate variance and mean as the input data u(k-1), and record the output y(k)=[y1(k)…y s (k)…y l (k)] T As the output data for identification, the y(k) is defined as y(k)=f(yc(k))+w(k), where w(k)=[w1(k)…w s (k)…w l (k)] T is the second white noise, where w s (k) represents the second white noise component at any k-th moment.

5. A method for reducing the number of parameters to be estimated based on the output parameter sensitivity matrix in nonlinear Wiener model identification according to claim 4, characterized in that: The step (4) specifically includes the following sub-steps: (4.1) According to the parameterized state space model, the characteristic polynomial of the matrix A is calculated. The characteristic polynomial is in the following form: l n +h1λ n-1 +…+h n-1 λ+h n =0; Among them, h1,…,h e ,…,h n are the coefficients of the characteristic polynomial. Any coefficient h e Both are determined by parameter a 1,1 ,…,a h,e ,…,a n,n The root λ of the characteristic polynomial is the pole of the system to be identified. In order to make the identified system stable, the modulus length of the root λ needs to be less than 1. The characteristic polynomial is applied to the coefficient criteria in the Jury table to obtain the constraint conditions that make the system stable. The form of the Jury table is as follows: The first and second rows of the table consist of 1 and coefficients h1,…,h n The coefficients starting from the third row in the table are calculated using the following formula: Among them, t=0,1,…,t,…,n; j=2,3,…,j,…,n-2; v=0,1,…,v,…,nj; Calculate the 3 coefficients of the 2n-3th row and until; Using the coefficients in the Jury table, the conditions for system stability are obtained as follows: (4.2) Parameter estimation for a Winer system with measurement noise is to use nonlinear optimization methods to find an estimate of the output error criterion function that minimizes the following and The constraints for minimizing the output error criterion function are: In estimating the parameters of matrices A, B, and C and the nonlinear part parameters When , starting from the initial value, using nonlinear optimization method, under the premise of satisfying the given constraints, iteratively update The parameter estimates of the matrices A, B, and C whose values ​​are reduced and and nonlinear part parameters Estimated value of Until I can no longer find The estimated value of and at this time and is the final parameter estimate; Indicates that the current estimate is used and The outputs y(1),…,y(k),…,y(N) at N moments are used to simulate and calculate the value of the nonlinear output; J(·) represents the objective function of optimization. When estimating parameters, it is to find the estimated value that minimizes the objective function. and Indicates that the current estimate is used and Calculated estimates; and They represent the estimated parameter matrices respectively; represents the state vector estimated at the kth moment; is the output vector representing the linear part of the estimate at the kth moment, is the output component representing the linear part estimated at any k-th moment; In the above formula, each parameter in the matrices A, B and C is expressed according to the β parameter, and the initial value is given during the optimization solution; After the estimation is completed, the parameters of matrices A, B and C in the state space model and the nonlinear part parameters are obtained The value of .

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