CN111930014B - Dynamic and static data hybrid-driven Hammerstein nonlinear industrial system simple grey box space identification method - Google Patents

Dynamic and static data hybrid-driven Hammerstein nonlinear industrial system simple grey box space identification method Download PDF

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CN111930014B
CN111930014B CN202010814214.8A CN202010814214A CN111930014B CN 111930014 B CN111930014 B CN 111930014B CN 202010814214 A CN202010814214 A CN 202010814214A CN 111930014 B CN111930014 B CN 111930014B
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侯杰
黄祎
谢昭贤
陈丰伟
刘井响
李鹏华
朱智勤
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Chongqing University of Post and Telecommunications
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Abstract

The invention relates to a dynamic and static data hybrid-driven Hammerstein nonlinear industrial system simple grey box space identification method, and belongs to the field of control theory and control engineering nonlinear system identification. The method comprises the following steps: s1: collecting system dynamic data and static data; s2: selecting and processing a Hammerstein nonlinear system model to obtain a prediction model; s3: constructing a dynamic simple model of a Hammerstein nonlinear system; s4: constructing a Hammerstein nonlinear system static simple model; s5: and solving the system parameter fusion identification by using a hierarchical Lagrange optimal weighting method. The method simultaneously adopts dynamic and static data, avoids estimating additional intermediate parameters based on a decomposed reduced model, reduces the variance of the estimation model and improves the precision of the model.

Description

Dynamic and static data hybrid-driven Hammerstein nonlinear industrial system simple gray box space identification method
Technical Field
The invention belongs to the field of control theory and control engineering nonlinear system identification, and relates to a dynamic/static data hybrid-driven Hammerstein nonlinear industrial system simple grey box space identification method.
Background
The Hammerstein nonlinear system is a modularized nonlinear system, consists of a static nonlinear module and a linear dynamic module, can effectively reflect the system characteristics of most practical input nonlinear industrial processes, and has been widely applied to describing practical industrial processes, such as a blast furnace iron-making system, a battery management system, a wireless power transmission system and the like. During the past decades, much research work has been carried out on Hammerstein nonlinear system identification, and a variety of mature identification methods, such as maximum likelihood estimation, least squares, stochastic approximation, and Subspace Identification (SIMs), have been proposed in succession.
The state space can effectively describe the internal dynamic characteristics of the system, so that the design of the controller is facilitated. The Hammerstein nonlinear system identification with the state space linear dynamic module becomes a research hotspot of the control field, and the corresponding subspace identification method obtains extensive research of the control field. In the last two decades, many Hammerstein nonlinear system subspace identification methods have been proposed in succession, and existing methods include two broad categories, namely non-iterative methods and iterative methods. (1) Non-iterative methods include a subspace identification Method (MOESP) based on a multivariate output error state space, a subspace method (N4SID) based on a numerical subspace state space system, and a subspace method based on a parity space. (2) The iteration method comprises a subspace identification method based on fixed point iteration and a subspace identification method based on Gaussian-Newton iteration. The method is well applied to mechanical systems, wireless power transmission systems and process industrial systems. However, the existing method is all dynamic data driven, namely only dynamic data is adopted to carry out Hammerstein nonlinear system model identification, and belongs to a black box space identification method. Because the dynamic data only contains high-frequency system information and can not provide low-frequency information of the system, the existing black box system identification method is difficult to establish a high-precision system model.
Disclosure of Invention
In view of the above, the present invention provides a novel gray box reduced subspace identification method driven by dynamic/static data in a hybrid manner to accurately identify a Hammerstein nonlinear system, so as to overcome the problem of low model accuracy caused by the fact that the existing black box space identification method cannot provide low-frequency information of the system.
In order to achieve the purpose, the invention provides the following technical scheme:
a Hammerstein nonlinear industrial system simple gray box space identification method driven by dynamic and static data in a mixed mode specifically comprises the following steps:
s1: collecting system dynamic data and static data;
s2: selecting and processing a Hammerstein nonlinear system model to obtain a prediction model;
s3: constructing a dynamic simple model of a Hammerstein nonlinear system;
s4: constructing a Hammerstein nonlinear system static simple model;
s5: and solving the system parameter fusion identification by using a hierarchical Lagrange optimal weighting method.
Further, the step S1 specifically includes: data acquisition is carried out on the actual industrial process to obtain the dynamic data of the system
Figure BDA0002632095520000021
And static data
Figure BDA0002632095520000022
Wherein u (t), y (t) are dynamic input/output data, N is the number of dynamic input/output data,
Figure BDA0002632095520000023
the data are static input and output data respectively, and M is the number of the static input and output data.
Further, the step S2 specifically includes the following steps:
s21: for the actual industrial process with the nonlinear characteristic, a Hammerstein nonlinear model with a state space module is adopted for description, and the specific form is as follows:
Figure BDA0002632095520000024
f(t)＝ωz(t) (2)
ω＝[ω1，ω2，…，ωr] (3)
z(t)＝[f1(u(t))，f2(u(t))，…，fr(u(t))]T (4)
wherein the content of the first and second substances,
Figure BDA0002632095520000025
u (t) E R, y (t) E R, e (t) E R respectively represent a state vector, collected input data, collected output data and an innovation vector. (A, B, C, K) represents a system matrix. n isxRepresenting model order, innovation being zero mean Gaussian noise, nonlinear input function f (t) epsilon R being known basis function fi(u (t)) a linear combination of ε R,. omega.ie.R denotes unknownAnd (4) the coefficient.
S22: the selected model is processed to obtain a prediction model so as to facilitate the subsequent model parameter identification, and the specific model form is as follows:
Figure BDA0002632095520000026
wherein:
Figure BDA0002632095520000027
while the following general assumptions are made about the legibility of the system:
the a1 linear system is observable and realizable.
A2
Figure BDA0002632095520000028
Lies within the unit circle.
The first non-zero element of A3 omega is positive, | omega | magnetism not calculation2＝1(||.||2Define the euclidean norm).
The invention estimates unknown parameters (A, B, C, omega) from dynamic and steady state data, and can improve the precision of estimating model parameters only by dynamic data.
Further, in step S3, constructing a Hammerstein nonlinear system dynamic simple model, specifically including the following steps:
s31: carrying out iteration processing on the innovation model for p times to obtain a new state equation:
Figure BDA0002632095520000031
wherein:
Figure BDA0002632095520000032
Figure BDA0002632095520000033
zp(t)＝[zT(t-p)，zT(t-p+1)，…，zT(t-1)]T (9)
yp(t)＝[yT(t-p)，yT(t-p+1)，…，yT(t-1)]T (10)
it should be noted that p can be arbitrarily selected according to the actual system characteristics, and is generally selected within the range of 10 to 60.
S32: bringing the new state equation into output according to
Figure BDA0002632095520000034
The output y (t) of the system is rewritten to obtain a new system output function:
(t)＝P0(t)θ0+P1(t)θ1+e(t) (11)
wherein:
P0(t)＝[zT(t-1)，zT(t-2)，…，zT(t-p)] (12)
P1(t)＝[yT(t-1)，yT(t-2)，…，yT(t-p)] (13)
Figure BDA0002632095520000035
Figure BDA0002632095520000036
due to theta0The number of unknown parameters is pr, and the system parameters
Figure BDA0002632095520000037
And the number of nonlinear parameter vectors ω is p + r. For most systems, including highly non-linear models, the over-parameterized model (11) and system parameters
Figure BDA0002632095520000038
Compared to ω, it contains redundant parameters, which can lead to unnecessarily high variance and model distortion, especially when the system uses small and noisy data sets.
S33: to theta0And (3) carrying out decomposition processing, rewriting the parameterized model (11) to obtain two dynamic simplified models:
y(t)＝P2(t)θ2+P1(t)θ1+e(t) (16)
＝P3(t)ωT+P1(t)θ1+e(t) (17)
wherein:
P2(t)＝[ωz(t-1)，ωz(t-2)，…，ωz(t-p)] (18)
Figure BDA0002632095520000039
Figure BDA0002632095520000041
and theta1And theta2Including all extended markov parameters (i ═ 1, …, p), as follows:
Figure BDA0002632095520000042
Figure BDA0002632095520000043
the corresponding system markov parameter is defined as:
hi＝CAi-1K (23)
gi＝CAi-1B (24)
it should be noted that such decomposition processing can avoid the estimation of redundant parameters, thereby improving the estimation accuracy.
S34: performing matrixing processing on the dynamic data and the dynamic reduction model:
Y＝[y(1)，y(2)，…，y(N)]T (25)
E＝[e(1)，e(2)，…，e(N)] (26)
Figure BDA0002632095520000044
Figure BDA0002632095520000045
Figure BDA0002632095520000046
Figure BDA0002632095520000047
and (3) performing iterative operation on (11), (16) and (17) on the basis of matrixing to obtain a new matrixing dynamic reduced model as follows:
Y＝φ0θ0+φ1θ1+E (31)
Y＝φ2θ2+φ1θ1+E (32)
Y＝φ3ωT+φ1θ1+E (33)
thus, a linear extended Markov parameter vector θ is estimated using a dynamic reduced model (32)2(ii) a Estimating a non-linear parameter ω with a dynamic reduced model (33); estimating redundant parameters, Markov parameters theta, in extensions (32) and (33) with an auxiliary model (31)1。
Further, in step S4, constructing a Hammerstein nonlinear system static simple model, specifically including the following steps:
s41: corresponding obtained static state outputThe input and output are substituted into the formula (11), and the steady state output is obtained after the arrangement
Figure BDA0002632095520000048
Another expression of (1):
Figure BDA0002632095520000049
wherein:
Figure BDA00026320955200000410
Figure BDA00026320955200000411
Figure BDA00026320955200000412
s42: performing matrixing processing on the static data:
Figure BDA0002632095520000051
Figure BDA0002632095520000052
obtaining a consistent least squares solution for steady state parameters G and Q:
Figure BDA0002632095520000053
s43: since the functional expressions (36) of the parameters G and (37) of Q contain over-parameterized model parameters, the over-parameterized model (37) is rewritten into two static reduced models:
Q＝K1ωT (41)
Figure BDA0002632095520000054
wherein:
Figure BDA0002632095520000055
K2＝ωi[1，…，1] (44)
wherein Q isiIs the ith parameter of Q, and can be arbitrarily selected from Q;
estimating a nonlinear parameter vector omega by using a static reduced model (41); estimating a linear extended Markov parameter vector θ using a static reduced model (42)2。
Further, in step S5, solving the system parameter fusion identification by using a hierarchical lagrangian optimal weighting method specifically includes the following steps:
s51: estimating theta in an auxiliary model (31) using a projection method1；
Orthographically projecting the output Y in the dynamic reduced model (31) to phi0Further performing least squares estimation to obtain theta1The estimated values of (c) are as follows:
Figure BDA0002632095520000056
wherein:
Figure BDA0002632095520000057
Figure BDA0002632095520000061
Figure BDA0002632095520000062
wherein, INAn identity matrix with dimension N;
s52: constructing a multi-regularization framework based on a weighting matrix reduction model:
Figure BDA0002632095520000063
Figure BDA0002632095520000064
wherein, tau1And τ2Is an unknown lagrange multiplier related to the equality constraint of the steady state data;
s53: and (3) derivation calculation: get
Figure BDA0002632095520000065
To pair
Figure BDA0002632095520000066
And τ1First derivative of (A), take
Figure BDA0002632095520000067
To pair
Figure BDA0002632095520000068
And τ2The first derivative of (a) yields:
Figure BDA0002632095520000069
Figure BDA00026320955200000610
Figure BDA00026320955200000611
Figure BDA00026320955200000612
wherein the content of the first and second substances,
Figure BDA00026320955200000613
and
Figure BDA00026320955200000614
the definitions of (A) are similar;
s54: estimating omega and theta by adopting hierarchical Lagrange iteration method2；
S55: linear system parameters are estimated.
Further, in step S54, estimating ω and θ by using a hierarchical lagrange iteration method2The method specifically comprises the following steps: in the (k +1) th iteration, the k-th iteration is used
Figure BDA00026320955200000615
The estimated value of time replaces phi in the (k +1) th iteration2And K2For estimating theta2By using
Figure BDA00026320955200000616
Instead of phi3And K1Unknown parameter vector θ in2To estimate ω; to pair
Figure BDA00026320955200000617
And (3) executing a normalized operation:
Figure BDA00026320955200000618
initial values ω (0) and θ2(0) Generated arbitrarily between 0 and 1.
Further, in step S55, estimating linear system parameters specifically includes:
s551: extracting system markov parameters from the estimated markov parameters:
Figure BDA00026320955200000619
Figure BDA00026320955200000620
s552: the following Singular Value Decomposition (SVD) is performed:
Figure BDA0002632095520000071
wherein the content of the first and second substances,
Figure BDA0002632095520000072
and
Figure BDA0002632095520000073
maximum n of SxEigenvector, left eigenvector sum V of UTRight feature vector of (a); since the system is observable and realizable, it is clear that SS＝0。
S553: from
Figure BDA0002632095520000074
And
Figure BDA0002632095520000075
extracting observable matrix from the image
Figure BDA0002632095520000076
And controllability matrix
Figure BDA0002632095520000077
Figure BDA0002632095520000078
Figure BDA0002632095520000079
S554: from
Figure BDA00026320955200000710
And
Figure BDA00026320955200000711
medium direct retrieval system matrix:
Figure BDA00026320955200000712
Figure BDA00026320955200000713
Figure BDA00026320955200000714
the invention has the beneficial effects that: the method adopts a reduced model based on decomposition, avoids estimating additional intermediate parameters, reduces the variance of the estimation model and improves the precision of the model; in addition, the method adopts an optimally weighted multiple regularization frame to fuse static data and dynamic data, and realizes the identification and solution of model parameters. Compared with the existing black box space identification, the method greatly improves the model precision of the Hammerstein nonlinear system and reduces the estimation parameter variance.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a block flow diagram of the method of the present invention;
FIG. 2 is a WPT circuit topology adopted in the present embodiment;
FIG. 3 is a simulation diagram of dynamic input/output data and steady-state input/output data of the WTP system;
FIG. 4 is a comparison of real WTP system training results to test results.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Referring to fig. 1 to 4, the method of the present invention is applied to a Wireless Power Transfer (WPT) system to illustrate the effectiveness and advantages of the present invention. The WPT system studied in this embodiment is composed of a phase-controlled full-bridge inverter and a receiver, and the circuit topology diagram and the physical structure diagram are respectively shown in fig. 2, where specific parameters are shown in table 1.
TABLE 1 WPT experimental setup essential parameters
Parameter(s) Explaining the meaning Value of
C1 Capacitance of transmission resonator 38.59nF
C2 Capacitance of receiving resonator 37.76nF
Cf Capacitor of output filter 471.2uF
fs Drive frequency of frequency converter 80kHz
L1 Inductance of transmission resonator 104.4uH
L2 Inductance of receiving resonator 104.3uH
M L1And L2Mutual inductance between 8.83uH
R1 L1、C1Equivalent resistance of branch 161.1mΩ
R2 L2、C2Branch circuitEquivalent resistance of 162.4mΩ
Ro Load resistance 5Ω
RS On-resistance of switch and diode 12.6mΩ
Vd Voltage of DC power supply 7V
Vr Forward voltage of diode 0.5V
The specific implementation process is as follows:
s1: and carrying out data acquisition on the wireless electric energy transmission system.
The input is WPT system inverter phase shift, and the output is load voltage. The sampling period is selected to be T ═ 0.1 ms. To ensure the system is continuously excited, the value is [0, 0.4 ]]Generating input variable u (t) arbitrarily in the range to obtain dynamic input/output data
Figure BDA0002632095520000081
In [0, 0.4 ]]At intervals of 0.01 and for 3.8s within the range as a variable
Figure BDA0002632095520000082
To obtain static input/output data
Figure BDA0002632095520000083
To verify the effectiveness of the method of the invention, the data is divided into training data and test data 2 parts. The training data includes 5000 dynamic data and 11780 steady-state data, and the test data includes 1000 dynamic data and 3800 sampling steady-state data. The training data and the test data are shown in fig. 3.
S2: for a wireless electric energy transmission system with nonlinear characteristics, a Hammerstein nonlinear model with a state space module is adopted for description, and a prediction model form is further obtained to facilitate subsequent identification. It should be noted that the basis functions of the Hammerstein model may select different non-linear functions according to the system characteristics, such as radial basis functions, hinge functions, polynomial functions, and the like. Because the unknown coefficients provide the degree of freedom, the nonlinear function can be ensured to be accurately approximated to the system characteristic, and therefore, the basis function can be simply selected as a polynomial function. Further, r can be selected from the range of [2, 11 ]. The optimal value can be obtained through a comparison experiment, namely under the condition of different values of r, the error between the static output of the system and the static output of the model is taken as a selection basis, and the r corresponding to the minimum error is the optimal value. The number of nonlinear basis functions in this example is optimally 7.
S3: the innovation model is subjected to iteration processing for p times, a dynamic simplified model of the Hammerstein nonlinear system can be obtained, and the model can effectively process dynamic data. Wherein p can be arbitrarily selected within the corresponding range of [10, 60 ]. The optimal value can be obtained through a comparison experiment, namely under the condition of different values of p, the error between the static output of the system and the static output of the model is taken as a selection basis, and the p corresponding to the minimum error is the optimal value. The number of nonlinear basis functions in this embodiment is most preferably 27.
S4: and processing the static input and output data obtained correspondingly, and after arrangement, obtaining matrixed static data and further obtaining a consistent least square solution of the steady-state parameters G and Q. Therefore, a Hammerstein nonlinear system static simple model can be obtained.
S5: first, θ is estimated by projection1Then, the output Y in the dynamic reduced model is orthogonally projected to phi0Further performing least square estimation to obtain theta1An estimate of (d). And then estimating omega and theta by adopting a recursive Lagrange iteration method with random initial values2. Where the estimated value of ω is shown in table 2.
TABLE 2 WPT System parameters
Figure BDA0002632095520000091
Then, θ is obtained from the estimation1And theta2System markov parameters are extracted and corresponding matrices are constructed, performing Singular Value Decomposition (SVD). And finally to get from
Figure BDA0002632095520000092
And
Figure BDA0002632095520000093
the system matrix parameters A, B, C are extracted. The results are shown in FIG. 4.
The model training output data and the model test output data of the WPT system obtained based on the obtained model are shown in fig. 4. The fit of the model training output data and the model test output data is shown in table 3. The degree of fit is defined as:
Figure BDA0002632095520000094
wherein y (T) and
Figure BDA0002632095520000101
respectively representing the sampled actual output and the model output,
Figure BDA0002632095520000102
is the average value of y (T). It can be seen that the method proposed in the present disclosure has a better effect on the parameter estimation of the WPT system. The model precision and the ideal value are basically coincident, the fitting degree is high, and a high-precision WPT model is established.
TABLE 3 WPT System Fit
Scope of adaptation GSIM
Training dynamic data fitness 84.36
Testing dynamic data fitness 83.35
Training steady state data fitness 99.46
Testing steady state data fitness 80.26
The novel ash box simple subspace identification method driven by dynamic/static data in a hybrid mode is used for accurately identifying the WPT system, a simple model based on decomposition is adopted, estimation of additional intermediate parameters is avoided, the variance of the estimation model is reduced, and the model accuracy is improved; in addition, the method adopts an optimally weighted multiple regularization frame to fuse static data and dynamic data, realizes identification and solution of model parameters, and solves the problem of low model precision caused by the fact that the existing black box space identification method cannot provide system low-frequency information. And obtaining an accurate WPT model. Can be popularized and applied to other complex systems.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (4)

1. A Hammerstein nonlinear industrial system simple grey box space identification method driven by dynamic and static data in a mixed mode is characterized by comprising the following steps:
s1: collecting system dynamic data and static data;
s2: selecting and processing a Hammerstein nonlinear system model to obtain a prediction model;
s3: constructing a dynamic simple model of a Hammerstein nonlinear system;
s4: constructing a Hammerstein nonlinear system static simple model;
s5: solving system parameter fusion identification by using a hierarchical Lagrange optimal weighting method;
the step S1 specifically includes: data acquisition is carried out on the actual industrial process to obtain the dynamic data of the system
Figure FDA0003520978270000011
And static data
Figure FDA0003520978270000012
Wherein u (t), y (t) are dynamic input/output data, N is the number of dynamic input/output data,
Figure FDA0003520978270000013
respectively static input and output data, wherein M is the number of the static input and output data;
in step S2, the prediction model is obtained as:
Figure FDA0003520978270000014
wherein:
Figure FDA0003520978270000015
ω＝[ω1,ω2,...,ωr]，z(t)＝[f1(u(t)),f2(u(t)),...,fr(u(t))]T；
Figure FDA0003520978270000016
u (t) epsilon R, y (t) epsilon R, e (t) epsilon R respectively represent a state vector, collected input data, collected output data and an innovation vector; (A, B, C, K) represents a system matrix; n isxRepresenting the model order; the innovation is zero mean Gaussian noise, and the nonlinear input function f (t) epsilon R is a known basis function fi(u (t)) a linear combination of ε R, [ omega ]ie.R represents an unknown coefficient;
in step S3, a Hammerstein nonlinear system dynamic simple model is constructed, which specifically includes the following steps:
s31: carrying out iteration processing on the innovation model for p times to obtain a new state equation:
Figure FDA0003520978270000017
wherein:
Figure FDA0003520978270000018
Figure FDA0003520978270000019
zp(t)＝[zT(t-p),zT(t-p+1),...,zT(t-1)]T (9)
yp(t)＝[yT(t-p),yT(t-p+1),…,yT(t-1)]T (10)
s32: bringing the new state equation into output according to
Figure FDA00035209782700000110
The output y (t) of the system is rewritten to obtain a new system output function:
y(t)＝P0(t)θ0+P1(t)θ1+e(t) (11)
wherein:
P0(t)＝[zT(t-1),zT(t-2),...,zT(t-p)] (12)
P1(t)＝[yT(t-1),yT(t-2),...,yT(t-p)] (13)
Figure FDA0003520978270000021
Figure FDA0003520978270000022
s33: to theta0And (3) carrying out decomposition processing, rewriting the parameterized model (11) to obtain two dynamic simplified models:
Figure FDA0003520978270000023
wherein:
P2(t)＝[ωz(t-1),ωz(t-2),...,ωz(t-p)] (18)
Figure FDA0003520978270000024
Figure FDA0003520978270000025
and theta1And theta2Including all of the extended markov parameters,as follows:
Figure FDA0003520978270000026
Figure FDA0003520978270000027
the corresponding system markov parameter is defined as:
hi＝CAi-1K (23)
gi＝CAi-1B (24)
wherein, i is 1.. multidot.p;
s34: performing matrixing processing on the dynamic data and the dynamic reduction model:
Y＝[y(1),y(2),…,y(N)]T (25)
E＝[e(1),e(2),...,e(N)] (26)
Figure FDA0003520978270000028
Figure FDA0003520978270000029
Figure FDA00035209782700000210
Figure FDA00035209782700000211
and (3) performing iterative operation on (11), (16) and (17) on the basis of matrixing to obtain a new matrixing dynamic reduced model as follows:
Y＝φ0θ0+φ1θ1+E (31)
Y＝φ2θ2+φ1θ1+E (32)
Y＝φ3ωT+φ1θ1+E (33)
thus, a linear extended Markov parameter vector θ is estimated using a dynamic reduced model (32)2(ii) a Estimating a non-linear parameter ω with a dynamic reduced model (33); estimating redundant parameters, i.e. Markov parameters theta, in extensions (32) and (33) using auxiliary models (31)1；
In step S4, a Hammerstein nonlinear system static simple model is constructed, which specifically includes the following steps:
s41: substituting the formula (11) corresponding to the obtained static input and output to obtain the steady state output after the arrangement
Figure FDA0003520978270000031
Another expression of (1):
Figure FDA0003520978270000032
wherein:
Figure FDA0003520978270000033
Figure FDA0003520978270000034
Figure FDA0003520978270000035
s42: performing matrixing processing on the static data:
Figure FDA0003520978270000036
Figure FDA0003520978270000037
obtaining a consistent least squares solution for steady state parameters G and Q:
Figure FDA0003520978270000038
s43: since the functional expressions (36) of the parameters G and (37) of Q contain over-parameterized model parameters, the over-parameterized model (37) is rewritten into two static reduced models:
Q＝K1ωT (41)
Figure FDA0003520978270000039
wherein:
Figure FDA00035209782700000310
K2＝ωi[1，…,1] (44)
wherein Q isiIs the ith parameter of Q;
estimating a nonlinear parameter vector omega by using a static reduced model (41); estimating a linear extended Markov parameter vector θ using a static reduced model (42)2。
2. The Hammerstein nonlinear industrial system simple gray box space identification method as claimed in claim 1, wherein in step S5, the system parameter fusion identification is solved by using a hierarchical Lagrangian optimal weighting method, specifically comprising the following steps:
s51: estimating theta in an auxiliary model (31) using a projection method1；
Orthographically projecting the output Y in the dynamic reduced model (31) to phi0Further performing least squares estimation to obtain theta1The estimated values of (c) are as follows:
Figure FDA0003520978270000041
wherein:
Figure FDA0003520978270000042
Figure FDA0003520978270000043
Figure FDA0003520978270000044
wherein, INRepresenting an identity matrix of dimension N;
s52: constructing a multi-regularization framework based on a weighting matrix reduction model:
Figure FDA0003520978270000045
Figure FDA0003520978270000046
wherein, tau1And τ2Is an unknown lagrange multiplier related to the equality constraint of the steady state data;
s53: and (3) derivation calculation: get
Figure FDA0003520978270000047
To pair
Figure FDA0003520978270000048
And τ1First derivative of (1), take
Figure FDA0003520978270000049
To pair
Figure FDA00035209782700000410
And τ2The first derivative of (a) yields:
Figure FDA00035209782700000411
Figure FDA00035209782700000412
Figure FDA00035209782700000413
Figure FDA00035209782700000414
wherein the content of the first and second substances,
Figure FDA00035209782700000415
and
Figure FDA00035209782700000416
the definitions of (A) are similar;
s54: estimating omega and theta by adopting hierarchical Lagrange iteration method2；
S55: linear system parameters are estimated.
3. The Hammerstein nonlinear industrial system simple gray box space identification method as claimed in claim 2, wherein in step S54, omega and theta are estimated by using a hierarchical Lagrangian iteration method2The method specifically comprises the following steps: in the (k +1) th iteration, the k-th iteration is used
Figure FDA0003520978270000051
The estimated value of time replaces phi in the (k +1) th iteration2And K2For estimating theta2By using
Figure FDA0003520978270000052
Instead of phi3And K1Unknown parameter vector θ in2To estimate ω; to pair
Figure FDA0003520978270000053
And (3) executing a normalized operation:
Figure FDA0003520978270000054
initial values ω (0) and θ2(0) Generated arbitrarily between 0 and 1.
4. The Hammerstein nonlinear industrial system simple gray box space identification method as claimed in claim 3, wherein in step S55, estimating linear system parameters specifically comprises:
s551: extracting system markov parameters from the estimated markov parameters:
Figure FDA0003520978270000055
Figure FDA0003520978270000056
s552: the following singular value decomposition is performed:
Figure FDA0003520978270000057
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003520978270000058
and
Figure FDA0003520978270000059
maximum n of SxEigenvector, left eigenvector sum V of UTRight feature vector of (a);
s553: from
Figure FDA00035209782700000510
And
Figure FDA00035209782700000511
extracting observable matrix from the image
Figure FDA00035209782700000512
And controllability matrix
Figure FDA00035209782700000513
Figure FDA00035209782700000514
Figure FDA00035209782700000515
S554: from
Figure FDA00035209782700000516
And
Figure FDA00035209782700000517
medium direct retrieval system matrix:
Figure FDA00035209782700000518
Figure FDA00035209782700000519
Figure FDA00035209782700000520
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