CN111427266A - Nonlinear system identification method aiming at disturbance - Google Patents

Nonlinear system identification method aiming at disturbance Download PDF

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CN111427266A
CN111427266A CN202010198543.4A CN202010198543A CN111427266A CN 111427266 A CN111427266 A CN 111427266A CN 202010198543 A CN202010198543 A CN 202010198543A CN 111427266 A CN111427266 A CN 111427266A
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杨晓冬
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Abstract

The invention discloses a method for identifying a non-linear system with disturbance, which comprises the following steps: A. converting an industrial control system to be identified into a non-linear system model with disturbance, wherein the non-linear system consists of a non-linear link and a linear link, namely a Hammerstein system of an output error type; B. decomposing the non-linear system model with disturbance into two sub-models: the system has no noise output submodel and disturbance submodel; C. updating system parameters
Figure DDA0002418508890000011
Construction of Pζ(k) And eζ(k) Updating the parameters
Figure DDA0002418508890000012
Let k equal to k +1, return to step A until the cut-off condition is satisfied
Figure DDA0002418508890000013
D. Identifying parameters and disturbances of the industrial control system. The invention can improve the defects of the prior art and has high convergence rate and high identification precision.

Description

Nonlinear system identification method aiming at disturbance
Technical Field
The invention relates to the technical field of industrial control, in particular to a method for identifying a non-linear system with disturbance.
Background
The nonlinear system is widely existed in an industrial system, the problems of Identification and Control of the nonlinear system are more and more emphasized by students and engineers, and the research is focused, the nonlinear system can be divided into a Hamiltein system, a wiener system and a Hamiltein system, and generally consists of a nonlinear link and a linear link, wherein the nonlinear link is composed of a plurality of forms, such as a dead zone, a set of a plurality of linear functions and the like, the linear link is mainly an output error model, the Hamiltein system of an output error type is the most widely researched, in the industrial process, the measurement noise is widely existed, when the output error model is converted into a regression equation, white noise is converted into colored noise, the Identification algorithm becomes an offset estimation and reduces the Identification precision, in the Identification process, the disturbance always pollutes output data and reduces the Identification precision, so the influence of the disturbance must be eliminated, foreign and foreign documents and patent mention the non-linear system Identification method with disturbance, such as the algorithm of the foreign and patent, such as the foreign person Y.ao in the patent, the field of the "A noise Identification system", the algorithm is capable of reducing the Identification precision of the linear algorithm, the algorithm of the noise reduction of the linear algorithm, the algorithm of the noise reduction, the algorithm of the algorithm, the algorithm of the algorithm with disturbance, the algorithm with disturbance, the noise of the algorithm with disturbance, the noise reduction of the noise reduction of the algorithm with disturbance, the algorithm with the algorithm of the algorithm with the noise of.
For the Hammerstein system with the output error type of disturbance, the existing method has the following defects: (1) the influence of measurement noise is not well processed, and when the output error model is converted into a regression equation, white noise can be converted into colored noise, so that the identification problem is more complicated; (2) the influence of disturbance is not considered, so that the disturbance pollutes output data in the identification process, the identification precision is reduced, or the system parameters and the disturbance are identified at the same time and are not distinguished; (3) the identification of system parameters adopts a single-information identification method, and only current data can be utilized. The identification of a hammerstein system for perturbed output error types has become a hotspot of research.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for identifying a non-linear system with disturbance, which can solve the defects of the prior art and has high convergence rate and identification precision.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows.
A method for identifying a non-linear system with disturbance comprises the following steps:
A. converting an industrial control system to be identified into a non-linear system model with disturbance, wherein the non-linear system consists of a non-linear link and a linear link, namely a Hammerstein system of an output error type; setting initial value
Figure BDA0002418508870000021
P(0)，Pξ(0)，
Figure BDA0002418508870000022
u(k)＝，p，γ1And gamma2Collecting system input and output data u (k) and y (k);
B. decomposing the non-linear system model with disturbance into two sub-models: the system noiseless output submodel and the disturbance submodel construct a system output vector Y (p, k) and an information vector
Figure BDA0002418508870000023
Information matrix
Figure BDA0002418508870000031
Disturbance vector
Figure BDA0002418508870000032
C. Updating system parameters
Figure BDA0002418508870000033
Construction of
Figure BDA0002418508870000034
And
Figure BDA0002418508870000035
updating parameters
Figure BDA0002418508870000036
Let k equal to k +1, return to step A until the cut-off condition is satisfied
Figure BDA0002418508870000037
Figure BDA0002418508870000038
Wherein the number is non-negative or reaches a certain sampling number;
D. identifying parameters and disturbances of the industrial control system.
Preferably, in step a, u (k) is defined to represent system input, F (u (k)) represents a nonlinear link function, and F (q) represents a linear link function-1) Representing a linear link function, x (k) representing the noise-free output of the system,
Figure BDA00024185088700000313
representing the system disturbance, v (k) representing the system output noise, and y (k) representing the system output, the perturbed nonlinear discrete model is as follows:
Figure BDA0002418508870000039
wherein the content of the first and second substances,
Figure BDA00024185088700000310
Figure BDA00024185088700000311
Figure BDA00024185088700000312
naand nbFor integer representation of the linear order of the elements, z represents a shifting factor, i.e.
Figure BDA0002418508870000041
The parameter vectors and information vectors defining the system model are as follows:
Figure BDA0002418508870000042
Figure BDA0002418508870000043
Figure BDA0002418508870000044
Figure BDA0002418508870000045
wherein n iss＝na+nb×nc；
The system noiseless output and the system output are respectively represented as,
Figure BDA0002418508870000046
Figure BDA0002418508870000047
preferably, in step B, the system noiseless output submodel and the disturbance submodel are expressed as,
y1(k)＝x(k)
Figure BDA0002418508870000048
the output of the system is represented as,
y(k)＝y1(k)+y2(k)+v(k)，
the prediction error function is expressed as,
Figure BDA0002418508870000049
a prediction error vector is defined which is,
Figure BDA0002418508870000051
wherein, p represents the length of the multiple information,
defining an information matrix, a noise vector, a disturbance vector and a system output vector,
Figure BDA0002418508870000052
Figure BDA0002418508870000053
Figure BDA0002418508870000054
Figure BDA0002418508870000055
the noiseless output sub-model vector, the perturbation sub-model vector and the system output vector are represented as,
Y1(p，k)＝Φ(p，k)Tθ(k)
Figure BDA0002418508870000056
Y(p，k)＝Y1(p，k)+Y2(p，k)+V(p，k)，
a loss function is defined that is a function of,
Figure BDA0002418508870000057
since the non-linear system is decomposed into two submodels, the loss function is updated,
Figure BDA0002418508870000058
wherein, γ1∈(0，1]And gamma2∈(0，1]A forgetting factor is represented, which is,
Figure BDA0002418508870000061
represents an estimated value of phi (p, i),
Figure BDA0002418508870000062
an estimated value of theta (theta) is represented,
Figure BDA0002418508870000063
to represent
Figure BDA0002418508870000064
An estimate of (d).
Preferably, in step C, the loss function is updated to,
Figure BDA0002418508870000065
when calculating E (p, k), the last estimated value is used
Figure BDA0002418508870000066
Instead of the former
Figure BDA0002418508870000067
Calculating e1(k) Using the last estimated value
Figure BDA0002418508870000068
Instead of the former
Figure BDA0002418508870000069
Establishing an auxiliary model, using the output of the auxiliary model to replace the unknown variable x (k),
Figure BDA00024185088700000610
Figure BDA00024185088700000611
to pair
Figure BDA00024185088700000612
The first-order derivation is carried out,
Figure BDA00024185088700000613
Figure BDA00024185088700000614
Figure BDA00024185088700000615
Figure BDA00024185088700000616
Figure BDA00024185088700000617
Figure BDA0002418508870000071
ψi(k)＝[fi(u(k-1))，L，fi(u(k-nb))]
Figure BDA0002418508870000072
Figure BDA0002418508870000073
Figure BDA0002418508870000074
to pair
Figure BDA0002418508870000075
The first-order derivation is carried out,
Figure BDA0002418508870000076
Figure BDA0002418508870000077
Figure BDA0002418508870000078
wherein E (p, k) is multi-information,
Figure BDA0002418508870000079
is single information.
Preferably, in step C, before updating the loss function, the method is applied to
Figure BDA00024185088700000710
The pre-treatment is carried out, and the pretreatment,
Figure BDA00024185088700000711
Figure BDA00024185088700000712
wherein e is a natural base number, and F is a preprocessing function.
Adopt the beneficial effect that above-mentioned technical scheme brought to lie in:
the invention establishes a parameterized model of the Hammerstein system under disturbance conditions, and determines input variables, output variables, intermediate variables, measurement noise and disturbance noise. The parameterized model is decomposed into two sub-models: the system has no noise output submodel and disturbance submodel, and divides the parameters to be identified into non-time-varying parameters and time-varying parameters. The noise-free output submodel consists of a nonlinear part and a linear part, the two parts are converted into a regression equation form of an input variable and a noise-free output, wherein the noise-free output is unknown. The noise-free output submodel of the system is deduced to be in a regression equation form, the system output is equal to the sum of the noise-free output, the disturbance and the measurement noise, wherein the disturbance is slow time-varying noise, and the measurement noise is white noise, so that the white noise is prevented from being converted into colored noise, and the least square is changed into biased estimation. And establishing a combined loss function of the noiseless output submodel and the disturbance submodel, and deducing a recursive least square algorithm, wherein the parameters of the noiseless output submodel adopt a multi-innovation theory, so that the convergence speed and the prediction precision of the identification algorithm are improved, and the parameters of the disturbance submodel adopt single innovation, so that the tracking performance of the algorithm is improved. Aiming at different characteristics of a noiseless output submodel and a disturbance submodel, two different forgetting factors are introduced, the introduction of forgetting can improve the convergence speed and the identification precision, and meanwhile, the tracking performance on the disturbance with slow time variation is better.
The method for identifying various nonlinear industrial control systems has the advantages of high identification precision, high identification speed, strong robustness and wide applicable scenes.
Drawings
FIG. 1 is a schematic diagram of a Hammerstein system of the type with perturbed output errors of the present invention.
FIG. 2 is a schematic diagram of the aiding model of the present invention.
FIG. 3 is a flow chart of the identification method of the present invention.
Fig. 4 is a diagram of the input signals of the system of the present invention.
FIG. 5 is a graph of a perturbation signal according to the present invention.
Fig. 6 is a graph of the output signal of the present invention.
FIG. 7 is a comparison of the recognition effect of the present invention and the aided model least squares method on the first parameter.
FIG. 8 is a comparison of the recognition effect of the least squares method of the present invention and the aided model on the second parameter.
FIG. 9 is a comparison graph of the recognition effect of the least square method of the invention and the auxiliary model on the third parameter.
FIG. 10 is a comparison graph of the recognition effect of the least squares method of the present invention and the auxiliary model on the fourth parameter.
FIG. 11 is a comparison graph of the recognition effect of the least square method of the invention and the auxiliary model on the fifth parameter.
FIG. 12 is a comparison graph of the recognition effect of the least square method of the invention and the auxiliary model on the sixth parameter.
Detailed Description
Referring to fig. 1-3, one embodiment of the present invention includes the steps of:
A. converting an industrial control system to be identified into a non-linear system model with disturbance, wherein the non-linear system consists of a non-linear link and a linear link, namely a Hammerstein system of an output error type; setting initial value
Figure BDA0002418508870000091
P(0)，Pξ(0)，
Figure BDA00024185088700000912
u(k)＝0，p，γ1And gamma2Collecting system input and output data u (k) and y (k);
B. decomposing the non-linear system model with disturbance into two sub-models: the system noiseless output submodel and the disturbance submodel construct a system output vector Y (p, k) and an information vector
Figure BDA0002418508870000093
Information matrix
Figure BDA0002418508870000094
Disturbance vector
Figure BDA0002418508870000095
C. Updating system parameters
Figure BDA0002418508870000096
Construction of
Figure BDA0002418508870000097
And
Figure BDA0002418508870000098
updating parameters
Figure BDA0002418508870000099
Let k equal to k +1, return to step A until the cut-off condition is satisfied
Figure BDA00024185088700000910
Figure BDA00024185088700000911
Wherein the number is non-negative or reaches a certain sampling number;
D. identifying parameters and disturbances of the industrial control system.
In step A, defining u (k) as system input, F (u (k)) as nonlinear link function, and F (q)-1) Representing a linear link function, x (k) representing the noise-free output of the system,
Figure BDA0002418508870000101
representing the system disturbance, v (k) representing the system output noise, and y (k) representing the system output, the perturbed nonlinear discrete model is as follows:
Figure BDA0002418508870000102
wherein the content of the first and second substances,
Figure BDA0002418508870000103
Figure BDA0002418508870000104
Figure BDA0002418508870000105
naand nbFor integer representation of the linear order of the elements, z represents a shifting factor, i.e.
Figure BDA0002418508870000106
The parameter vectors and information vectors defining the system model are as follows:
Figure BDA0002418508870000107
Figure BDA0002418508870000108
Figure BDA0002418508870000109
Figure BDA0002418508870000111
wherein n iss＝na+nb×nc；
The system noiseless output and the system output are respectively represented as,
Figure BDA0002418508870000112
Figure BDA0002418508870000113
in the step B, the noiseless output submodel and the disturbance submodel of the system are expressed as,
y1(k)＝x(k)
Figure BDA0002418508870000114
the output of the system is represented as,
y(k)＝y1(k)+y2(k)+v(k)，
the prediction error function is expressed as,
Figure BDA0002418508870000115
a prediction error vector is defined which is,
Figure BDA0002418508870000116
wherein, p represents the length of the multiple information,
defining an information matrix, a noise vector, a disturbance vector and a system output vector,
Figure BDA0002418508870000117
Figure BDA0002418508870000118
Figure BDA0002418508870000121
Figure BDA0002418508870000122
the noiseless output sub-model vector, the perturbation sub-model vector and the system output vector are represented as,
Y1(p，k)＝Φ(p，k)Tθ(k)
Figure BDA0002418508870000123
Y(p，k)＝Y1(p，k)+Y2(p，k)+V(p，k)，
a loss function is defined that is a function of,
Figure BDA0002418508870000124
since the non-linear system is decomposed into two submodels, the loss function is updated,
Figure BDA0002418508870000125
wherein, γ1∈(0，1]And gamma2∈(0，1]A forgetting factor is represented, which is,
Figure BDA0002418508870000126
represents an estimated value of phi (p, i),
Figure BDA0002418508870000127
represents an estimated value of theta (k),
Figure BDA0002418508870000128
to represent
Figure BDA0002418508870000129
An estimate of (d).
In step C, the loss function is updated to,
Figure BDA00024185088700001210
when calculating E (p, k), the last estimated value is used
Figure BDA00024185088700001211
Instead of the former
Figure BDA00024185088700001212
Calculating e1(k) Using the last estimateValue of
Figure BDA0002418508870000131
Instead of the former
Figure BDA0002418508870000132
Establishing an auxiliary model, using the output of the auxiliary model to replace the unknown variable x (k),
Figure BDA0002418508870000133
Figure BDA0002418508870000134
to pair
Figure BDA0002418508870000135
The first-order derivation is carried out,
Figure BDA0002418508870000136
Figure BDA0002418508870000137
Figure BDA0002418508870000138
Figure BDA0002418508870000139
Figure BDA00024185088700001310
Figure BDA00024185088700001311
ψi(k)＝[fi(u(k-1))，L，fi(u(k-nb))]
Figure BDA00024185088700001312
Figure BDA00024185088700001313
Figure BDA0002418508870000141
to pair
Figure BDA0002418508870000142
The first-order derivation is carried out,
Figure BDA0002418508870000143
Figure BDA0002418508870000144
Figure BDA0002418508870000145
wherein E (p, k) is multi-information,
Figure BDA0002418508870000149
is single information.
In step C, before updating the loss function, the pair
Figure BDA0002418508870000146
The pre-treatment is carried out, and the pretreatment,
Figure BDA0002418508870000147
Figure BDA0002418508870000148
wherein e is a natural base number, and F is a preprocessing function.
After the system model is identified, the system model needs to be verified, the input and output data of the system are collected again, the effectiveness of the identified model is verified by using new data, when the effect is not good, the initial value of the algorithm can be adjusted, and the identification is carried out again until the system model meeting the requirements is obtained.
The advantages of the method provided by the invention are illustrated by taking a hammerstein system model with disturbed output error types converted by a power plant reheater control system as an example. The system model is as follows:
Figure BDA0002418508870000151
wherein, the parameter to be identified is a [ -1.22, 0.93], B [ -0.81, 0.73], C [ -0.51, 0.22], and the input signal u (k) adopts a gaussian random sequence with a mean value 0 and a variance of 1 as shown in fig. 4; the perturbation signal is shown in FIG. 5; the output signal y (k) is shown in FIG. 6.
The method has an initial value of P (0) to 106I8×8，
Figure BDA0002418508870000152
p is 6 and N is 6000. The method provided by the invention and the university of Ding F, Shi Y, Chen T and the like are disclosed in the document "Auxiliary model-based least-squares identification methods for Hammerstein output-errors [ J].Systems&The auxiliary model least square method mentioned in Control L etters,2007,56(5): 373-.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (5)

1. A method for identifying a non-linear system with disturbance is characterized by comprising the following steps:
A. converting an industrial control system to be identified into a non-linear system model with disturbance, wherein the non-linear system consists of a non-linear link and a linear link, namely a Hammerstein system of an output error type; setting initial value
Figure FDA0002418508860000011
P(0)，
Figure FDA00024185088600000112
u(k)＝0，p，γ1And gamma2Collecting system input and output data u (k) and y (k);
B. decomposing the non-linear system model with disturbance into two sub-models: the system noiseless output submodel and the disturbance submodel construct a system output vector Y (p, k) and an information vector
Figure FDA0002418508860000012
Information matrix
Figure FDA0002418508860000013
Disturbance vector
Figure FDA0002418508860000014
C. Updating system parameters
Figure FDA0002418508860000015
Construction of
Figure FDA0002418508860000016
And
Figure FDA0002418508860000017
updating parameters
Figure FDA0002418508860000018
Let k equal to k +1, return to step A until the cut-off condition is satisfied
Figure FDA0002418508860000019
Figure FDA00024185088600000110
Wherein the number is non-negative or reaches a certain sampling number;
D. identifying parameters and disturbances of the industrial control system.
2. The method for perturbed nonlinear system identification according to claim 1, wherein: in step A, defining u (k) as system input, F (u (k)) as nonlinear link function, and F (q)-1) Representing a linear link function, x (k) representing the noise-free output of the system,
Figure FDA00024185088600000111
representing the system disturbance, v (k) representing the system output noise, and y (k) representing the system output, the perturbed nonlinear discrete model is as follows:
Figure FDA0002418508860000021
wherein the content of the first and second substances,
Figure FDA0002418508860000022
Figure FDA0002418508860000023
Figure FDA0002418508860000024
naand nbFor integer representation of the linear order of the elements, z represents a shifting factor, i.e.
Figure FDA0002418508860000025
The parameter vectors and information vectors defining the system model are as follows:
Figure FDA0002418508860000026
Figure FDA0002418508860000027
Figure FDA0002418508860000028
Figure FDA0002418508860000029
wherein n iss＝na+nb×nc；
The system noiseless output and the system output are respectively represented as,
Figure FDA00024185088600000210
Figure FDA0002418508860000031
3. the method for perturbed nonlinear system identification according to claim 2, wherein: in the step B, the noiseless output submodel and the disturbance submodel of the system are expressed as,
y1(k)＝x(k)
y2(k)＝ζ(k)，
the output of the system is represented as,
y(k)＝y1(k)+y2(k)+v(k)，
the prediction error function is expressed as,
Figure FDA0002418508860000032
a prediction error vector is defined which is,
Figure FDA0002418508860000033
wherein, p represents the length of the multiple information,
defining an information matrix, a noise vector, a disturbance vector and a system output vector,
Figure FDA0002418508860000034
Figure FDA0002418508860000035
Figure FDA0002418508860000036
Figure FDA0002418508860000037
the noiseless output sub-model vector, the perturbation sub-model vector and the system output vector are represented as,
Y1(p，k)＝Φ(p，k)Tθ(k)
Figure FDA0002418508860000041
Y(p，k)＝Y1(p，k)+Y2(p，k)+V(p，k)，
a loss function is defined that is a function of,
Figure FDA0002418508860000042
since the non-linear system is decomposed into two submodels, the loss function is updated,
Figure FDA0002418508860000043
wherein, γ1∈(0，1]And gamma2∈(0，1]A forgetting factor is represented, which is,
Figure FDA0002418508860000044
represents an estimated value of phi (p, i),
Figure FDA0002418508860000045
represents an estimated value of theta (k),
Figure FDA0002418508860000046
to represent
Figure FDA0002418508860000047
An estimate of (d).
4. The method for perturbed nonlinear system identification according to claim 3, wherein: in step C, the loss function is updated to,
Figure FDA0002418508860000048
when calculating E (p, k), the last estimated value is used
Figure FDA0002418508860000049
Instead of the former
Figure FDA00024185088600000410
Calculating e1(k) Using the last estimated value
Figure FDA00024185088600000411
Instead of the former
Figure FDA00024185088600000412
Establishing an auxiliary model, using the output of the auxiliary model to replace the unknown variable x (k),
Figure FDA0002418508860000051
Figure FDA0002418508860000052
to pair
Figure FDA0002418508860000053
The first-order derivation is carried out,
Figure FDA0002418508860000054
Figure FDA0002418508860000055
Figure FDA0002418508860000056
Figure FDA0002418508860000057
Figure FDA0002418508860000058
Figure FDA0002418508860000059
ψi(k)＝[fi(u(k-1))，L，fi(u(k-nb))]
Figure FDA00024185088600000510
Figure FDA00024185088600000511
Figure FDA00024185088600000512
to pair
Figure FDA00024185088600000513
The first-order derivation is carried out,
Figure FDA00024185088600000514
Figure FDA0002418508860000061
Figure FDA0002418508860000062
wherein E (p, k) is multi-information,
Figure FDA0002418508860000063
is single information.
5. The method for perturbed nonlinear system identification according to claim 4, wherein: in step C, before updating the loss function, the pair
Figure FDA0002418508860000064
The pre-treatment is carried out, and the pretreatment,
Figure FDA0002418508860000065
F＝∫0 +∞e-kθ(k)dk，
wherein e is a natural base number, and F is a preprocessing function.
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