CN114566227A - PH neutralization process model identification method based on Newton iteration algorithm - Google Patents

PH neutralization process model identification method based on Newton iteration algorithm Download PDF

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CN114566227A
CN114566227A CN202110878223.8A CN202110878223A CN114566227A CN 114566227 A CN114566227 A CN 114566227A CN 202110878223 A CN202110878223 A CN 202110878223A CN 114566227 A CN114566227 A CN 114566227A
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李俊红
芮佳丽
蒋泽宇
朱建红
宗天成
袁银龙
李政
李磊
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Abstract

The invention provides a method for identifying a PH neutralization process model based on a Newton iteration algorithm, which comprises the following steps: step 1) constructing a fractional order Hammerstein CAR model in the PH neutralization process, and acquiring an identification model of the PH neutralization process according to the constructed system model; and 2) constructing an identification process of a Newton iterative algorithm. The invention has the beneficial effects that: the parameter identification result of the pH value neutralization process model identification method of the Newton iterative algorithm shows that the identification precision of the method is high, and the output parameter estimation error is small; meanwhile, the identification method has better applicability to the PH neutralization process model.

Description

PH neutralization process model identification method based on Newton iteration algorithm
Technical Field
The invention relates to the technical field of sewage treatment system identification, in particular to a fractional order model identification method for a PH neutralization process based on a Newton iteration algorithm.
Background
Environmental protection and sustainable development are important links in national development strategy. In chemical and three-waste treatment, environmental protection departments will focus on monitoring the pH value in industrial wastewater discharge. It is generally strongly nonlinear during PH neutralization for wastewater treatment. In order to better analyze, predict and control the PH neutralization process, a system model must be established for PH neutralization process control, and parameters of the established model must be identified. Therefore, a plurality of scholars establish an integer order model of the PH neutralization process and perform parameter identification by applying algorithms such as a least square method, a gradient iteration method and the like.
When a model is established for an actual system, a commonly used integer model is sometimes not accurate enough, so that the identification accuracy is not high enough. In addition, the nonlinear part is mostly a polynomial of a known order, and all nonlinear phenomena cannot be fully described. In the identification method, the least square method is deficient in identification precision, and data saturation is easy to occur when identification parameters are many. The gradient iteration method has a low convergence rate, so that the identification result is easy to fall into local optimum, and the problem of divergence of the identification result needs to be considered when the iteration step is selected.
How to solve the above technical problems is the subject of the present invention.
Disclosure of Invention
The invention aims to provide a method for identifying a PH neutralization process model based on a Newton iterative algorithm; the fractional order Hammerstein CAR model is different from a polynomial commonly used in a nonlinear part, the nonlinear part of the model is composed of two sections of piecewise nonlinearity, a fractional order model which is more accurate than an integer order model is adopted, and a Newton iterative algorithm is adopted, so that the model has higher identification precision and high convergence speed, and can be better suitable for modeling and parameter identification in the PH neutralization process.
The invention is realized by the following measures: a PH neutralization process model identification method based on a Newton iteration algorithm specifically comprises the following steps:
step 1) constructing a fractional order Hammerstein CAR model in the PH neutralization process, and acquiring an identification model of the PH neutralization process according to the constructed system model;
and 2) constructing an identification process of a Newton iterative algorithm.
As a further optimization scheme of the PH neutralization process model identification method based on the Newton iterative algorithm, the concrete modeling steps of the step 1) are as follows:
step 1-1) constructing a fractional Hammerstein CAR model of PH neutralization process: given the general form of the fractional order Hammerstein CAR system, as in equation (4), u (t) is the input to the system, y (t) is the output of the system, x (t) is the output of the nonlinear element, and v (t) is white noise. Wherein x (t) is two piece piecewise non-linearity:
Figure BDA0003188538660000021
introducing a switching function:
Figure BDA0003188538660000022
formula (1) can be written as:
x(t)＝lu(t)+(m-l)u(t)h(t) (3)
the general form of the model that can be obtained is:
Figure BDA0003188538660000023
step 1-2) the relationship between the output y (t) and the input u (t), the error v (t) can be obtained according to equations (5) to (12), wherein the relationship can be obtained according to the definition of Grnwald-Letnikov (G-L):
Figure BDA0003188538660000024
where α is the fractional order, when t ═ kh, h is the sampling time set to 1, and k is the number of samples for which the derivative approximation is calculated.
In the formula (5)
Figure BDA0003188538660000025
Is Newton's binomial formula:
Figure BDA0003188538660000026
Γ is the Euler function:
Figure BDA0003188538660000027
formula (4) can be written as:
Figure BDA0003188538660000031
Figure BDA0003188538660000032
Figure BDA0003188538660000033
Figure BDA0003188538660000034
Figure BDA0003188538660000035
as a further optimization scheme of the PH neutralization process model identification method based on the Newton iterative algorithm, the model in the step 1) is a fractional Hammerstein CAR model.
As a further optimization scheme of the PH neutralization process model identification method based on the Newton iteration algorithm, the specific steps of the step 2) for constructing the identification process of the Newton iteration algorithm are as follows:
step 2-1) defining the data length as L, the output vector as Y (L) and the information matrix as phi (L),
Figure BDA0003188538660000036
Figure BDA0003188538660000037
Figure BDA0003188538660000038
obtaining an identification model of
Y(L)＝Φ(L)θ+V(L) (13)
The criterion function is:
J(θ):＝||Y(L)-Φ(L)θ||2 (14)
step 2-2) obtaining a valve position current signal for regulating acid flow in the PH neutralization process as an input signal, taking the PH value of the neutralization solution as an output signal, and recording data;
step 2-3) calculating Delta according to the formulas (9) and (10)αy(t-i),Δαx(t-j)；
Step 2-4) construction
Figure BDA0003188538660000041
And
Figure BDA0003188538660000042
the number of iterations is denoted by k and,
Figure BDA0003188538660000043
an iterative estimate of theta is expressed, will
Figure BDA0003188538660000044
Unknown term Δ in (1)αx (t-j) using the iterative estimation of the k-1 th time
Figure BDA0003188538660000045
Instead of, after substitution
Figure BDA0003188538660000046
To be recorded as
Figure BDA0003188538660000047
Then
Figure BDA0003188538660000048
Is composed of
Figure BDA0003188538660000049
Step 2-5) calculation
Figure BDA00031885386600000410
And
Figure BDA00031885386600000411
Figure BDA00031885386600000412
Figure BDA00031885386600000413
step 2-6) update parameter estimation
Figure BDA00031885386600000414
Using the newton's iterative algorithm is as follows:
Figure BDA00031885386600000415
wherein, mu is a convergence factor,
Figure BDA00031885386600000416
is about (10)
Figure BDA00031885386600000417
The gradient of (a) of (b) is,
Figure BDA00031885386600000418
is about (10)
Figure BDA00031885386600000419
The sea plug matrix.
Step 2-7) calculation
Figure BDA00031885386600000420
And
Figure BDA00031885386600000421
Figure BDA00031885386600000422
Figure BDA00031885386600000423
step 2-8) judging whether the maximum iteration times are reached, if not, skipping the program to step 2-4), and if so, entering step 2-9);
and 2-9) outputting a result to finish identification.
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention establishes a Hammerstein CAR fractional order model for the PH neutralization process in sewage treatment, wherein the model is formed by connecting a static nonlinear system with a linear dynamic system in series; the method comprises the steps of establishing a dynamic part, establishing a static nonlinear part, establishing a dynamic part, and establishing a dynamic part, wherein the static nonlinear part is described by adopting two-segment piecewise nonlinearity, and the linear dynamic part adopts a CAR model which is different from a common integer order model.
(2) The identification method adopted by the Hammerstein CAR fractional order model established by the invention is a Newton iteration algorithm, and the parameter identification result after the algorithm is used shows that the Newton iteration algorithm applied to the model has high convergence speed, can obtain high identification precision only by iterating for a plurality of times, reduces the running time of the practical application algorithm, and meanwhile, the parameter estimation value after the Newton iteration algorithm is adopted tends to be stable, and the corresponding estimation error value is small.
(3) Aiming at the PH neutralization process in sewage treatment, the established Hammerstein CAR fractional order model has a good identification effect by adopting a Newton iterative algorithm, which shows that the identification method has good applicability to the established model and has engineering value.
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The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
FIG. 1 is a flow chart of a Newton's iterative algorithm of the present invention.
FIG. 2 is a schematic diagram of a PH neutralization control process system according to the present invention.
FIG. 3 is a schematic diagram of the error between the identification parameter and the actual value according to the present invention.
FIG. 4 is a diagram of a general model of the Hammerstein CAR fractional order system of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative of the invention and are not intended to be limiting.
Example 1
Referring to fig. 1 to 4, the technical solution provided by the present invention is a method for identifying a PH neutralization process model based on a newton iteration algorithm, wherein the method specifically includes the following steps:
step 1) constructing a fractional order Hammerstein CAR model in the PH neutralization process, and acquiring an identification model of the PH neutralization process according to the constructed system model;
and 2) constructing an identification process of a Newton iterative algorithm.
Specifically, the specific modeling steps of step 1) are as follows:
step 1-1) constructing a fractional Hammerstein CAR model of PH neutralization process: given the general form of the fractional order Hammerstein CAR system, as in equation (4), u (t) is the input to the system, y (t) is the output of the system, x (t) is the output of the nonlinear element, and v (t) is white noise. Wherein x (t) is two piece piecewise non-linearity:
Figure BDA0003188538660000061
introducing a switching function:
Figure BDA0003188538660000062
formula (1) can be written as:
the general form of the model available from x (t) + (m-l) u (t) h (t) (3) is:
Figure BDA0003188538660000071
Figure BDA0003188538660000072
A(z)y(t)＝B(z)x(t)+v(t) (4)
step 1-2) the relationship between the output y (t) and the input u (t), the error v (t) can be obtained according to equations (5) to (12), wherein the relationship can be obtained according to the definition of Grnwald-Letnikov (G-L):
Figure BDA0003188538660000073
where α is the fractional order, and when t ═ kh, h is the sampling time set to 1, and k is the number of samples for which the derivative approximation is calculated. In the formula (5)
Figure BDA0003188538660000074
Is Newton's binomial formula:
Figure BDA0003188538660000075
Γ is the Euler function:
Figure BDA0003188538660000076
formula (4) can be written as:
Figure BDA0003188538660000077
Figure BDA0003188538660000078
Figure BDA0003188538660000079
Figure BDA00031885386600000710
Figure BDA00031885386600000711
specifically, the model of the step 1) is a fractional Hammerstein CAR model.
Specifically, the specific steps of constructing the identification process of the newton iteration algorithm in step 2) are as follows:
step 2-1) defining the data length as L, the output vector as Y (L) and the information matrix as phi (L),
Figure BDA0003188538660000081
Figure BDA0003188538660000082
Figure BDA0003188538660000083
obtain an identification model of
Y(L)＝Φ(L)θ+V(L) (13)
The criterion function is:
J(θ):＝||Y(L)-Φ(L)θ||2 (14)
step 2-2) obtaining a valve position current signal for regulating acid flow in the PH neutralization process as an input signal, taking the PH value of the neutralization solution as an output signal, and recording data;
step 2-3) calculating Delta according to the formulas (9) and (10)αy(t-i),Δαx(t-j)；
Step 2-4) construction
Figure BDA0003188538660000084
And
Figure BDA0003188538660000085
the number of iterations is denoted by k,
Figure BDA0003188538660000086
an iterative estimate of theta is expressed, will
Figure BDA0003188538660000087
Unknown term Δ in (1)αx (t-j) using the (k-1) th iterative estimate
Figure BDA0003188538660000088
Instead of, after substitution
Figure BDA0003188538660000089
To be recorded as
Figure BDA00031885386600000810
Then
Figure BDA00031885386600000811
Is composed of
Figure BDA00031885386600000812
Step 2-5) calculation
Figure BDA00031885386600000813
And
Figure BDA00031885386600000814
Figure BDA0003188538660000091
Figure BDA0003188538660000092
step 2-6) update parameter estimation
Figure BDA0003188538660000093
Using the newton iteration algorithm is as follows:
Figure BDA0003188538660000094
wherein, mu is a convergence factor,
Figure BDA0003188538660000095
is about (10)
Figure BDA0003188538660000096
The gradient of (a) of (b) is,
Figure BDA0003188538660000097
is about (10)
Figure BDA0003188538660000098
The sea plug matrix.
Step 2-7) calculation
Figure BDA0003188538660000099
And
Figure BDA00031885386600000910
Figure BDA00031885386600000911
Figure BDA00031885386600000912
step 2-8) judging whether the maximum iteration times are reached, if not, skipping the program to step 2-4), and if so, entering step 2-9);
and 2-9) outputting a result to finish identification.
A schematic diagram of a PH neutralization process control system used in this embodiment is shown in fig. 2. Wherein, the input signal u (t) is a valve position current signal for regulating the acid flow, and the output signal y (t) is the PH value of the neutralization solution.
With the above-mentioned fractional order Hammerstein CAR model, the present embodiment can be modeled as follows:
y(t)+1.64Δ0.12y(t-1)+0.89Δ0.12y(t-2)＝0.22Δ0.12x(t-1)-1.58Δ0.12x(t-2)+1.07u(t)+0.26u(t)h(t)+v(t)
comparing the above model with step 1), it is possible to obtain
a1＝1.64,a2＝0,89,b1＝0.22,b2＝-1.58,
m＝1.33,l＝1.07,α＝1.12。
Let the parameter vector to be identified be
θ＝[a1,a2,b1,b2,l,(m-l)]T
The information vector is
Figure BDA00031885386600000913
Collecting input and output data according to the step 2-1) and the step 2-2), and constructing Y (L);
calculating Delta according to step 2-3)αy(t-i),Δαx(t-j)；
Construction according to step 2-4)
Figure BDA0003188538660000101
And
Figure BDA0003188538660000102
calculation according to step 2-5)
Figure BDA0003188538660000103
And
Figure BDA0003188538660000104
updating the parameter estimation values according to the step 2-6)
Figure BDA0003188538660000105
Calculation according to step 2-7)
Figure BDA0003188538660000106
And
Figure BDA0003188538660000107
the cycle is completed according to steps 2-8) and 2-9).
Wherein the convergence factor μ is set by selecting a constant greater than zero. In addition, the number k of iterations is selected with care not to be too large, otherwise the computation time will increase.
The parameter identification result by using the PH neutralization process model identification method based on the newton iteration algorithm of the present invention is shown in fig. 3. It can be seen that the algorithm adopted by the model has high identification result precision, and the output estimated value is closer to the true value. Meanwhile, the model and the adopted algorithm have better applicability to the PH neutralization process model.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (4)

1. A PH neutralization process model identification method based on a Newton iteration algorithm is characterized by comprising the following steps:
step 1) constructing a fractional order Hammerstein CAR model in the PH neutralization process, and acquiring an identification model of the PH neutralization process according to the constructed system model;
and 2) constructing an identification process of a Newton iterative algorithm.
2. The method for identifying a PH neutralization process model based on Newton's iterative algorithm as claimed in claim 1, wherein the specific modeling steps of the step 1) are as follows:
step 1-1) constructing a fractional Hammerstein CAR model of PH neutralization process: given the general form of the fractional order Hammerstein CAR system, as in equation (4), u (t) is the input to the system, y (t) is the output of the system, x (t) is the output of the nonlinear element, v (t) is white noise, where x (t) is two piecewise nonlinearity:
Figure FDA0003188538650000011
introducing a switching function:
h(t)＝h[u(t)]＝0.5{1+sgn[u(t)]}，
Figure FDA0003188538650000012
formula (1) can be written as:
x(t)＝lu(t)+(m-l)u(t)h(t) (3)
the general form of the model that can be obtained is:
Figure FDA0003188538650000013
Figure FDA0003188538650000014
A(z)y(t)＝B(z)x(t)+v(t) (4)
step 1-2) the relationship between the output y (t) and the input u (t), the error v (t) can be obtained according to equations (5) to (12), wherein the relationship can be obtained according to the definition of Grnwald-Letnikov (G-L):
Figure FDA0003188538650000015
wherein, α is a fractional order, when t ═ kh, h is a sampling time set to 1, and k is a sample number for calculating a derivative approximation;
in the formula (5)
Figure FDA0003188538650000021
Is Newton's binomial formula:
Figure FDA0003188538650000022
Γ is the Euler function:
Figure FDA0003188538650000023
formula (4) can be written as:
Figure FDA0003188538650000024
Figure FDA0003188538650000025
Figure FDA0003188538650000026
Figure FDA0003188538650000029
Figure FDA0003188538650000027
3. the newton's iterative algorithm-based PH neutralization process model of claim 1, wherein the model of step 1) is a fractional order Hammerstein CAR model.
4. The method for identifying a PH neutralization process model based on Newton's iterative algorithm as claimed in claim 1, wherein the specific steps of the step 2) for constructing the identification process of the Newton's iterative algorithm are as follows:
step 2-1) defining the data length as L, the output vector as Y (L) and the information matrix as phi (L),
Figure FDA0003188538650000028
Figure FDA0003188538650000031
Figure FDA0003188538650000032
obtain an identification model of
Y(L)＝Φ(L)θ+V(L) (13)
The criterion function is:
J(θ):＝||Y(L)-Φ(L)θ||2 (14)
step 2-2) obtaining a valve position current signal for regulating acid flow in the PH neutralization process as an input signal, taking the PH value of the neutralization solution as an output signal, and recording data;
step 2-3) calculating Delta according to the formulas (9) and (10)αy(t-i),Δαx(t-j)；
Step 2-4) construction
Figure FDA0003188538650000033
And
Figure FDA0003188538650000034
the number of iterations is denoted by k,
Figure FDA0003188538650000035
an iterative estimate of theta is expressed, will
Figure FDA0003188538650000036
Unknown term Δ in (1)αx (t-j) using the (k-1) th iterative estimate
Figure FDA0003188538650000037
Instead of, after substitution
Figure FDA0003188538650000038
To be recorded as
Figure FDA0003188538650000039
Then
Figure FDA00031885386500000310
Is composed of
Figure FDA00031885386500000311
Step 2-5) calculation
Figure FDA00031885386500000312
And
Figure FDA00031885386500000313
Figure FDA00031885386500000314
Figure FDA00031885386500000315
step 2-6) update parameter estimation
Figure FDA00031885386500000316
Using the newton iteration algorithm is as follows:
Figure FDA0003188538650000041
wherein, mu is a convergence factor,
Figure FDA0003188538650000042
is about (10)
Figure FDA0003188538650000043
The gradient of (a) is determined,
Figure FDA0003188538650000044
is about (10)
Figure FDA0003188538650000045
The sea plug matrix of;
step 2-7) calculation
Figure FDA0003188538650000046
And
Figure FDA0003188538650000047
Figure FDA0003188538650000048
Figure FDA0003188538650000049
step 2-8) judging whether the maximum iteration times are reached, if not, skipping the program to step 2-4), and if so, entering step 2-9);
and 2-9) outputting a result to finish identification.
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Title
JUNHONG LI: ""Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration"", 《APPLIED MATHEMATICS LETTERS》 *
KARIMA HAMMER ETC.: ""Fractional Hammerstein CAR system identification"", 《PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON SYSTEMS AND CONTROL》 *

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