CN117236060A - Novel recursion estimation method of continuous stirring reaction kettle based on auxiliary model - Google Patents

Abstract

The application discloses a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model, which belongs to the field of model parameter estimation and comprises the following steps: constructing a nonlinear system of a continuous stirring reaction kettle, and obtaining output data based on the nonlinear system and input data, wherein the output data comprises: measurable data and non-measurable data; constructing a maximum likelihood variable interval recursion least square algorithm; constructing an auxiliary model based on the measurable data, and replacing the unmeasurable data with the output data of the auxiliary model to obtain unmeasurable output and unmeasurable noise; and estimating the unmeasurable output and the unmeasurable noise based on the maximum likelihood variable interval recursive least square algorithm to obtain a parameter estimation function. Aiming at the identification difficulty of the non-measurable variable, the application changes the sampling interval, replaces the non-measurable variable by using the auxiliary model output, and introduces the maximum likelihood principle to directly identify the known and unknown parameters of the system aiming at the colored noise.

Description

Novel recursion estimation method of continuous stirring reaction kettle based on auxiliary model

Technical Field

The application belongs to the technical field of model parameter estimation, and particularly relates to a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model.

Background

Accurate modeling is important for real-time monitoring and efficient control of industrial processes, and model parameter estimation is important for control system analysis and design. In practical application, due to the limitation of hardware equipment, economic conditions and environmental factors, observation data at some sampling points are difficult to obtain, so that missing data is caused, and recognition of a missing data system is widely focused.

Many missing data system identification methods, such as polynomial transformation technology and lifting technology, propose a variational Bayesian method and a multi-step long gradient iteration method based on improved Kalman filtering aiming at ARX models with missing data, chen Dengfen respectively, and a likelihood function-based Expectation Maximization (EM) identification method is used for identifying parameters of the missing data system, but the algorithm can generate redundant parameters, and the calculated amount is large; in this respect, zhang et al propose the EM method based on likelihood function to the discrete distribution system with missing data, the nonlinear system of block structure is widely used, inspired by decomposing the complex system into simple system thought, a dynamic nonlinear system can be decomposed into dynamic linear subsystem and static nonlinear subsystem, thus the research of the corresponding nonlinear system becomes simple and intuitive.

In summary, the following disadvantages exist in the prior art: the polynomial technology causes redundant parameters, the calculated amount is large, the ARX model is not flexible and changeable without a block structure model, the generalization capability is strong, other algorithms are limited by a regression model, the noise parameters cannot be directly estimated, and the processing of the non-measurable variables is complex.

Disclosure of Invention

The application provides a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model, wherein the maximum likelihood method can directly process colored noise.

In order to achieve the above purpose, the application provides a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model, which comprises the following steps:

constructing a nonlinear system of a continuous stirring reaction kettle, and obtaining output data based on the nonlinear system and input data, wherein the output data comprises: measurable data and non-measurable data;

constructing a maximum likelihood variable interval recursion least square algorithm;

constructing an auxiliary model based on the measurable data, and replacing the non-measurable data with the output data of the auxiliary model to obtain non-measurable variables;

and estimating the unmeasurable variable based on the maximum likelihood variable interval recursion least square algorithm to obtain a parameter estimation function.

Preferably, the nonlinear system formula is:

wherein y (t) is the output of the nonlinear system, θ is the parameter vector, v (t) is the noise,is the transpose of the information vector.

Preferably, the process of constructing the maximum likelihood based variable interval recursive least squares algorithm comprises:

obtaining a maximum likelihood estimation function based on the nonlinear system;

constructing an estimation function of parameters to be identified based on the maximum likelihood estimation function;

based on the estimation function and the nonlinear system function, calculating to obtain an unmeasurable variable, wherein the unmeasurable variable comprises: undetectable output and undetectable noise.

Preferably, the maximum likelihood estimation function formula is:

wherein the method comprises the steps ofIs a maximum likelihood estimate of θ, let ∈ ->

Preferably, the estimation function formula of the parameter to be identified is:

wherein,is the parameter to be identified.

Preferably, the formula for calculating the non-measurable output is:

wherein,is an undetectable output;

the formula of the undetectable noise is:

wherein,is an undetectable noise.

Preferably, the measurable data is a data set { y (t) _s ):s＝0，1，2，···}。

Preferably, the non-measurable data is { y (t _s +1)，y(t _s +2)，···，y(t _s+1 -1):s＝0，1，2，···}。

Preferably, before estimating the non-measurable variable, further comprises:

and adding measurement noise to the output signal of the nonlinear system, and verifying the variable interval recursive least square algorithm based on the measurement noise.

Preferably, the process of obtaining the parameter estimation function includes:

and estimating the unmeasurable variable by adopting a maximum likelihood principle based on the variable interval recursive least square algorithm to obtain a parameter estimation function.

Compared with the prior art, the application has the following advantages and technical effects:

the application provides a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model, which comprises the following steps: constructing a nonlinear system of a continuous stirring reaction kettle, and obtaining output data based on the nonlinear system and input data, wherein the output data comprises: measurable data and non-measurable data; constructing a variable interval recursion least square algorithm based on maximum likelihood; constructing an auxiliary model based on the measurable data, and replacing the non-measurable data with the output data of the auxiliary model to obtain non-measurable variables; and estimating the unmeasurable variable based on the maximum likelihood variable interval recursion least square algorithm to obtain a parameter estimation function.

Aiming at the identification difficulty of the non-measurable variable, the application changes the sampling interval, replaces the non-measurable variable by using the auxiliary model output, and introduces the maximum likelihood principle to directly identify the known and unknown parameters of the system aiming at the colored noise.

Drawings

The accompanying drawings, which are included to provide a further understanding of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the application. In the drawings:

FIG. 1 is a graph showing the variation of the ML-IV-RLS estimation error with s according to the embodiment of the present application;

FIG. 2 is a graph showing the variation of the ML-IV-RLS and the IV-RLS estimation error delta with s according to the embodiment of the present application;

FIG. 3 is a ML-IV-RLS estimation according to an embodiment of the present applicationSchematic diagram of variation curve with s;

FIG. 4 shows a sigma of an embodiment of the application ² ＝0.11 ² At this time, ML-IV-RLS estimationSchematic diagram of variation curve with s;

FIG. 5 shows a sigma of an embodiment of the application ² ＝0.05 ² When the ML-IV-RLS algorithm predicts the value along with t _s Schematic diagram of a change curve;

FIG. 6 is a schematic diagram of simulation input/output data of a CSTR process in accordance with an embodiment of the present application;

FIG. 7 is a graph showing the variation of the parameter estimation error delta with s according to an embodiment of the present application;

FIG. 8 is an ML-IV-RLS estimation according to an embodiment of the present applicationSchematic diagram of variation curve with s;

FIG. 9 is a flow chart of a method according to an embodiment of the application.

Detailed Description

It should be noted that, without conflict, the embodiments of the present application and features of the embodiments may be combined with each other. The application will be described in detail below with reference to the drawings in connection with embodiments.

It should be noted that the steps illustrated in the flowcharts of the figures may be performed in a computer system such as a set of computer executable instructions, and that although a logical order is illustrated in the flowcharts, in some cases the steps illustrated or described may be performed in an order other than that illustrated herein.

Example 1

The embodiment provides a novel recursion estimation method of a continuous stirring reaction kettle based on an auxiliary model, which comprises the following steps:

constructing a nonlinear system of a continuous stirring reaction kettle, and obtaining output data based on the nonlinear system and input data, wherein the output data comprises: measurable data and non-measurable data;

constructing a maximum likelihood variable interval recursion least square algorithm;

constructing an auxiliary model based on the measurable data, and replacing the non-measurable data with the output data of the auxiliary model to obtain non-measurable variables;

and estimating the unmeasurable variable based on the maximum likelihood variable interval recursion least square algorithm to obtain a parameter estimation function.

In this embodiment, the input data is the condensing flow rate q _c (t)

In this embodiment, the method specifically includes the following steps:

1. system description and identification model

Consider a sparse measurement output nonlinear system

y(t)＝A(z)f(y(t))+B(z)u(t)+D(z)v(t) (1)

Where y (t) is the system output, u (t) is the input, v (t) is the noise, and the back shift operator z ^-1 The polynomials a (z), B (z), and D (z), defined as:

let n be _a ，n _b And n _d When t is equal to or less than 0, y (t) =0, u (t) =0, and v (t) =0.

Defining a parameter vector θ and an information vectorThe method comprises the following steps:

equation (1) can be written as:

for sparse measurement system, u (t) is fully measurable, defining { t } _s S=0, 1,2, ·· } satisfies:

0＝t ₀ <t ₁ <t ₂ <t ₃ <···<t _s-1 <t _s <···，

t ^* s:＝t _s -t _s-1 gtoreq 1, assuming y (t) is only at t=t _s (s=0, 1,2, ··.) is measurable, or dataset { y (t) _s ) S=0, 1,2, ·· } contains all measurable outputs, the non-measurable outputs { y (t) _s +1)，y(t _s +2)，···，y(t _s+1 -1): s=0, 1,2, ·· } is the missing data, with t _s Substituting t in formulas (2) and (3) gives:

or (b)

v(t _s )＝D ^-1 (z)[y(t _s )-A(z)f(y(t _s ))-B(z)u(t _s )] (5)

In the information vectorContains the undetectable missing data f (y (ts-i)), and the system parameters cannot be identified by the traditional least squares method. The present embodiment derives a maximum likelihood based variable interval recursive least squares method using only measurable inputs and sparse outputs y (t _s ) And directly identifying system parameters.

2. Variable interval recursion least square algorithm based on maximum likelihood

The maximum likelihood estimate of equation (4) can be obtained by minimizing the following criterion function:

wherein the method comprises the steps ofIs a maximum likelihood estimate of θ, let ∈ ->Thus->Andcan be written as:

constructionThe estimation is:

observing (4) by combining θ andwith its corresponding estimate->and/>Can calculate and get->The method comprises the following steps:

v (t) in the formula (5) _s ) Regarding a _i ，b _i and d _i At the pointPartial derivative at:

definition:

thus (t) _s ) The method can be written as follows:

deriving a maximum likelihood-based variable interval recursive least squares (ML-IV-RLS) algorithm:

the relevant steps of the ML-IV-RLS Algorithm are listed in Algorithm 1, as shown in FIG. 9;

algorithm 1ML-IV-RLS Algorithm

Initializing:

let t=1, give the data length Land t ^* _s Input/output data u (t), y (t) are collected _s )；

Setting initial data;

P(0)＝p _O I _n j is +.0 +.>

and/>p ₀ ＝10 ⁶ ；

For t≤L do

Constructed by the formulae (14) and (13), respectivelyAnd->

Calculated by the formulas (18) and (19), respectivelyAnd->

Calculated by using (15) - (17) respectivelyAnd->

The gain vector L (t) is calculated by applying (11) and (12) respectively _s )

And covariance matrix P (t) _s )；

Updating parameter estimates using (9)

Collecting missing parameter estimates using (10)from；

Calculation using (20)

if t≤L

t:＝t+1；

else interrupt;

end

end for

in the embodiment, the identification difficulty caused by processing missing data by a variable interval recursion identification method based on an auxiliary model is deduced, the basic method is to construct the auxiliary model by using measurable data or calculated data, and then the identification problem is solved by using the output of the auxiliary model to replace an non-measurable variable; the colored noise is directly processed by applying the principle of maximum likelihood, and parameter estimation is obtained under the condition that the model is not limited to only one parameter vector regression model.

In this example, a Continuous Stirred Tank Reactor (CSTR) is taken as an example to further verify the ML-IV-RLS algorithm, u (t) is the condensing rate q _c (t), y (t) is the concentration C of product A _A (t) data length t=l=1500, to verify the proposed algorithm, adding a mean value of 0 to the output signal and variance of σ ² ＝0.0005 ² Based on training data q _c ∈[98，101，104]The simulated input output dataset for L/min is shown in FIG. 6.

Nonlinear model t of CSTR process ^* _s =2 is

y(t)＝-0.02568y(t-1) ² +0.06539u(t-1)+v(t)-0.31776v(t-1)，

θ＝＝[-0.02568，0.06539，-0.31776]T.

The proposed method is applied to identify the process, parameter estimation and errorCSTR process ML-IV-RLS estimation and error, as shown in Table 1; the simulation results are shown in fig. 7-8, demonstrating the effectiveness of the proposed algorithm.

TABLE 1

Consider the following output nonlinear system t ^* s＝2:

y(t)＝0.38y ² (t-1)+0.91u(t-1)+v(t)+0.05v(t-1)，

The parameter vector to be estimated is:

θ＝[a ₁ ，b ₁ ，d ₁ ]T＝[0.38，0.91，0.05]T.

in this embodiment, the input u (t) is an independent continuous excitation signal vector sequence satisfying u (t) to N (0, 1), and v (t) isThe Gaussian white noise sequence satisfies v (t) to N (0, sigma) ² ) Y (t) is the sequence of simulation model parameter generation.

First, sigma ² ＝0.05 ² ，t ^* _s =1 and t ^* _s When=2, the parameter estimation error of the ML-IV-RLS algorithm is shown in fig. 1;

second, sigma ² ＝0.05 ² When the ML-IV-RLS algorithm and the parameter estimation error of the IV-RLS algorithm are shown in figure 2;

finally, sigma ² ＝0.05 ² Sum sigma ² ＝0.11 ² When the ML-IV-RLS algorithm and the IV-RLS algorithm of formulas (9) - (20) are applied, the exemplary system is estimated, the parameters are estimated and the errors are estimated

Wherein IV-RLS estimation and error (σ) ² ＝0.05 ² ) As shown in Table 2, ML-IV-RLS estimation and error (σ ² ＝0.05 ² ) As shown in Table 3, ML-IV-RLS estimation and error (σ ² ＝0.11 ² ) As shown in Table 4, the parameter versus s curve is shown in FIGS. 3-4, and the ML-IV-RLS algorithm predicted value versus t _s The variation is shown in fig. 5.

TABLE 2

TABLE 3 Table 3

TABLE 4 Table 4

Comparison conclusion of this example:

FIG. 1 shows that the parameter estimation error of the ML-IV-RLS algorithm decreases with the increase of s, and the influence of the missing data on the convergence speed is not great;

the values in tables 1-2 and FIG. 2 illustrate that the ML-IV-RLS algorithm performs better than the IV-RLS algorithm;

the values in tables 2-3 show that the ML-IV-RLS algorithm performs better as the noise variance increases;

FIGS. 3-4 show that the ML-IV-RLS parameter estimation is very close to the true value;

FIG. 5 illustrates predicted outputAnd true output y (t _s ) The estimation model is substantially consistent and well describes the dynamics of the system.

The beneficial effects of the embodiment are that:

in the embodiment, aiming at the identification difficulty of the non-measurable variable, the sampling interval is changed, the non-measurable variable is replaced by the auxiliary model output, and the known and unknown parameters of the system are directly identified by introducing the maximum likelihood principle aiming at colored noise.

In order to further improve the parameter estimation precision and speed up the convergence rate, the embodiment provides a variable interval recursive least square method based on maximum likelihood, and compared with a variable interval recursive least square (IV-RLS) method, a simulation result verifies theoretical discovery, and a simulation result of an application example shows that the proposed method can generate reliable parameter estimation.

The present application is not limited to the above-mentioned embodiments, and any changes or substitutions that can be easily understood by those skilled in the art within the technical scope of the present application are intended to be included in the scope of the present application. Therefore, the protection scope of the present application should be subject to the protection scope of the claims.

Claims (10)

1. The novel recursion estimation method of the continuous stirring reaction kettle based on the auxiliary model is characterized by comprising the following steps of:

constructing a nonlinear system of a continuous stirring reaction kettle, and obtaining output data based on the nonlinear system and input data, wherein the output data comprises: measurable data and non-measurable data;

constructing a maximum likelihood variable interval recursion least square algorithm;

constructing an auxiliary model based on the measurable data, and replacing the non-measurable data with the output data of the auxiliary model to obtain non-measurable variables;

and estimating the unmeasurable variable based on the maximum likelihood variable interval recursion least square algorithm to obtain a parameter estimation function.

2. The novel recursion estimation method of the continuous stirred tank reactor based on the auxiliary model as set forth in claim 1, wherein the nonlinear system formula is:

wherein,is the output of the nonlinear system, θ is the parameter vector, v (t) is noise, ++>Is the transpose of the information vector.

3. The continuous stirred tank reactor novel recursion estimation method based on the auxiliary model as claimed in claim 1, wherein the process of constructing the maximum likelihood-based variable interval recursion least square algorithm comprises the following steps:

obtaining a maximum likelihood estimation function based on the nonlinear system;

constructing an estimation function of parameters to be identified based on the maximum likelihood estimation function;

based on the estimation function and the nonlinear system function, calculating to obtain an unmeasurable variable, wherein the unmeasurable variable comprises: undetectable output and undetectable noise _。

4. The novel recursive estimation method of a continuous stirred tank reactor based on an auxiliary model as in claim 3, wherein the maximum likelihood estimation function formula is:

wherein the method comprises the steps ofIs a maximum likelihood estimate of θ, let ∈ ->

5. The novel recursive estimation method of a continuous stirred tank reactor based on an auxiliary model according to claim 3, wherein the estimation function formula of the parameters to be identified is:

wherein,is the parameter to be identified.

6. The continuous stirred tank reactor novel recursive estimation method based on the auxiliary model according to claim 3, wherein the formula for calculating the unmeasurable output is:

wherein,is an undetectable output;

the formula of the undetectable noise is:

wherein,is an undetectable noise.

7. The novel recursion estimation method of continuous stirred tank reactor based on auxiliary model as claimed in claim 1, wherein the measurable data is a data set

8. The novel recursion estimation method of the continuous stirred tank reactor based on the auxiliary model as set forth in claim 1, wherein the unmeasurable data is

9. The method for novel recursive estimation of a continuous stirred tank reactor based on an auxiliary model according to claim 1, further comprising, before estimating the non-measurable quantity:

and adding measurement noise to the output signal of the nonlinear system, and verifying the variable interval recursive least square algorithm based on the measurement noise.

10. The continuous stirred tank reactor novel recursive estimation method based on the auxiliary model according to claim 9, wherein the process of obtaining the parameter estimation function comprises the following steps:

and estimating the unmeasurable variable by adopting a maximum likelihood principle based on the variable interval recursive least square algorithm to obtain a parameter estimation function.