A method for selecting estimable parameter subsets for reaction process mechanism modeling

Abstract

The present invention provides a method for selecting an estimable parameter subset for reaction process mechanism modeling. First, according to the mechanism analysis, a model of the reaction process is established, the nominal value of the parameter is obtained in the prior knowledge, and the simplified Hessian matrix of the parameter is calculated at the nominal value of the parameter. Then, the original parameter is converted based on the simplified Hessian matrix of the parameter, a Monte Carlo simulation is designed, and the simplified Hessian matrix is ​​used to calculate the mean square error of the model output prediction, and the parameter subset that minimizes the mean square error of the model output prediction is selected as the optimal parameter subset. The present invention avoids the problem of solving the sensitivity matrix, thereby saving a lot of calculation time.

Description

Technical Field

[0001] The invention relates to the technical field of reaction process mechanism modeling, and in particular to a method for selecting an estimable parameter subset for reaction process mechanism modeling. Background Art

[0002] Parameter estimation is an important part of the reaction process mechanism modeling. However, due to the estimability of parameters and the correlation between parameters, it is usually impossible to estimate all parameters. An effective approach is to select a subset of estimable parameters to minimize the prediction error of the established mechanism model. Among them, using the mean square error (MSE) to select a subset of estimable parameters is a more common method, but this method must use the sensitivity matrix when calculating the mean square error of the estimated parameters or system output. There are generally two methods for obtaining the sensitivity matrix: one is a solution method based on the parameter sensitivity equation, and the other is a solution method based on adaptive step difference. The former method has high calculation accuracy, but the method is complicated and the solution efficiency is low; the latter method is easy to implement, but the calculation accuracy is not high and the solution efficiency is also average. Summary of the invention

[0003] In view of the problems existing in the prior art, the present invention proposes a method for selecting an estimable parameter subset for reaction process mechanism modeling. The method selects the optimal parameter subset by minimizing the prediction mean square error of the model, avoids the problem of directly solving the sensitivity matrix, and equivalently replaces the sensitivity matrix by solving the simplified Hessian matrix. The prediction mean square error obtained by the method of the present invention is consistent with the prediction mean square error based on the sensitivity matrix, and can effectively improve the calculation efficiency of the prediction mean square error.

[0004] To achieve the above object, the present invention adopts the following technical solution: a method for selecting a subset of estimable parameters for reaction process mechanism modeling, comprising the following steps:

[0005] (1) Based on the mechanism analysis, a model of the reaction process is established and the nominal values ​​of the model parameters are obtained based on prior knowledge;

[0006] (2) according to the model of the reaction process, calculating the simplified Hessian matrix of the model parameters at the nominal values ​​of the model parameters;

[0007] (3) Transform the model parameters based on the simplified Hessian matrix of the model parameters, design a Monte Carlo simulation, and use the simplified Hessian matrix to calculate the MSE of the model output prediction;

[0008] (4) According to the parameter selection criteria, select the optimal subset of estimable parameters.

[0009] Furthermore, the simplified Hessian matrix Hr It is expressed as:

[0010]

[0011] Among them, Z is the null space matrix of the constrained gradient, C y is a diagonal matrix where the diagonal elements are the noise variance values ​​of each measurement.

[0012] Furthermore, the null space Z of the constrained gradient is expressed as:

[0013]

[0014] Among them, β is the model parameter vector, γ represents the state at the sampling point, c represents the equality constraint, h represents the functional relationship between the state and the output, and I is the unit matrix.

[0015] Furthermore, the calculation process of the MSE is:

[0016]

[0017]

[0018] Among them, MSE(β * ) is the mean square error of the model parameter estimation, MSE(y*) is the mean square error of the model parameter prediction, α2 0 is the calibration value, δ 2 is the measurement noise variance, λ is the eigenvalue of the simplified Hessian matrix, and p is the number of estimated model parameters.

[0019] Compared with the prior art, the invention has the following beneficial effects: when solving the MSE, the estimable parameter subset selection method of the invention uses the simplified Hessian matrix to replace the sensitivity matrix, thereby saving a lot of solution time while ensuring that accurate results can be obtained. In addition, when dealing with large-scale reaction process model parameter estimation problems, it is difficult to calculate or even impossible to obtain the sensitivity matrix, and this method has better applicability. DETAILED DESCRIPTION

[0020] In the process of estimating parameters, the method of the present invention converts the parameters to the direction of the principal component of the covariance matrix of the parameter estimation in order to eliminate the correlation between the parameters. The converted parameters are estimated, and then the optimal parameter subset is selected based on the mean square error. In the expression of the mean square error, the information provided by the sensitivity matrix can be obtained in the simplified Hessian matrix, and the simplified Hessian matrix can be solved instead of solving the sensitivity matrix.

[0021] The present invention provides a method for selecting a subset of estimable parameters for reaction process mechanism modeling, which specifically comprises the following steps:

[0022] (1) Based on the mechanism analysis, a model of the reaction process is established and the nominal values ​​of the model parameters are obtained based on prior knowledge;

[0023] (2) according to the model of the reaction process, calculating the simplified Hessian matrix of the model parameters at the nominal values ​​of the model parameters;

[0024] For a nonlinear differential algebraic equation (DAE) model:

[0025] dx(t) / dt=f(x(t),z(t),β),x(t0)=x0

[0026] g(x(t),z(t),β)=0

[0027] y(t)=h(x(t),z(t),β)

[0028] y m (t) = y(t) + ε

[0029] Among them, x(t) and z(t) represent differential state variables and algebraic state variables, y(t) and y m (t) represents the predicted and measured values ​​of the model, β is the parameter vector, ε is the mean of 0, and the variance is a constant δ 2 After discretizing the above equation, the parameter estimation problem can be expressed as:

[0030]

[0031]

[0032] Among them, γ represents the state at the sampling point, c represents the equality constraint, and C y Is a diagonal matrix whose diagonal elements are the variances of the output. The null space Z of the constrained gradient can be expressed as:

[0033]

[0034] Therefore, the simplified Hessian matrix H r It is expressed as:

[0035]

[0036] Among them, h represents the functional relationship between state and output, I is the unit matrix, C y is a diagonal matrix where the diagonal elements are the variance values ​​of the measurement noise.

[0037] (3) Based on the simplified Hessian matrix of model parameters, the model parameters are transformed, a Monte Carlo simulation is designed, and the simplified Hessian matrix is ​​used to calculate the MSE of the model output prediction. The specific process is as follows:

[0038] The system model can be linearized at the nominal values ​​of the parameters as:

[0039] y=y 0 +S(β-β 0 )

[0040] =Sβ+b

[0041] b=y 0 -Sβ 0

[0042] y m =y+ε

[0043] Among them, y and y m Represent the actual value and measured value of the output vector, y 0 represents the output obtained at the nominal value of the parameter, S is the sensitivity matrix at the nominal value of the parameter, β 0 is the nominal value of the parameter, ε is zero, and the variance is a constant δ 2 White noise. Decompose the sensitivity matrix S singular value:

[0044] S=UQV T

[0045] The decomposed orthogonal matrices U, V and diagonal elements are S singular values The diagonal matrix Q with i=1..n. The model output can be expressed as:

[0046] y=SVV T β+b

[0047] =Wα=b

[0048] =W1α1+W2α2+b

[0049] Wherein, y represents the output vector, V is the orthogonal matrix obtained by singular value decomposition (S=UQV) of the sensitivity matrix S, b is the constant term, and α is the conversion parameter, including the selected parameter subset α1 and the unselected parameter subset α2.

[0050] W=SV=UQ

[0051] α=V T β

[0052]

[0053]

[0054] where u i is the i-th column vector of the orthogonal matrix U, is the i-th singular value of the sensitivity matrix.

[0055] The MSE for parameter estimation and model prediction can be expressed as:

[0056]

[0057]

[0058] Where: p is the number of parameters in the estimable parameter subset, β * and * They represent the optimal parameter estimation value and model prediction value obtained in parameter estimation respectively. The above is a method for solving MSE through sensitivity matrix. The present invention utilizes the relationship between the simplified Hessian matrix and the sensitivity matrix to avoid the calculation of the sensitivity matrix. From the existing knowledge, it can be known that:

[0059]

[0060] Where Q Hr It contains H r Eigenvalue λ i , a diagonal matrix with i=1..n and In the MSE equation It can be replaced by λ according to the above relationship:

[0061]

[0062]

[0063]

[0064] Therefore:

[0065]

[0066]

[0067] The orthogonal matrix V and singular values ​​required in the sensitivity matrix in the mean square error equation are It can be obtained by solving the simplified Hessian matrix. The present invention avoids the problem of directly solving the sensitivity matrix. Note that calculating MSE requires a priori information about the given measurement noise variance and uncertainty distribution and range. Taking into account the uncertainty of the true value, when calculating MSE using the above equation, each sample of the true parameter value is evaluated by Monte Carlo simulation of the true parameter value of the given distribution, and then the average MSE is obtained.

[0068] (4) According to the parameter selection criteria, select the optimal subset of estimable parameters. The specific process includes the following:

[0069] The focus of the present invention is to obtain accurate predictions rather than exact parameters. In the proposed criteria, it is necessary to give a priori information on the measurement noise variance and uncertainty distribution and range to minimize the MSE average value of the model prediction to determine the optimal value of p. The criteria are as follows:

[0070] p = argminMSE(y * )

[0071] When determining the optimal p, the effect of uncertainty in the true parameter value should be considered. Each sample of the true parameter value is evaluated by Monte Carlo simulation of a given distribution of the true parameter value, and then the average MSE is obtained. The overall procedure for Monte Carlo simulation using the criterion is as follows:

[0072] Step 1: Sample the true parameter value based on the parameter range and distribution.

[0073] Step 2: For each value of p (p = 1..n, n is the number of parameters), calculate the MSE.

[0074] Step 3: Repeat steps 1-2 for all samples, then calculate the average MSE of the model predictions for each p-value, choosing the p corresponding to the minimum average MSE of the model predictions.

[0075] Once the optimal p value is determined, the optimal subset of estimable transformation parameters is determined, that is, the first p transformation parameters. The transformation parameters in the selected subset are estimated, and the unselected transformation parameters are fixed to the nominal value to obtain a set of optimized transformation parameters. Since the characteristic matrix V is reversible, β = Vα. The original parameters can be obtained from the optimized transformation parameters.

[0076] Example

[0077] Selection of a subset of estimable parameters is performed in the CSTR system model.

[0078] Step 1: Modeling based on the reaction process mechanism;

[0079] The CSTR system model is as follows:

[0080]

[0081]

[0082]

[0083] R A =ke -E / (RT) c A

[0084] Among them, the feed rate F is 0.1m 3 / hr, coolant flow rate Fc 0.15m 3 / hr, initial state C A0 1000mol / m 3 , the initial temperature of the reactor T0 and the initial temperature of the cooling liquid T c 293.15K and 293.15K respectively, and the reactor volume V is 0.2m 3 , cooling jacket volume V c 0.055m 3 , the heat transfer area A is 4.5m 3 , coolant heat capacity C pc 1.2 kJ kg -1 K -1 , fluid heat capacity C p 1.55 kJ kg -1 K -1 .

[0085] The parameters to be estimated and their true values ​​are shown in Table 1:

[0086] Table 1 shows the parameters to be estimated and their true values

[0087]

[0088] Step 2: Calculate the simplified Hessian matrix of the parameter at the nominal value of the parameter;

[0089] The experiment selected the reactor temperature T as the output, and took 150 sampling points within 0-5h. The nominal value of the experimental setting parameters is 1.2 times its true value (each parameter nominal value is equal to the true value multiplied by 1.2), and the simplified Hessian matrix and sensitivity matrix are obtained at the nominal value. The calculation was solved under the Windows Server 2012R2Standard operating system using the processor Inter(R)Xeon(R)Silver 4112CPU, 2.60GHz, 64GRAM configuration. The calculation results show that the time to solve the simplified Hessian matrix of the CSTR system is 0.50980s, and the time to solve the sensitivity matrix using adaptive step size is 4.51949s.

[0090] Step 3: Design a Monte Carlo experiment and calculate the MSE of the model predictions;

[0091] When calculating MSE, the prior information of measurement noise variance and parameter distribution and range is given. Considering the uncertainty of the true value, 10,000 Monte Carlo simulation experiments are performed when calculating the mean square error. The experimental setting parameters are evenly distributed between 0.5 and 1.5 times the true value of the parameter. Each experiment randomly samples the true value of the parameter according to the distribution of the parameter within the parameter range. The variance of the parameter and the prediction is obtained, the mean is calculated, and the average mean square error is obtained.

[0092] The parameters and prediction mean square errors obtained by the two methods are shown in Table 2:

[0093] Table 2 is a comparison of the parameters and prediction mean square errors obtained by the simplified Hessian matrix and the adaptive finite difference method.

[0094]

[0095] Step 4: According to the parameter selection criteria, select the optimal subset of estimable parameters, estimate the parameters in the selected subset, and fix the unselected parameters to the nominal values. Solve the optimized model and calculate the prediction error of the model.

[0096] The results in Table 2 show that the mean square error calculated by solving the simplified Hessian matrix is ​​basically consistent with the result obtained by calculating the sensitivity matrix, and the calculation time required is less than that of calculating the sensitivity matrix. According to the principle of selecting the optimal subset proposed in equation S4, MSEY obtains the minimum value when p = 4, so the first 4 conversion parameters of α are selected for estimation, and the remaining 5 conversion parameters are fixed at the nominal value.

[0097] The contents described in the embodiments of this specification are merely an enumeration of the implementation forms of the inventive concept. The protection scope of the present invention should not be regarded as limited to the specific forms described in the embodiments. The protection scope of the present invention also involves equivalent technical means that can be thought of by those skilled in the art based on the inventive concept.

Claims

1. A method for selecting a subset of estimable parameters for reaction process mechanism modeling, characterized in that: The following steps are involved: (1) Based on the mechanism analysis, a model of the reaction process is established and the nominal values ​​of the model parameters are obtained based on prior knowledge; (2) according to the model of the reaction process, calculating the simplified Hessian matrix of the model parameters at the nominal values ​​of the model parameters; (3) Transform the model parameters based on the simplified Hessian matrix of the model parameters, design a Monte Carlo simulation, and use the simplified Hessian matrix to calculate the MSE of the model output prediction; The calculation process of the MSE is: Among them, MSE(β * ) is the mean square error of the model parameter estimation, β * represents the optimal parameter estimate obtained in parameter estimation, MSE(y*) is the mean square error of the model parameter prediction, y * They represent the model prediction values ​​obtained in parameter estimation, α2 0 is the calibration value, δ 2 is the measurement noise variance, λ is the eigenvalue of the simplified Hessian matrix, p is the number of parameters in the selected parameter subset to be estimated, and n is the number of all parameters; u i is the i-th column vector of the orthogonal matrix U, is the i-th singular value of the sensitivity matrix; α2＝V T β In the formula, V is the feature matrix, β is the model parameter vector; (4) According to the parameter selection criteria, select the optimal subset of estimable parameters.

2. The method for selecting a subset of estimable parameters for reaction process mechanism modeling according to claim 1, characterized in that: The simplified Hessian matrix H r It is expressed as: Among them, Z is the null space matrix of the constrained gradient, C y is a diagonal matrix where the diagonal elements are the noise variance values ​​of each measurement.

3. The method for selecting a subset of estimable parameters for reaction process mechanism modeling as claimed in claim 2, characterized in that: The null space matrix Z of the constrained gradient is expressed as: Among them, β is the model parameter vector, γ represents the state at the sampling point, c represents the equality constraint, h represents the functional relationship between the state and the output, and I is the unit matrix.

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