Parameter Identification Method for Continuous Stirred Tank Reactor Based on Two-Stage Least Squares

Abstract

The present invention provides a parameter identification method for a continuous stirred tank reactor based on the two-stage least squares method, belonging to the technical field of fine chemical process identification, and solves the technical problem of low parameter identification accuracy of the continuous stirred tank reactor. The technical solution is as follows: It includes the following steps: Step 1) Establish a fractional-order Wiener model of the continuous stirred tank reactor; Step 2) Construct an identification process of the two-stage weighted recursive least squares algorithm. The beneficial effect of the present invention is that the two-stage weighted recursive least squares algorithm proposed by the present invention has a fast convergence speed and a high convergence accuracy, and can be well applied to the parameter identification of the continuous stirred tank reactor.

Description

Technical Field

[0001] The present invention relates to the technical field of fine chemical process identification, and particularly to a parameter identification method for a continuous stirred tank reactor based on two-stage least squares method. Background Art

[0002] With the rapid development of science and technology, the industrial process will inevitably put forward more stringent and urgent requirements for the control of the actual production process. In order to meet such control requirements and achieve better control effects, it is necessary to establish an accurate and effective mathematical model for the actual production process. The continuous stirred tank reactor belongs to the field of fine chemicals and is particularly widely used in the industrial world. The continuous stirred tank reactor is a commonly used device in chemical production. When the reaction raw materials enter the reactor at a stable flow rate, the reaction materials in the reactor flow out at a stable flow rate. Since the continuous stirred tank reactor is a typical non-linear system, modeling it has become the key and difficult point of production control. In order to better analyze and predict the production process, it is necessary to establish an accurate mathematical model for the continuous stirred tank reactor and identify the parameters of the established model at the same time.

[0003] For this reason, researchers have also proposed different identification methods, such as: stochastic gradient algorithm, Newton iteration algorithm, particle swarm algorithm, etc. The stochastic gradient algorithm has problems of slow convergence speed and low identification accuracy in parameter identification; the Newton iteration algorithm is an iterative algorithm, and the inverse matrix of the Hessian matrix of the objective function needs to be solved at each step, with complex calculations and large memory occupancy; although the particle swarm algorithm as a swarm intelligence algorithm can be better applied to different working conditions, the selection of genetic operators is sometimes troublesome and there is a problem of low identification accuracy. In the literature "Identification of Hammerstein System Based on Improved Differential Evolution", although the improved differential evolution algorithm is used to identify the parameters of the continuous stirred tank reactor, the improved differential evolution algorithm as an intelligent algorithm has problems of excessive calculation amount and difficult selection of initial parameters, which is not conducive to the parameter identification of the continuous stirred tank reactor in the actual production process.

[0004] How to solve the above technical problems is the subject faced by the present invention. Summary of the Invention

[0005] The purpose of the present invention is to provide a parameter identification method for a continuous stirred tank reactor based on two-stage least squares method. The proposed two-stage weighted recursive least squares algorithm has a faster convergence speed and higher convergence accuracy, and can be better applied to the modeling and parameter identification of the continuous stirred tank reactor.

[0006] The present invention is implemented by the following measures: A parameter identification method for a continuous stirred tank reactor based on two-stage least squares method specifically includes the following steps:

[0007] Step 1) Establish a fractional-order Wiener model for a continuous stirred tank reactor;

[0008] Step 2) Construct an identification process for a two-stage weighted recursive least squares algorithm.

[0009] As a further optimized solution for the least squares-based parameter identification method of a continuous stirred tank reactor provided by the present invention, the specific modeling steps of the above Step 1) are as follows:

[0010] Step 1-1) Construct a fractional-order Wiener model for a continuous stirred tank reactor, and give the general form of the fractional-order Wiener system. u(t) is the system input signal, y(t) is the system output signal, v(t) is a white noise with a mean of 0 and a variance of σ 2 and satisfying a Gaussian distribution, and the intermediate variable m(t) is an intermediate unmeasurable signal. Among them, N(·) is a non-linear polynomial, and the general form of the model can be obtained:

[0011]

[0012] y(t) = N(·) + v(t), (1)

[0013] where A(z) and B(z) are constant polynomials, and the polynomial factors a i , b j and p i are parameters to be estimated, and α i and β j are the fractional orders of the polynomial denominator. The present invention considers a fractional commensurate system, that is, the fractional orders are multiples of the same base and are known. Then the intermediate signal m(t) is expressed as:

[0014]

[0015] Step 1-2) The present invention uses the Grünwald Letnikov (GL) definition to solve the fractional derivative. The GL definition is:

[0016]

[0017] where Δ is the discrete fractional difference operator, and Δ α x(kh) is the α-order fractional derivative of the function x(k). Let t = kh, where h is the sampling interval and k is the number of samples for calculating the derivative approximation. Substitute Equation (3) into Equation (2), and the intermediate signal m(t) is rewritten as:

[0018]

[0019] Assume that the first coefficient p1 of the non - linear function is 1. The key term m(t) of the non - linear output of the system is separated by using the key - term separation technique. Then the non - linear part can be described as:

[0020]

[0021] Step 1 - 3) Obtain the identification model of the fractional - order Wiener model of the continuous - stirred tank reactor:

[0022]

[0023] In the above formula, is the information vector of the system, expressed as:

[0024]

[0025] θ is the parameter vector of the system, expressed as:

[0026]

[0027] θ2 = [p2, p3, …, p r T ,

[0028] As a further optimization scheme of the least - squares - based parameter identification method for continuous - stirred tank reactors provided by the present invention, the specific process of the algorithm in step 2) includes the following steps:

[0029] Step 2 - 1) Initialization. Given the number of loops h, define the data length L d , the weight factor ω, and let P1(0) = 1 / 10 6 and P2(0) = 1 / 10 6 , t = 1;

[0030] Step 2 - 2) Take the coolant flow rate of the continuous - stirred tank reactor as the input data u(t) and the fluid concentration as the output data y(t), and record the data;

[0031] Step 2 - 3) Replace the intermediate variable m(t) and the fractional - order derivative Δ α m(t) in the information vector with their estimated values and Construct the estimated information vectors and

[0032]

[0033] ​Step 2-4) Calculate the gain vectors L1(t) and L2(t) according to Equations (9) and (10), and calculate the covariance matrices P1(t) and P2(t) according to Equations (11) and (12);

[0034]

[0035]

[0036] Step 2-5) Calculate the parameter vector estimates according to Equations (13) and (14) and

[0037]

[0038] Step 2-6) Calculate according to Equation (15) and calculate the fractional derivative

[0039]

[0040] Step 2-7) Determine whether the maximum number of loops is reached. If not, t = t + 1 and jump to Step 2-3). If so, enter Step 2-8);

[0041] Step 2-8) Output the identification result The identification is completed.

[0042] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0043] (1) The present invention establishes a fractional-order Wiener model for a continuous stirred tank reactor, which is composed of a linear dynamic system in series with a static nonlinear system; different from common integer-order models, the present invention establishes a model with a fractional order when describing the system. When describing a real system, establishing a fractional-order model is more accurate than an integer-order model. From Figure 4 and Figure 5 it can be seen that the algorithm can well identify the internal parameters of the model.

[0044] (2) Compared with the least squares algorithm, the two-stage weighted least squares algorithm introduces the hierarchical identification theory and the weighted identification idea, which improves the convergence speed. The two-stage weighted least squares algorithm can better identify nonlinear systems, and the identification accuracy is also higher, and the obtained estimation error is smaller; at the same time, it also shows that the present identification method has good applicability to continuous stirred tank reactors. Description of the Drawings

[0045] The drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation to the present invention.

[0046] Figure 1 Schematic diagram of the continuous stirred reactor provided by the present invention.

[0047] Figure 2 Overall flowchart of the two-stage weighted least squares algorithm provided by the present invention.

[0048] Figure 3 General schematic diagram of the Wiener fractional-order system of the present invention.

[0049] Figure 4 Schematic diagram of the error curve between the identified parameters and the true values of the present invention.

[0050] Figure 5 Schematic diagram of the error curve between the identified parameters and the true values of the present invention. Detailed implementation manners

[0051] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0052] Embodiment 1

[0053] Refer to Figures 1 to 4 , the technical solution provided in this embodiment is a method for identifying the parameters of a continuous stirred reactor based on the two-stage least squares method, and the specific steps are as follows:

[0054] Step 1) Establish a fractional-order Wiener model of the continuous stirred reactor;

[0055] Step 2) Construct an identification process of the two-stage weighted recursive least squares algorithm.

[0056] As a further optimized solution of the method for identifying the parameters of a continuous stirred reactor based on the least squares provided by the present invention, the specific modeling steps of step 1) are as follows:

[0057] Step 1-1) Construct a fractional-order Wiener model of the continuous stirred reactor, and give the general form of the fractional-order Wiener system. u(t) is the system input signal, y(t) is the system output signal, v(t) is a white noise with a mean of 0 and a variance of σ 2 and satisfying the Gaussian distribution, and the intermediate variable m(t) is an intermediate unmeasurable signal. Among them, N(·) is a non-linear polynomial, and the general form of the model can be obtained:

[0058]

[0059] y(t) = N(·) + v(t), (1)

[0060] where \(A(z)\) and \(B(z)\) are constant polynomials, and the polynomial factors \(a\) i , \(b\) j and \(p\) i are parameters to be estimated, \(\alpha\) i and \(\beta\) j are the fractional orders of the polynomial denominator. Considering a fractional commensurate system, i.e., the fractional orders are multiples of the same base and are known, the intermediate signal \(m(t)\) is expressed as:

[0061]

[0062] Step 1-2) The present invention uses the Grünwald Letnikov (GL) definition to solve the fractional derivative. The GL definition is:

[0063]

[0064] where \(\Delta\) is the discrete fractional difference operator, \(\Delta\) α \(x(kh)\) is the \(\alpha\) - order fractional derivative of the function \(x(k)\). Let \(t = kh\), where \(h\) is the sampling interval and \(k\) is the number of samples for calculating the derivative approximation. Substituting Equation (3) into Equation (2), the intermediate signal \(m(t)\) is rewritten as:

[0065]

[0066] Assume that the first coefficient \(p_1 = 1\) of the nonlinear function. Using the key - term separation technique to separate the key term \(m(t)\) of the system's nonlinear output, the nonlinear part can be described as:

[0067]

[0068] Step 1-3) Obtain the identification model of the fractional - order Wiener model for the continuous stirred - tank reactor:

[0069]

[0070] In the above formula, is the information vector of the system, expressed as:

[0071]

[0072] \(\theta\) is the parameter vector of the system, expressed as:

[0073]

[0074] \(\theta_2=[p_2,p_3,\cdots,p\) r T ,

[0075] ​As a further optimization scheme of the parameter identification method for the continuous stirred tank reactor based on least squares, the specific process of the algorithm in step 2) includes the following steps:

[0076] Step 2-1) Initialization: Given the number of loops h, define the data length L d , the weight factor ω, and let P1(0) = 1 / 10 6 , P2(0) = 1 / 10 6 , t = 1;

[0077] Step 2-2) Take the coolant flow rate of the continuous stirred tank reactor as the input data u(t), and the fluid concentration as the output data y(t), and record the data;

[0078] Step 2-3) Replace the intermediate variable m(t) and the fractional derivative Δ α m(t) in the information vector with their estimated values and Construct the estimation of the information vector according to equations (7) and (8) and

[0079]

[0080] Step 2-4) Calculate the gain vectors L1(t) and L2(t) according to equations (9) and (10), and calculate the covariance matrices P1(t) and P2(t) according to equations (11) and (12);

[0081]

[0082] Step 2-5) Calculate the parameter vector estimates and

[0083]

[0084] Step 2-6) Calculate according to equation (15) and calculate the fractional derivative

[0085]

[0086] Step 2-7) Judge whether the maximum number of loops is reached. If not, t = t + 1 and jump to step 2-3). If so, enter step 2-8);

[0087] Step 2-8) Output the identification result Complete the identification.

[0088] The schematic diagram of the continuous stirred tank reactor applied in this embodiment is as follows Figure 2 shown. Among them, the input signal u(t) is the coolant flow rate of the continuous stirred tank reactor, and the output signal y(t) is the fluid concentration.

[0089] Through the fractional-order Wiener model mentioned above, the following model can be established for this Embodiment 1:

[0090] y(t) = -0.16Δ 0.3 m(t - 1) - 0.17Δ 0.3 m(t - 2) + 0.60Δ 0.3 u(t - 1) - 0.28Δ 0.3 u(t - 2) - 0.29m 2 (t) + 0.71m 3 (t),

[0091] Comparing the above model with step 1), we can obtain

[0092] a1 = 0.16, a2 = 0.17, b1 = 0.60, b2 = -0.28, p2 = -0.29, p3 = 0.71,

[0093] To facilitate substituting the parameters to be identified into the gradient iteration algorithm, the parameters to be identified are composed into a parameter vector θ, and let the parameters to be identified be as follows:

[0094] θ = [a1, a2, b1, b2, p2, p3] T ,

[0095] According to the initialization in step 2-1), a loop count h is given, and the data length L d , and the weight factor ω are defined;

[0096] According to step 2-2), input and output data are collected;

[0097] According to step 2-3), the estimation of the information vector and

[0098] According to step 2-4), the gain vectors L1(t) and L2(t) are calculated, and the covariance matrices P1(t) and P2(t) are calculated;

[0099] According to step 2-5), the estimated values of the parameter vector and

[0100] According to step 2-6), is calculated and the fractional-order derivative

[0101] Complete the loop according to steps 2-7) and 2-8), and output the result.

[0102] Among them, when setting the weight factor ω and the data length L d several issues need to be considered: If the weight factor is too small or too large, the identification error will be too large to obtain an accurate identification result. If the data length is too small, the identification result will be unsatisfactory, leading to a problem of low identification accuracy; if the data length is too large, it will cause a large amount of calculation.

[0103] The parameter identification result obtained by using the parameter identification method of the continuous stirred tank reactor based on least squares in this embodiment is as Figure 4 shown; it can be seen that the identification accuracy of this method is relatively high, and the estimated value of the parameter to be identified is very close to the true value. At the same time, it also shows that this identification method has good applicability for the parameter identification of the continuous stirred tank reactor.

[0104] Embodiment 2

[0105] According to the parameter identification method of the continuous stirred tank reactor based on two-stage least squares in Embodiment 1, the following model can be established in this Embodiment 2:

[0106] y(t) = -0.123Δ 0.43 m(t - 1) - 0.177Δ 0.43 m(t - 2) - 0.124Δ 0.43 m(t - 3) + 0.672Δ 0.43 u(t - 1) - 0.252Δ 0.43 u(t - 2) - 0.306m 2 (t) + 0.828m 3 (t),

[0107] Comparing the above model with step 1), we can obtain

[0108] a1 = 0.123, a2 = 0.177, a3 = 0.124, b1 = 0.672, b2 = -0.252, p2 = -0.306, p3 = 0.828,

[0109] In order to conveniently substitute the parameters to be identified into the gradient iteration algorithm, the parameters to be identified are composed into a parameter vector θ, and let the parameters to be identified be as follows:

[0110] θ = [a1, a2, a3, b1, b2, p2, p3] T ,

[0111] Initialize according to step 2-1), given the number of loops h, define the data length L d , the weight factor ω;

[0112] Collect input and output data according to step 2-2);

[0113] Construct the estimation of the information vector according to step 2-3) and

[0114] Calculate the gain vectors L1(t) and L2(t), and calculate the covariance matrices P1(t) and P2(t) according to step 2-4);

[0115] Calculate the estimated value of the parameter vector according to step 2-5) and

[0116] Calculate according to step 2-6) and calculate the fractional derivative

[0117] Complete the loop according to steps 2-7) and 2-8), and output the result.

[0118] The parameter identification result obtained by using the least-squares-based continuous stirred tank reactor parameter identification method of this embodiment is as Figure 5 shown; it can be seen that the identification accuracy of this method is relatively high, and the estimated values of the parameters to be identified are very close to the true values. At the same time, it also further shows that this identification method has good applicability for the parameter identification of continuous stirred tank reactors.

[0119] The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for identifying parameters of a continuous stirred reactor based on a two-stage least squares method, characterized in that: The following steps are involved: Step 1) establishing a continuous stirred reactor fractional order Wiener model; The step 1) comprises the following steps: Step 1-1) Construct a fractional-order Wiener model for a continuous stirred reactor and give the general form of a fractional-order Wiener system, where u(t) is the input signal, y(t) is the output signal, and v(t) is a signal with a mean of 0 and a variance of σ 2 And the white noise satisfies the Gaussian distribution, the intermediate variable m(t) is the intermediate unmeasurable signal, where N(·) is a nonlinear polynomial, and the general form of the model is obtained: Where A(z) and B(z) are constant polynomials, and the polynomial factor a i ,b j and p i is the parameter to be estimated, α i and β j is the fractional order of the denominator of the polynomial. Considering the fractional symmetric system, that is, the fractional order is a multiple of the same basis and is known, the intermediate signal m(t) is expressed as: Step 1-2) uses the Grünwald Letnikov, GL definition, to solve the fractional derivatives. GL is defined as: where Δ is the discrete fractional difference operator, Δ α x(kh) is the α-order fractional derivative of the function x(k). Let t = kh, where h is the sampling interval and k is the number of samples for calculating the derivative approximation. Substituting equation (3) into equation (2), the intermediate signal m(t) is rewritten as: Assuming that the first coefficient p1 of the nonlinear function is 1, the key term separation technique is used to separate the key term m(t) of the nonlinear output of the system. The nonlinear part is described as: Step 1-3) obtains the identification model of the continuous stirred reactor fractional order Wiener model: In the above formula, is the information vector of the system, expressed as: is the parameter vector of the system, expressed as: Step 2) Construct the identification process of the two-stage weighted recursive least squares algorithm.

2. The method for identifying parameters of a continuous stirred reactor based on a two-stage least squares method according to claim 1, characterized in that: The specific process of the step 2) algorithm includes the following steps: Step 2-1) Initialization, given the number of loops h, define the data length L d , weight factor ω, let P1(0)=1 / 10 6 , P2(0)=1 / 10 6 , t = 1; Step 2-2) Using the coolant flow rate of the continuous stirred tank reactor as input data u(t), and the fluid concentration as output data y(t), and recording the data; Step 2-3) The intermediate variable m(t) and fractional derivative Δ in the information vector α m(t) is replaced by its estimated value and According to equations (7) and (8), the estimation of the information vector is constructed and Step 2-4) Calculate the gain vectors L1(t) and L2(t) according to equations (9) and (10), and calculate the covariance matrices P1(t) and P2(t) according to equations (11) and (12); Step 2-5) Calculate the estimated value of the parameter vector according to equations (13) and (14) and Step 2-6) Calculate according to formula (15) And calculate the fractional derivative Step 2-7) Determine whether the maximum number of cycles has been reached. If not, t=t+1 and jump to step 2-3). If reached, go to step 2-8); Step 2-8) Output the recognition results Complete identification.

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