Electric heating furnace system identification method based on hierarchical least squares

Abstract

The invention provides an electric heating furnace system identification method based on hierarchical least squares, belongs to the technical field of industrial control process system identification, and solves the technical problem of low parameter identification precision of an electric heating furnace system. According to the technical scheme, the hierarchical least square-based electric heating furnace system identification method comprises the following steps of (1) establishing a fractional order state space model of an electric heating furnace system; and 2) constructing an identification process of a hierarchical least square algorithm based on Kalman filtering. The beneficial effects of the invention are that the hierarchical least square algorithm based on Kalman filtering provided by the invention has a fast convergence rate and high convergence precision, and can be well suitable for parameter identification of the electric heating furnace system.

Description

Technical Field

[0001] The invention relates to the technical field of industrial control process system identification, and in particular to a hierarchical least squares electric heating furnace system identification method. Background Art

[0002] With the continuous expansion of the scale of national infrastructure, the demand for steel products has increased dramatically. Electric heating furnaces play an important role in steel rolling production lines and have a significant impact on the entire steel rolling production process. As rolling mill technology develops towards continuity, high speed, large-scale and precision, the requirements for the heating quality of steel billets are becoming increasingly stringent. The key to improving the heating quality of steel billets is to achieve high-precision control of the heating furnace. Electric heating furnaces are common equipment in industrial heating processes, and the stability of the temperature in the furnace is crucial to ensure the smooth progress of subsequent processing and product quality. Its main production task is to heat the steel to the temperature and quality level required by the process according to the operating rhythm of the rolling mill. Due to the large inertia and large hysteresis characteristics of the furnace temperature of the electric heating furnace, modeling becomes a major problem in controlling its production process. In order to accurately control the temperature in the production process, it is necessary to establish an accurate mathematical model and identify the model parameters.

[0003] To this end, researchers have also proposed different identification methods, such as step response method, least squares algorithm and genetic algorithm. The step response method usually relies on graphical analysis of the output data, which limits its accuracy. The analysis of the graphics relies on visual judgment and may be affected by subjective factors, resulting in inaccurate identification results. The least squares method is effective in dealing with linear system parameter identification, but its ability is limited for electric heating furnace systems with nonlinear characteristics. The nonlinear dynamics of the electric heating furnace system may cause the least squares method to be unable to accurately capture all key system behaviors. The genetic algorithm requires an evaluation of each individual in the population, which means that multiple system simulations are required in the electric heating furnace parameter identification, and each simulation may be very time-consuming. Therefore, the application of genetic algorithms in the identification of electric heating furnace systems will result in significant consumption of computing resources.

[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 an electric heating furnace system identification method based on hierarchical least squares. The proposed hierarchical least squares algorithm based on Kalman filtering has a faster convergence speed and higher convergence accuracy, and can be well applied to the modeling and parameter identification of the electric heating furnace system.

[0006] The present invention is achieved by the following measures: an electric heating furnace system identification method based on hierarchical least squares, specifically comprising the following steps:

[0007] Step 1) establishing a fractional-order state space model of an electric heating furnace;

[0008] Step 2) Construct an identification process of a hierarchical least squares algorithm based on Kalman filtering.

[0009] As a further optimization scheme of the electric heating furnace system identification method based on hierarchical least squares provided by the present invention, the specific modeling steps of step 1) are as follows:

[0010] Step 1-1) Construct a fractional-order state-space model of the electric heating furnace system and give the general form of the fractional-order discrete state-space model. u(k) is the system input signal, y(k) is the system output signal, and v(k) is a signal with a mean of 0 and a variance of σ 2 , covariance is R k And the white noise satisfies Gaussian distribution, x(k)=[x1(k)x2(k)…x n (k)] T is the state vector of the system, and the general form of the model is obtained:

[0011]

[0012] C=[1,0,…,0],

[0013] Among them, the matrix and is the system matrix, a n and b n is the parameter to be estimated, α n is a fractional order. Considering a fractional commensurate system, that is, the fractional order is a multiple of the same basis and is known, the fractional order discrete state space model is expressed as:

[0014]

[0015] Step 1-2) uses the Grünwald Letnikov, GL definition, to solve the fractional derivatives. GL is defined as:

[0016]

[0017] 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, k is the k-th sampling, and h = 1. Definition Substituting it into equation (3), the fractional derivative is rewritten as:

[0018]

[0019] According to formula (4), we can get:

[0020]

[0021] Substituting equation (5) into equation (2), we can obtain the fractional-order discrete state space equation based on GL:

[0022]

[0023] Step 1-3) obtains the identification model of the fractional-order discrete state space model of the electric heating furnace:

[0024]

[0025] In the above formula, is the information vector of the system, expressed as:

[0026]

[0027] φ a (k)=[x1(kn),x2(kn),…,x n (kn)] T ,

[0028] φ b (k)=[u(k-1),u(k-2),…,u(kn)] T ,

[0029] θ is the parameter vector of the system, expressed as:

[0030]

[0031] a=[-a1,-a2,…,-a n ] T ,

[0032] b=[b1,b2,…,b n ] T ,

[0033] When the fractional order α is known, Q(k) is a known term, expressed as:

[0034] Q(k)=-w(1)x1(k-1)-…-w(k-1)x1(1)-w(1)x2(k-2)-…-w(k-2)x2(1)-…-w(1)x n (kn)-…-w(kn)x n (1).

[0035] As a further optimization scheme of the electric heating furnace system identification method based on hierarchical least squares provided by the present invention, the specific process of the algorithm of step 2) includes the following steps:

[0036] Step 2-1) Initialization, given the number of loops h, defined data length k max ,make P1(0)=10 6 I n , P2(0)=10 6 I n , P k (0) = I n / 10 2 , k = 1;

[0037] Step 2-2) The heating power of the electric heating furnace system is used as input data u(k), and the temperature inside the furnace is used as output data y(k), and the data is recorded;

[0038] Step 2-3) Replace the state vector x(k) in the information vector with its estimated value According to equations (8) and (9), the estimation of the information vector is constructed and φ b (k);

[0039]

[0040] Step 2-4) Calculate the gain vectors L1(k) and L2(k) according to equations (10) and (11), calculate the covariance matrices P1(k) and P2(k) according to equations (12) and (13), and calculate

[0041]

[0042] Step 2-5) Calculate the estimated value of the parameter vector according to equations (15) and (16) and

[0043]

[0044] Step 2-6) Construct the system estimation matrix according to equations (17) and (18) and

[0045]

[0046] Step 2-7) Calculate the state prediction value according to equations (19) and (20) and covariance predicted values

[0047]

[0048] Step 2-8) Calculate the Kalman gain K according to equations (21) and (22):k (k) and the state update value Update the state estimation covariance matrix P according to formula (23): k (k);

[0049]

[0050] Step 2-9) determines whether the maximum number of cycles has been reached. If not, k=k+1 and jumps to step 2-3). If reached, proceed to step 2-10);

[0051] Step 2-10) Output the recognition result Complete identification.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] (1) The present invention establishes a fractional-order discrete state space model for the electric heating furnace system, with heating power as the input data of the electric heating furnace system and furnace temperature as the output data of the electric heating furnace system. Different from the common integer-order model, the present invention establishes a model with a fractional order when describing the system. The fractional-order model can describe the actual system more accurately. Figure 4 It can be seen that the algorithm can identify the model parameters very well.

[0054] (2) The Kalman filter algorithm is used for state estimation, and the hierarchical least squares algorithm is used for parameter identification. Compared with the least squares algorithm, the hierarchical least squares algorithm divides the system into two subsystems and identifies the parameters of each subsystem separately, thereby reducing the amount of calculation required for identification and improving the identification accuracy. The parameter estimation error is smaller; at the same time, it also shows that this identification method has good applicability to the electric heating furnace system. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0056] Figure 1 This is a schematic diagram of the electric heating furnace system provided by the present invention.

[0057] Figure 2 The present invention provides an overall flow chart of the hierarchical least squares algorithm based on Kalman filtering.

[0058] Figure 3 Schematic diagram of the error curve between the identification parameters and the true values ​​of Example 1 of the present invention.

[0059] Figure 4 Schematic diagram of the error curve between the identification parameters and the true values ​​of Example 2 of the present invention. DETAILED DESCRIPTION

[0060] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. Of course, the specific embodiments described here are only used to explain the present invention and are not used to limit the present invention.

[0061] Example 1

[0062] See also Figures 1 to 4 The technical solution provided in this embodiment is a method for identifying an electric heating furnace system based on hierarchical least squares, and the specific steps are as follows:

[0063] Step 1) establishing a fractional-order discrete state space model of the electric heating furnace system;

[0064] Step 2) Construct an identification process of a hierarchical least squares algorithm based on Kalman filtering.

[0065] As a further optimization scheme of the electric heating furnace system identification method based on hierarchical least squares provided by the present invention, the specific modeling steps of step 1) are as follows:

[0066] Step 1-1) Construct a fractional-order state-space model of the electric heating furnace system and give the general form of the fractional-order discrete state-space model. u(k) is the system input signal, y(k) is the system output signal, and v(k) is a signal with a mean of 0 and a variance of σ 2 , covariance is R k And the white noise satisfies Gaussian distribution, x(k)=[x1(k),x2(k),…,x n (k)] T is the state vector of the system, and the general form of the model is obtained:

[0067]

[0068] C=[1,0,…,0],

[0069] Among them, the matrix and is the system matrix, a n and b n is the parameter to be estimated, α n is a fractional order. Considering a fractional commensurate system, that is, the fractional order is a multiple of the same basis and is known, the fractional order discrete state space model is expressed as:

[0070]

[0071] Step 1-2) uses the Grünwald Letnikov, GL definition, to solve the fractional derivatives. GL is defined as:

[0072]

[0073] 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, k is the k-th sampling, and h = 1. Definition Substituting it into equation (3), the fractional derivative is rewritten as:

[0074]

[0075] According to formula (4), we can get:

[0076]

[0077] Substituting equation (5) into equation (2), we can obtain the fractional-order discrete state space equation based on GL:

[0078]

[0079] Step 1-3) obtains the identification model of the fractional-order discrete state space model of the electric heating furnace:

[0080]

[0081] In the above formula, is the information vector of the system, expressed as:

[0082]

[0083] φ a (k)=[x1(kn),x2(kn),…,x n (kn)] T ,

[0084] φ b (k)=[u(k-1),u(k-2),…,u(kn)] T ,

[0085] θ is the parameter vector of the system, expressed as:

[0086]

[0087] a=[-a1,-a2,…,-a n ] T ,

[0088] b=[b1,b2,…,b n ] T ,

[0089] When the fractional order α is known, Q(k) is a known term, expressed as:

[0090] Q(k)=-w(1)x1(k-1)-…-w(k-1)x1(1)-w(1)x2(k-2)-…-w(k-2)x2(1)-…-w(1)x n (kn)-…-w(kn)x n (1).

[0091] As a further optimization scheme of the electric heating furnace system parameter identification method based on hierarchical least squares provided by the present invention, the specific process of the algorithm of step 2) includes the following steps:

[0092] Step 2-1) Initialization, given the number of loops h, defined data length k max ,make P1(0)=10 6 I n , P2(0)=10 6 I n , P k (0) = I n / 10 2 , k = 1;

[0093] Step 2-2) The heating power of the electric heating furnace system is used as input data u(k), and the temperature inside the furnace is used as output data y(k), and the data is recorded;

[0094] Step 2-3) Replace the state vector x(k) in the information vector with its estimated value According to equations (8) and (9), the estimation of the information vector is constructed and φ b (k);

[0095]

[0096] Step 2-4) Calculate the gain vectors L1(k) and L2(k) according to equations (10) and (11), calculate the covariance matrices P1(k) and P2(k) according to equations (12) and (13), and calculate

[0097]

[0098] Step 2-5) Calculate the estimated value of the parameter vector according to equations (15) and (16) and

[0099]

[0100] Step 2-6) Construct the system estimation matrix according to equations (17) and (18) and

[0101]

[0102] Step 2-7) Calculate the state prediction value according to equations (19) and (20) and covariance predicted values

[0103]

[0104] Step 2-8) Calculate the Kalman gain K according to equations (21) and (22): k (k) and the state update value Update the state estimation covariance matrix P according to formula (23): k (k);

[0105]

[0106] Step 2-9) determines whether the maximum number of cycles has been reached. If not, k=k+1 and jumps to step 2-3). If reached, proceed to step 2-10);

[0107] Step 2-10) Output the recognition result Complete identification.

[0108] The schematic diagram of the electric heating furnace used in this embodiment is as follows Figure 1 As shown in the figure, the input data u(k) is the heating power of the electric heating furnace system, and the output data y(k) is the temperature inside the furnace.

[0109] Through the above-mentioned fractional-order discrete state space model, the following model can be established for this embodiment 1:

[0110]

[0111] The system matrix is:

[0112] C=[1,0],

[0113] Considering the fractional order α = 0.3, we can get:

[0114] a1=0.54, a2=0.67, b1=-0.37, b2=-0.11,

[0115] In order to conveniently substitute the parameters to be identified into the hierarchical least squares algorithm, the parameters to be identified are combined into a parameter vector θ, and the parameters to be identified are as follows:

[0116] θ=[a1,a2,b1,b2] T

[0117] Initialize according to step 2-1), give the number of loops h, and define the data length k max ;

[0118] Collect input and output data according to step 2-2);

[0119] According to step 2-3) construct an estimate of the information vector and φ b (k);

[0120] According to step 2-4), calculate the gain vectors L1(k) and L2(k), calculate the covariance matrices P1(k) and P2(k), and calculate

[0121] Calculate the parameter vector estimate according to steps 2-5) and

[0122] According to steps 2-6) construct the system estimation matrix and

[0123] Calculate the state prediction value according to step 2-7) and covariance predicted values

[0124] According to step 2-8) calculate the Kalman gain K k (k) and the state update value Update the state estimation covariance matrix P k (k);

[0125] Complete the loop according to step 2-9) and step 2-10) and output the result.

[0126] Among them, when setting the estimated value of the initial state Several issues need to be considered when calculating the initial covariance matrix P(k): the estimated value of the initial state should be as close to the true value as possible. An incorrect initial estimate may cause the filter to take longer to converge or even cause the filter to diverge. The initial covariance matrix should be large enough to allow the algorithm to learn and adjust to the real system dynamics, but an initial covariance that is too large may cause the algorithm to be overly sensitive to measurement noise in the early stages.

[0127] The parameter identification results of the electric heating furnace system identification method based on hierarchical least squares in this embodiment are as follows: Figure 3The identification accuracy of the method in this embodiment is high, the parameter identification error is small, the estimated value of the parameter to be identified is very close to the true value, and it also has a high convergence accuracy, which also shows that the identification method has good applicability for parameter identification of electric heating furnace system.

[0128] Example 2

[0129] According to the electric heating furnace system identification method based on hierarchical least squares in Example 1, the following model can be established in Example 2:

[0130]

[0131] The system matrix is:

[0132] C=[1,0,0],

[0133] Considering the fractional order α = 0.2, we can get:

[0134] a1=-0.76, a2=0.18, a3=0.20, b1=0.52, b2=-0.41, b3=-0.75,

[0135] In order to conveniently substitute the parameters to be identified into the hierarchical least squares algorithm, the parameters to be identified are combined into a parameter vector θ, and the parameters to be identified are as follows:

[0136] θ=[a1,a2,a3,b1,b2,b3] T ,

[0137] Initialize according to step 2-1), give the number of loops h, and define the data length k max ;

[0138] Collect input and output data according to step 2-2);

[0139] According to step 2-3) construct an estimate of the information vector and φ b (k);

[0140] According to step 2-4), calculate the gain vectors L1(k) and L2(k), calculate the covariance matrices P1(k) and P2(k), and calculate

[0141] Calculate the parameter vector estimate according to steps 2-5) and

[0142] According to steps 2-6) construct the system estimation matrix and

[0143] Calculate the state prediction value according to step 2-7) and covariance predicted values

[0144] According to step 2-8) calculate the Kalman gain K k (k) and the state update value Update the state estimation covariance matrix P k (k);

[0145] Complete the loop according to step 2-9) and step 2-10) and output the result.

[0146] Among them, when setting the estimated value of the initial state Several issues need to be considered when calculating the initial covariance matrix P(k): the estimated value of the initial state should be as close to the true value as possible. An incorrect initial estimate may cause the filter to take longer to converge or even cause the filter to diverge. The initial covariance matrix should be large enough to allow the algorithm to learn and adjust to the real system dynamics, but an initial covariance that is too large may cause the algorithm to be overly sensitive to measurement noise in the early stages.

[0147] The parameter identification results of the electric heating furnace system identification method based on hierarchical least squares in this embodiment are as follows: Figure 4 The identification accuracy of the method in this embodiment is high, the amount of calculation is less than the least squares algorithm, the estimated value of the parameter to be identified is very close to the true value, and it also has a high convergence accuracy, which also shows that the identification method has good applicability for parameter identification of electric heating furnace system.

[0148] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for identifying parameters of an electric heating furnace system based on hierarchical least squares, characterized in that: The following steps are involved: Step 1) establishing a fractional-order discrete state space model of the electric heating furnace system; Step 2) Construct an identification process of a hierarchical least squares algorithm based on Kalman filtering.

2. The electric heating furnace system parameter identification method based on hierarchical least squares according to claim 1 is characterized in that: The step 1) comprises the following steps: Step 1-1) Construct a fractional-order state-space model of the electric heating furnace system and give the general form of the fractional-order discrete state-space model. u(k) is the system input signal, y(k) is the system output signal, and v(k) is a signal with a mean of 0 and a variance of σ 2 , covariance is R k And the white noise satisfies Gaussian distribution, x(k)=[x1(k),x2(k),…,x n (k)] T is the state vector of the system, and the general form of the model is obtained: Among them, the matrix and is the system matrix, a n and b n is the parameter to be estimated, α n is a fractional order. Considering a fractional commensurate system, that is, the fractional order is a multiple of the same basis and is known, the fractional order discrete state space model 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 k-th sampling. Assume h = 1 and define Substituting it into equation (3), the fractional derivative is rewritten as: According to formula (4), we can get: Substituting equation (5) into equation (2), we get the fractional-order discrete state space equation based on GL: Step 1-3) obtains the identification model of the fractional-order discrete state space model of the electric heating furnace: In the above formula, is the information vector of the system, expressed as: φ a (k)=[x1(kn),x2(kn),…,x n (kn)] T ,φ b (k)=[u(k-1),u(k-2),…,u(kn)] T ,θ is the parameter vector of the system, expressed as: <h2 style=";text-align:left;direction:ltr">a = [-a1,-a2,…,-a<h2 style=";text-align:left;direction:ltr"> n <h2 style=";text-align:left;direction:ltr"> ]<h2 style=";text-align:left;direction:ltr"> T <h2 style=";text-align:left;direction:ltr"> , b=[b1,b2,…,b n ] T , When the fractional order α is known, Q(k) is a known term, expressed as: Q(k)=-w(1)x1(k-1)-…-w(k-1)x1(1)-w(1)x2(k-2)-…-w(k-2)x2(1)-…-w(1)x n (kn)-…-w(kn)x n (1)。 3. The electric heating furnace system parameter identification method based on hierarchical least squares according to claim 1 is 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, defined data length k max ,make P1(0)=10 6 I n , P2(0)=10 6 I n , P k (0) = I n / 10 2 , k = 1; Step 2-2) The heating power of the electric heating furnace system is used as input data u(k), and the temperature inside the furnace is used as output data y(k), and the data is recorded; Step 2-3) Replace the state vector x(k) in the information vector with its estimated value According to equations (8) and (9), the estimation of the information vector is constructed and φ b (k); Step 2-4) Calculate the gain vectors L1(k) and L2(k) according to equations (10) and (11), calculate the covariance matrices P1(k) and P2(k) according to equations (12) and (13), and calculate Step 2-5) Calculate the estimated value of the parameter vector according to equations (15) and (16) and Step 2-6) Construct the system estimation matrix according to equations (17) and (18) and Step 2-7) Calculate the state prediction value according to equations (19) and (20) and covariance predicted values Step 2-8) Calculate the Kalman gain K according to equations (21) and (22): k (k) and the state update value Update the state estimation covariance matrix P according to formula (23): k (k); Step 2-9) determines whether the maximum number of cycles has been reached. If not, k=k+1 and jumps to step 2-3). If reached, proceed to step 2-10); Step 2-10) Output the recognition result Complete identification.

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