Data-driven modeling and predictive control method for electroslag remelting process with time delay characteristics

Abstract

A data-driven modeling and predictive control method for ESR process with time-delay characteristics is provided, which is based on the mechanism relationship between the servo motor encoder stroke and the consumable electrode output position to identify and model the ESR process by data-driven method; the transfer function model is converted into a standard linear regression form; a one-dimensional search algorithm is used to estimate the nonlinear time-delay parameters, and an adaptive forgetting factor is constructed by using the estimated parameter change rate; a history control input sequence is introduced into the augmented state vector to convert the original time-delay system into an equivalent time-delay-free augmented state space model; a rolling optimization mechanism is used to dynamically compensate the phase lag caused by multi-source time delay, and combined with feedback correction to effectively suppress the interference of uncertain factors, so as to obtain the optimal control sequence, and make the controller control the ESR system with strong nonlinear and multi-time delay coupling dynamic characteristics in real time with the optimal control sequence. The method can have excellent tracking accuracy and good control stability in the process of processing ESR tracking control.

Description

Technical Field

[0001] This invention belongs to the field of intelligent control technology, specifically relating to a data-driven modeling and predictive control method for an electroslag remelting process with time delay characteristics. Background Technology

[0002] Electroslag remelting (ESR) technology, a core process in special metallurgy, is a key technology for producing high-end special steels such as aero-engine blades and nuclear reactor components. The ESR process controls the remelting current by adjusting the immersion depth of the consumable electrode in the slag pool, thereby achieving precise control of the melting rate. However, in the control of the ESR process, the thermal inertia of the slag pool leads to a lag in the temperature response. The electrode mechanical transmission system exhibits electromechanical delays, and the droplet transition process also involves random time delays. This superposition of multiple time delays significantly increases the difficulty of modeling and reduces the stability of the controller, resulting in a series of problems such as excessive melting rate fluctuations and increased segregation within the crystallizer. It is worth noting that the system's strong nonlinearity and multi-time-delay coupled dynamic characteristics place stringent requirements on control accuracy and safety performance.

[0003] Among the technologies addressing the aforementioned control challenges, model-based control methods exhibit unique advantages. Existing control strategies can be categorized into robust adaptive control, data-driven control, and model-based control. However, these methods have limited ability to rapidly adapt to dynamic time delays, and adaptive control design often sacrifices dynamic performance. While intelligent control can handle nonlinear dynamics, it requires a large amount of training data, and its computational complexity hinders its real-time deployment. In contrast, model-based control methods explicitly handle time delay effects by constructing precise mathematical models, ensuring control accuracy while possessing clear physical interpretability. Comparative analysis shows that model-based control technology has better theoretical completeness and ease of engineering implementation in time-delay systems. Designing a model-based controller to handle the effects of system time delays, using a benchmark model established through data-driven methods and comprehensively considering unmodeled dynamics and external disturbances, is a key problem that needs to be solved to achieve precise tracking control of the position of electroslag remelted consumable electrodes.

[0004] In summary, accurate system modeling is crucial for achieving precise control of consumable electrodes in electroslag remelting. For the electroslag remelting process, either mechanistic modeling or data-driven methods can be chosen. Mechanistic modeling relies on complex calculus calculations and involves a large number of undetermined parameters, making its application in actual production less convenient. In contrast, data-driven identification modeling methods have the advantages of theoretical simplicity and ease of implementation. The identified model can significantly improve the control performance and robust stability of the production system. Since different control modes of servo motors can cause the system to operate in stable, nonlinear, or integral models, modeling and control studies under appropriate control modes based on the operational characteristics of electroslag remelting are of great significance for ensuring the quality of special steel refining. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a data-driven modeling and predictive control method for electroslag remelting process with time delay characteristics. This method is simple to implement and has low implementation cost. It can achieve excellent tracking accuracy and good control stability in the process of electroslag remelting tracking control, and can ensure the acquisition of high-quality special steel.

[0006] To achieve the above objectives, this invention provides a data-driven modeling and predictive control method for an electroslag remelting process with time-delay characteristics, comprising the following steps:

[0007] Step 1: For the electroslag remelting process in position control mode, based on the mechanism relationship between the servo motor encoder stroke and the output position of the consumable electrode, a third-order polynomial function model is used to model the electroslag remelting process.

[0008] Step 2: Convert the transfer function model of the electroslag remelting process into a standard linear regression form;

[0009] Step 3: Based on the two-dimensional recursive least squares algorithm, the nonlinear time delay parameters are estimated using a one-dimensional search algorithm, and an adaptive forgetting factor is constructed using the rate of change of the estimated parameters to improve the estimation accuracy and robustness to noise of the model.

[0010] Step 4: Introduce augmented state vectors into the historical control input sequence to transform the original time-delay system into an equivalent time-delay-free augmented state-space model;

[0011] Step 5: Dynamically compensate for phase lag caused by multi-source time delay through rolling optimization mechanism, and effectively suppress the interference of uncertain factors by combining feedback correction to obtain the optimal control sequence, and enable the controller to perform real-time control of the electroslag remelting system with strong nonlinearity and multi-time delay coupling dynamic characteristics using the optimal control sequence.

[0012] As a preferred option, in step one, a transfer function model is established according to formula (1);

[0013] A m (z -1 )y(k)=B m (z -1 )z -d u(k)+v(k)(1);

[0014] In the formula, y(k) represents the actual output of the electroslag remelting process at discrete time point k, u(k) represents the control input applied at time point k, and A m (z -1 The ) represents the influence of historical values ​​of y(k) on the current value.

[0015] z -1 Represents the unit delay operator; n represents the order; a i B represents the inherent dynamic characteristics of the system. m (z -1 The ) represents the effect of the historical values ​​of u(k) on the output. b j The input represents the weights; v(k) is the random perturbation. As a preferred option, in step two, the process of converting the transfer function model into the standard linear regression form is as follows: S21: Define the parameter vector θ according to formula (2); define the information vector according to formula (3).

[0016]

[0017] In the formula, n0=n a +n b ; d is the time delay parameter;

[0018] S22: Combining formulas (2) and (3), we transform formula (1) into formula (4);

[0019]

[0020] As a preferred option, the specific process of step three is as follows:

[0021] S31: Based on the least squares criterion and combined with the electroslag remelting process, construct the target loss function according to formula (5);

[0022]

[0023] In the formula, λ∈(0,1] is the forgetting factor.

[0024] S32: d is set according to the electroslag remelting process. min and d max And obtain the optimal time delay parameters according to formula (6).

[0025]

[0026] S33: Construct the least squares identification algorithm according to formula (7), and obtain the optimal estimated parameter vector according to formula (8).

[0027] In the formula, P(k) is the covariance matrix.

[0028]

[0029] In the formula, Y(k) represents the output vector;

[0030] S34: Obtain the adaptive forgetting factor λ(k) according to formula (9);

[0031]

[0032] In the formula, λ(k)≥λ min , λ min A lower threshold set by humans.

[0033] As a preferred option, the specific process of step four is as follows:

[0034] S41: Construct the state-space equation of the time-delay system according to formula (10), and construct the output sequence of the time-delay system according to formula (11);

[0035] x(k+1)=Ax(k)+Bu(kd)(10);

[0036] y(k)=Cx(k)+Du(kd)(11);

[0037] In the formula, x(k) is the system state vector; A is the system matrix; B is the control matrix; C is the output matrix; and D is the direct transfer matrix.

[0038] S42: Construct the augmented state vector ξ(k) according to formula (12);

[0039] ξ(k)=[x(k),u(kd),u(k-d+1),…,u(k-1)](12);

[0040] S43: Introduce the augmented state vector ξ(k) into the state space equation and output sequence of the time-delay system to obtain formula (13) and formula (14) respectively;

[0041]

[0042] In the formula, Aaug For augmented system matrix; B aug For augmented control matrix; C aug To augment the output matrix.

[0043] As a preferred option, the specific process of step five is as follows:

[0044] S51: Construct the optimal control cost function according to formula (15);

[0045]

[0046] In the formula, R(k) is the reference trajectory vector, ΔU(k) ​​is the control increment term, Q is the output error weight matrix, and R is the control increment weight matrix;

[0047] S52: Construct a prediction time domain N based on formula (16) p and control time domain N c The state-space equation is constructed according to formula (17) to form the prediction time domain N. p and control time domain N c The output sequence;

[0048]

[0049] S53: Convert formulas (16) and (17) into matrix forms, as shown in formula (18);

[0050]

[0051] S54: Substituting Δu(k+j)=u(k+j)-u(k+j-1) into formula (18) yields formula (19);

[0052]

[0053] S55: Transform formula (19) to obtain formula (20);

[0054] Y(k)=S x ξ(k)+S u ΔU(k)(20);

[0055] In the formula, S x and S u As an intermediate variable;

[0056]

[0057] S56: Substitute formula (20) into formula (15) to obtain formula (21);

[0058]

[0059] S57: Order Equation (21) is transformed to obtain equation (22);

[0060]

[0061] S58: Order The simplified formula (22) yields the optimal control sequence, as shown in formula (23);

[0062] u(k)=u(k-1)+Kξ(k)+K r R(k)(23).

[0063] This invention provides a data-driven modeling and predictive control method for electroslag remelting processes with time-delay characteristics. First, data-driven identification modeling is performed on the electroslag remelting process to precisely control the position of the consumable electrode in the crystallizer. Next, the transfer function model is converted into a standard linear regression form, which is better suited for the least squares identification algorithm. Furthermore, when estimating nonlinear time-delay parameters, a one-dimensional search strategy is used to determine them, effectively overcoming the high-dimensional search challenge of nonlinear time-delay systems and significantly improving parameter identification accuracy. Simultaneously, an adaptive forgetting factor is constructed using the rate of change of the estimated parameters. This allows for increasing λ to accelerate tracking when the system changes rapidly and decreasing λ to improve estimation accuracy in steady-state conditions. This avoids the contradiction between convergence speed and stability caused by a fixed forgetting factor and significantly improves robustness under non-stationary noise environments. Then, the historical control input sequence is introduced into the augmented state vector, effectively compensating for the time lag caused by electromechanical drive delay and thermal inertia lag. This transforms the original time-delay system into an equivalent time-delay-free augmented state-space model while fully preserving the original system's dynamic response characteristics and input delay effects. By constructing multi-step predictive equations to handle the time delay effect, the negative impact of time delay on control performance is fundamentally resolved, achieving optimal control of the electroslag remelting process. This structured augmented strategy not only avoids the complex predictor design in traditional time delay compensation methods but also enables model predictive control to be directly optimized based on linear time-invariant system theory, significantly reducing the complexity of the control algorithm. Finally, the phase lag caused by multi-source time delay is dynamically compensated through the rolling optimization mechanism, and feedback correction effectively suppresses the interference of uncertainties such as slag pool thermal inertia.

[0064] This method is simple to implement and has low implementation costs. It has excellent tracking accuracy and good control stability in the process of electroslag remelting tracking control, and can ensure the acquisition of high-quality special steel. Attached Figure Description

[0065] Figure 1 This is a flowchart of the present invention;

[0066] Figure 2 This is a block diagram illustrating the principle of melting rate and current coupling control in this invention.

[0067] Figure 3 This is a schematic diagram illustrating the relationship between motor speed and electrode position transmission in this invention.

[0068] Figure 4 This is a block diagram of the state-augmented time-delay MPC control principle.

[0069] Figure 5 This is a graph showing the input and output data of the modeling experiment for the electroslag remelting process of this invention.

[0070] Figure 6 A graph showing the parameter identification results of the modeling experiment for the electroslag remelting process of this invention.

[0071] Figure 7 This is a recursive change curve of the adaptive forgetting factor in this invention;

[0072] Figure 8 This is a comparison chart of the prediction results of the identification model and the cross-validation results in this invention;

[0073] Figure 9 The graphs show the trajectory tracking results of different algorithms in this invention.

[0074] Figure 10 This is a tracking error curve of the trajectory set in this invention. Detailed Implementation

[0075] The present invention will be further described below with reference to the embodiments and accompanying drawings.

[0076] Electroslag remelting is a key technology for refining high-quality special steels. During the electroslag remelting process, the remelting current is controlled by changing the position of the consumable electrode within the slag resistance, thereby controlling the melting rate of the consumable electrode. In existing technologies, the electroslag remelting process is controlled using an internal and external dual closed-loop system, such as... Figure 2As shown, the main aspects involved include smelting parameter detection, consumable electrode lifting and lowering control, trolley movement, centering control, weighing detection, pneumatic control, internal cooling water system detection and control, protective atmosphere control, and slag feeding control. The outer loop forms a melting rate feedback loop through the weighing system, while the inner loop uses the oscillation of the consumable electrode to change the remelting current for feedback control. The outer loop uses the melting rate as an evaluation index to correct the current setpoint of the inner loop, while the inner loop uses a correction algorithm for the deviation between the consumable electrode feed rate and the current. The basic feed rate is calculated by the melting rate prediction model. After the current enters the slag pool from the electrode, it passes through the molten metal pool and the solidified steel ingot, then returns to the transformer via the bottom water tank and short network. The magnitude of the current is controlled by adjusting the electrode descent speed. Based on the melting rate changes fed back from the electrode weighing system, a lifting motor controls the oscillation of the consumable electrode in the electroslag to change the current. During the electroslag remelting process, a servo motor drives a connecting rod to reciprocate the consumable electrode in the crystallizer. Common consumable electrode motion control methods include position mode control, speed mode control, torque mode control, and hybrid mode control. In speed control mode, the dynamic operation of the electrode exhibits an integral form; in torque control mode, due to the difference in reciprocating torque, the dynamic operation of the electrode exhibits dead zone characteristics; in position operation mode, the dynamic operation of the electrode exhibits stable characteristics. Based on the operating characteristics of the electroslag remelting furnace, in position mode, data-driven modeling and predictive controller design are performed for the electrode operation process during electroslag remelting. The input signal is the position of the servo motor encoder, and the output signal is the specific position of the consumable electrode, such as... Figure 3 As shown. Since signal transmission requires a time delay, the impact of time lag on model dynamics needs to be considered during modeling and control.

[0077] like Figure 1 As shown, this invention provides a data-driven modeling and predictive control method for electroslag remelting processes with time-delay characteristics, comprising the following steps:

[0078] Step 1: The position of the consumable electrode in the crystallizer determines the slag resistance and current. The magnitude of the current directly affects the melting rate of the electrode. The electrode melting rate is crucial to the quality of the steel ingot and solidification segregation. However, the electrode melting rate can only be measured by a weighing system and cannot be fully excited. Therefore, a model is established between the motor speed and the position of the consumable electrode. Using this model, a controller is designed to precisely control the position of the electrode in the crystallizer. Specifically, in the discrete domain, for the electroslag remelting process in position-controlled operation mode, a third-order polynomial function model is used to model the electroslag remelting process based on the mechanistic relationship between the servo motor encoder stroke and the output position of the consumable electrode.

[0079] Specifically, a transfer function model for identifying the electroslag remelting process is established based on formula (1);

[0080] A(z-1 )y(k)=B(z -1 )z -d u(k)+v(k)(1);

[0081] In the formula, y(k) represents the actual output of the electroslag remelting process at discrete time point k, u(k) represents the control input applied at time point k, and A m (z -1 The ) represents the influence of historical values ​​of y(k) on the current value.

[0082] z -1 Represents the unit delay operator; n represents the order; a i B represents the inherent dynamic characteristics of the system. m (z -1 The ) represents the effect of the historical values ​​of u(k) on the output. b j The input represents the weight; v(k) is the random perturbation.

[0083] Step 2: Define the parameter vector and information vector, and convert the transfer function model of the electroslag remelting process into a standard linear regression form to be suitable for identification algorithms such as least squares.

[0084] Specifically, the process of converting the transfer function model into the standard linear regression form is as follows:

[0085] S21: Define the parameter vector θ according to formula (2); define the information vector according to formula (3).

[0086]

[0087]

[0088] In the formula, n0=n a +n b ; d is the time delay parameter;

[0089] S22: Combining formulas (2) and (3), we transform formula (1) into formula (4);

[0090]

[0091] Step 3: Based on the two-dimensional recursive least squares algorithm, the nonlinear time delay parameters are estimated using a one-dimensional search algorithm, and an adaptive forgetting factor is constructed using the rate of change of the estimated parameters to improve the estimation accuracy and robustness to noise of the model.

[0092] The specific process is as follows:

[0093] S31: Since the time delay parameters of a discrete system must be integers, the range of time delay parameters for a specific actual system can usually be estimated by a step signal. The time delay parameters can be determined by a one-dimensional search algorithm and by comparing and evaluating the magnitude of the prediction error. Specifically, based on the least squares criterion and combined with the electroslag remelting process, the target loss function is constructed according to formula (5);

[0094]

[0095] In the formula, λ∈(0,1] is the forgetting factor.

[0096] S32: d is set according to the electroslag remelting process. min and d max And obtain the optimal time delay parameters according to formula (6).

[0097]

[0098] S33: Construct the least squares identification algorithm according to formula (7), and obtain the optimal estimated parameter vector according to formula (8). The identification algorithm involves the problem of assigning initial values ​​to the parameters to be estimated. Assign a small initial parameter; assign a large initial value to the covariance matrix P(k) to facilitate algorithm convergence; initialize the forgetting factor λ to 1, with its minimum value being λ. min The lower threshold value set manually can generally be between 0.95 and 0.99. The adaptive forgetting factor will be automatically adjusted as the algorithm converges.

[0099]

[0100] In the formula, P(k) is the covariance matrix.

[0101]

[0102] In the formula, Y(k) represents the output vector;

[0103] S34: To improve the estimation accuracy and convergence rate of the algorithm, the adaptive forgetting factor λ(k) is obtained according to formula (9);

[0104]

[0105] In the formula, λ(k)≥λ min , λ min A lower threshold set by humans.

[0106] Step 4: Introduce augmented state vectors into the historical control input sequence to transform the original time-delay system into an equivalent time-delay-free augmented state-space model;

[0107] The specific process is as follows:

[0108] S41: When the system has a time lag of d steps, the actual control input takes effect with a delay. Therefore, the state space equation of the time-delay system is constructed according to formula (10), and the output sequence of the time-delay system is constructed according to formula (11).

[0109] x(k+1)=Ax(k)+Bu(kd)(10);

[0110] y(k)=Cx(k)+Du(kd)(11);

[0111] In the formula, x(k) is the system state vector;

[0112] S42: Construct the augmented state vector ξ(k) according to formula (12);

[0113] ξ(k)=[x(k),u(kd),u(k-d+1),…,u(k-1)](12);

[0114] In the formula, x(k) is the system state vector; A is the system matrix; B is the control matrix; C is the output matrix; and D is the direct transfer matrix.

[0115] S43: Introduce the augmented state vector ξ(k) into the state space equation and output sequence of the time-delay system to obtain formula (13) and formula (14) respectively;

[0116]

[0117]

[0118] In the formula, A aug For augmented system matrix; B aug For augmented control matrix; C aug To augment the output matrix.

[0119] The state augmentation method transforms the original time-delay system into an equivalent time-delay-free augmented state-space model by incorporating historical control input sequences into the augmented state vector, fundamentally solving the negative impact of time-delay elements on control performance. This method explicitly embeds the dynamic characteristics of the time-delay element into the augmented system matrix A by constructing an extended state vector ξ(k) = [x(k), u(kd), ..., u(k-1)]. augIn this approach, the dynamic response characteristics and input delay effects of the original system are fully preserved. This structured augmentation strategy not only avoids the complex predictor design required by traditional time delay compensation methods, but also enables model predictive control to be directly optimized based on linear time-invariant system theory, significantly reducing the complexity of the control algorithm.

[0120] Step 5: Dynamically compensate for phase lag caused by multi-source time delay through rolling optimization mechanism, and effectively suppress the interference of uncertain factors by combining feedback correction to obtain the optimal control sequence, and enable the controller to perform real-time control of the electroslag remelting system with strong nonlinearity and multi-time delay coupling dynamic characteristics using the optimal control sequence.

[0121] The specific process is as follows:

[0122] S51: Construct the optimal control cost function according to formula (15);

[0123]

[0124] In the formula, R(k) is the reference trajectory vector, ΔU(k) ​​is the control increment term, Q is the output error weight matrix, and R is the control increment weight matrix; where Q and R determine the weights of the error and input during the control process. For a larger Q, the controller will quickly converge the system output to the ideal signal. For a larger R, the controller allows the system output sequence to converge to the ideal signal at a slower rate.

[0125] S52: Model predictive control includes prediction time-domain parameters and control time-domain parameters. The prediction time domain represents the degree to which the controller predicts future states. When the prediction time domain is large, the controller can predict output information further into the future, but this will increase the prediction error and reduce control accuracy. When the prediction time domain is small, the control performance of changing signals will decrease, and the control may be unstable. Specifically, a prediction time domain N is constructed according to formula (16). p and control time domain N c The state-space equation is constructed according to formula (17) to form the prediction time domain N. p and control time domain N c The output sequence;

[0126]

[0127] S53: Convert formulas (16) and (17) into matrix forms, as shown in formula (18);

[0128]

[0129] S54: Substituting Δu(k+j)=u(k+j)-u(k+j-1) into formula (18) yields formula (19);

[0130]

[0131] S55: Transform formula (19) to obtain formula (20);

[0132] Y(k)=S x ξ(k)+S u ΔU(k)(20);

[0133] In the formula, S x and S u As an intermediate variable;

[0134]

[0135] S56: Substituting formula (20) into formula (15) to solve for the optimal control law, we can obtain a quadratic programming problem, which yields formula (21);

[0136]

[0137] S57: Order Equation (21) is transformed to obtain equation (22);

[0138]

[0139] S58: Order The simplified formula (22) yields the optimal control sequence, as shown in formula (23);

[0140] u(k)=u(k-1)+Kξ(k)+K r R(k)(23).

[0141] Based on state augmentation time-delay MPC control block diagram, such as Figure 4 As shown. The time-delay MPC algorithm in this invention is based on a state augmented model. By constructing a multi-step prediction equation to handle the time-delay effect, it achieves optimal control of the electroslag remelting process. The core of the algorithm lies in establishing a prediction model under the augmented state space. By solving a constrained quadratic programming problem, the optimal control sequence u(k) = u(k-1) + Kξ(k) + K r The analytical solution form of R(k) ensures real-time computational efficiency. This algorithm dynamically compensates for phase lag caused by multi-source time delays through a rolling optimization mechanism, and effectively suppresses interference from uncertainties such as thermal inertia of the slag pool by combining feedback correction.

[0142] I. To verify the data-driven modeling and predictive control method for electroslag remelting process with time-delay characteristics proposed in this invention, the following stability and convergence properties analysis is performed:

[0143] Time delay introduces a fixed phase lag in the feedback loop, reducing the system's phase margin. When this lag is superimposed on the system's inherent dynamic phase, it can easily lead to closed-loop instability. To verify the asymptotic stability of the proposed control algorithm, a rigorous proof is performed using the Lyapunov direct method, the specific process of which is as follows:

[0144] Step 1: Substitute the formula (23) of the optimal control sequence into the formula (13) of the dynamic equation of the closed-loop system to obtain formula (24):

[0145]

[0146] In the formula, A cl and B cl A is an intermediate variable; cl =A aug +B aug K; B cl =B aug K r ;

[0147] Step 2: Assuming the ideal state of the system is as shown in formula (25), the system deviation error e(k) is as shown in formula (26), and the error dynamic equation is as shown in formula (27).

[0148] ξ ref (k+1)=A cl ξ ref (k)+B cl R(k)(25);

[0149] e(k)=ξ(k)-ξ ref (k)(26);

[0150] e(k+1)=A cl e(k)+B cl ΔR(k)(27);

[0151] In the formula, ΔR(k)=R(k)-R(k-1);

[0152] Step 3: Define the Lyapunov function V(k) according to formula (28);

[0153] V(k)=e T (k)Pe(k)(28);

[0154] In the formula, It is a symmetric positive definite matrix;

[0155] Step 4: Based on the theory of stability of discrete-time systems, prove that the Lyapunov difference function satisfies formula (29);

[0156] ΔV(k)=e(k+1)T Pe(k+1)-e(k) T Pe(k)≤0(29);

[0157] Step 5: Substitute the formula (27) of the error dynamic equation into the formula (29) of the difference function to obtain formula (30); if there exists a positive definite matrix P that satisfies the inequality in formula (31), then the Lyapunov difference equation satisfies the inequality in formula (32);

[0158]

[0159] Step 6: Define the constant as follows: γ1=λ min (Q2), Since the change of the reference signal is bounded, the upper bound of the reference signal is defined as ||ΔR(k)||≤δ, then formula (33) holds;

[0160] ΔV(k)≤-γ1‖e(k)‖ 2 +2γ3‖e(k)‖δ+γ2δ 2 (33);

[0161] Step 7: Use Young's inequality Pick Formula (34) is obtained; when δ is sufficiently small, it satisfies Then formula (35) holds true;

[0162]

[0163] In summary, to ensure the asymptotic stability of the system, the following conditions must be met:

[0164] 1) The weight matrices Q and R must satisfy the following conditions: Positive definiteness ensures the convexity of the optimization problem;

[0165] 2) Augmented system matrix A aug The controllability condition must be met;

[0166] 3) The rate of change δ of the tracking signal must meet the following requirements.

[0167] The above conditions can be met by reasonably selecting the prediction time domain and weighting coefficients, which ensures stability while taking into account engineering feasibility.

[0168] II. Modeling and Control Experiment of Electroslag Remelting Process;

[0169] 1. Modeling experiment of electroslag remelting process, the specific process is as follows:

[0170] A1. Based on the analysis of the system, the formation of the servo motor encoder and the position of the consumable electrode in the electroslag remelting process are fitted by a third-order system to obtain formula (36).

[0171] (1+a1z -1 +a2z -2 +a3z -3 )y(k)=(b1z -1 +b2z -2 )z -d u(k)(36);

[0172] A2. In order to fully excite the system and obtain the main dynamic characteristics of the system, the input excitation signal is composed of multiple sinusoidal signals, as shown in formula (37);

[0173] u(k)=1.2sin(200k)+0.4sin(400k)+0.9sin(300k)+1.1sin(160k)+0.9cos(200k)(37);

[0174] A3. Sampling period set T = 250ms. Initial position of consumable electrode set to y(0) = 300mm. Data acquisition length is N = 2000. Input and output data, such as... Figure 5 As shown. The initial parameters of the algorithm are taken as... P(0) = 10 10 I 5×5 d min =0,d max =10 and λ min =0.995. Parameter identification results, such as... Figure 6 As shown. The identification parameters are: The estimated time delay parameter is The identification model is shown in formula (38);

[0175] y(k)=1.0372y(k-1)-0.9231y(k-2)+0.8860y(k-3)-0.0001u(k-1)z -5 +0.0004u(k-2)z -5 (38); During the algorithm recursion process, the adaptive forgetting factor estimation result, such as Figure 7 As shown. The model's predictive ability is validated using the outputs of the identification algorithm and the outputs obtained from repeated experiments. The prediction results are output as follows: Figure 8 As shown in the cross-validation results above, the model established by the identification algorithm has good estimation accuracy and adaptability.

[0176] 2. Tracking and control of the electroslag remelting process, the specific process is as follows:

[0177] To illustrate the advantages of the proposed algorithm in handling the electroslag remelting tracking control problem, the proposed algorithm is compared with PID control, model-free adaptive control (MFAC), and RBF adaptive neural network control. The relevant parameters of the comparison algorithms are shown in Table 1.

[0178] Table 1: Initial parameters for different algorithms

[0179]

[0180]

[0181] Four control algorithms are used to define the tracking trajectory and tracking error of the set tracking curve, such as... Figure 9 and Figure 10 As shown. The average tracking error of the four control algorithms. and variance As shown in Table 2.

[0182] Table 2: Tracking Errors of Different Algorithms

[0183]

[0184] Figure 10 The tracking error analysis in Table 2 shows that, compared to other algorithms, this algorithm faces larger tracking estimation errors and fluctuations when dealing with time delay issues. This algorithm effectively compensates for the time lag caused by electromechanical drive delay and thermal inertia lag through a state augmentation mechanism. The augmented state space, by memorizing historical control inputs, uses rolling optimization to pre-compensate for phase shifts caused by time delays, reducing the tracking error to 0.77%. Figure 9 The control results show that the algorithm proposed in this invention has good tracking accuracy and control stability in handling the electroslag remelting tracking control problem.

[0185] In summary, for the electroslag remelting process in position-controlled operation mode, the system model structure was determined by analyzing the mechanistic relationship between the servo motor encoder stroke and the output position of the consumable electrode. The constructed identification algorithm was applied to consistently estimate the model parameters and time-delay parameters. An adaptive forgetting factor was designed to improve the robustness of the identification algorithm. Addressing the strong nonlinearity and time-delay dynamic characteristics exhibited by the electroslag remelting system, the original time-delay system was transformed into an equivalent time-delay-free augmented state-space model by introducing the historical control input sequence into the augmented state vector. A rolling optimization mechanism was used to dynamically compensate for the phase lag caused by the time delay. Combined with a feedback correction mechanism, uncertainties such as the thermal inertia of the slag pool were effectively suppressed. Finally, the effectiveness and superiority of the proposed identification algorithm and model predictive control trajectory tracking control were verified using the electroslag remelting process.

Claims

1. A data-driven modeling and predictive control method for an electroslag remelting process with time-delay characteristics, characterized in that, The method comprises the following steps: Step one: modeling the ESR process based on the mechanism relationship between the servo motor encoder stroke and the consumable electrode output position in the position control mode, using a third-order polynomial function model; Step two: converting the transfer function model of the ESR process into a standard linear regression form; Step three: estimating the nonlinear time delay parameter using a one-dimensional search algorithm based on the two-dimensional recursive least squares algorithm, and using the estimated parameter change rate to construct an adaptive forgetting factor to improve the estimation accuracy of the model and the robustness to noise; Step four: introducing an augmented state vector into the historical control input sequence to convert the original time delay system into an equivalent time delay-free augmented state space model; Step five: dynamically compensating for the phase lag caused by multiple time delays through a rolling optimization mechanism, and effectively suppressing the interference of uncertain factors combined with feedback correction to obtain an optimal control sequence, and making the controller control the ESR system with strong nonlinear and multi-time delay coupling dynamic characteristics in real time with the optimal control sequence; The specific process is as follows: S51: construct the optimal control cost function according to formula (15); (15)； wherein is a reference trajectory vector, is a control increment term, is an output error weight matrix, is a control increment weight matrix; S52: Construct state space equations with prediction horizon and control horizon according to equation (16) and output sequence with prediction horizon and control horizon according to equation (17); (16)； (17)； S53: convert formula (16) and formula (17) into a matrix form, as shown in formula (18); (18)； S54: Will Substituting into formula (18) yields formula (19); (19)； S55: deform formula (19) to obtain formula (20); (20)； wherein and are intermediate variables; ; ； S56: bring formula (20) into formula (15) to obtain formula (21); (21)； S57: Let Equation (22) is obtained by transforming Equation (21). (22)； S58: Let , Simplify equation (22) to get the optimal control sequence as shown in equation (23). (23)。 2. The data-driven modeling and predictive control method for the electroslag remelting process with time delay characteristics according to claim 1, characterized in that, In step one, the transfer function model is established according to formula (1); (1)； wherein represents the actual output quantity of the electroslag remelting process at the discrete time point k, represents the control input quantity applied at the time point k, represents the influence of the history value of ; represents the unit delay operator; represents the order; represents the system-inherent dynamic behavior; represents the influence of the history value of ; represents the weight of the input action; is a random disturbance.

3. The data-driven modeling and predictive control method for the electroslag remelting process with time delay characteristics according to claim 2, characterized in that, In step two, the process of converting the transfer function model into a standard linear regression form is as follows: S21 : Define the parameter vector according to equation (2) ; The information vector is defined according to equation (3) ; (2)； (3)； In the formula, ; is a time delay parameter; S22: combine formula (2) and formula (3), and convert formula (1) into formula (4); (4)。 4. The data-driven modeling and predictive control method for the electroslag remelting process with time delay characteristics according to claim 3, characterized in that, The specific process of step three is as follows: S31: based on the least squares criterion, combine the ESR process, and construct the target loss function according to formula (5); (5)； In the formula, is a forgetting factor; S32: setting according to electroslag remelting process and and the optimal time delay parameter is obtained according to formula (6) , ; (6)； S33: Construct a least square identification algorithm according to formula (7), and obtain an optimal estimated parameter vector according to formula (8) ; (7)； In the formula, is a covariance matrix, ; (8)； In the formula, represents an output vector; S34: Obtain the adaptive forgetting factor according to formula (9) ; (9)； In the formula, , is a lower limit threshold set artificially.

5. The data-driven modeling and predictive control method for the electroslag remelting process with time delay characteristics according to claim 4, characterized in that, The specific process of step four is as follows: S41: construct the state space equation of the time delay system according to formula (10), and construct the output sequence of the time delay system according to formula (11); (10)； (11)； wherein is the system state vector; is the system matrix; is the control matrix; is the output matrix; is the direct transmission matrix; S42: Construct the augmented state vector according to formula (12) ; (12)； S43: Augmented state vector The state-space equation and the output sequence of the time-delay system are introduced into the formula (13) and the formula (14), respectively. (13)； (14)； wherein is the augmented system matrix; is the augmented control matrix; is the augmented output matrix.

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