Electrode system model-free sliding mode adaptive control and damping estimation enhancement method

Abstract

This invention belongs to the field of process control technology in the metallurgical industry, specifically relating to a model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems. Based on the Partial Pseudo-Scheme Dynamic Linearization (PFDL) model-free adaptive control (MFAC) method, this invention introduces a sliding mode control (SMC) mechanism to enhance system robustness. Addressing the issue of pseudo-gradient estimation under sliding mode control being susceptible to high-frequency chattering, a sliding mode damped pseudo-gradient estimation method is further proposed. By dynamically adjusting the estimation gain, the stability and accuracy of the estimation process are effectively improved. Furthermore, this invention constructs a sliding mode disturbance observer (SMDO) for real-time estimation and compensation of unknown disturbances, thereby significantly enhancing the system's anti-interference capability. Simulation results show that this composite control strategy can significantly improve the tracking accuracy of electrode current, control smoothness, and anti-interference robustness.

Description

Technical Field

[0001] This invention belongs to the field of metallurgical industrial process control technology, specifically relating to a model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems. Background Technology

[0002] The stable control of electrode current in an electric molten magnesium furnace directly affects arc stability, electrode life, and final product quality. The system faces challenges including strong nonlinearity, parameter uncertainty, and complex internal disturbances (such as arc fluctuations) and external disturbances (such as power grid interference), placing extremely high demands on the control system's performance. Traditional PID controllers, due to their fixed structure, poor performance in strongly coupled and rapidly changing systems, and limited disturbance rejection capabilities, are no longer sufficient to meet the stringent control performance requirements of modern metallurgical industries.

[0003] To circumvent the complexity of electric molten magnesium furnace control system design and effectively address the control requirements of nonlinear time-varying systems, model-free control methods offer a feasible solution. Professor Hou Zhongsheng proposed Model-free Adaptive Control (MFAC), a data-driven control algorithm. Its core idea is to establish an equivalent data model at each operating point of the system. Based on system complexity, these models can be categorized into Compact Format Dynamic Linearization (CFDL), Partial Format Dynamic Linearization (PFDL), and Full Format Dynamic Linearization (FFDL). These data models rely solely on the input and output data of the controlled object. By online estimation of the system's pseudo-gradient (PG) vector (or pseudo-Jacobi matrix) and minimizing the system's one-step lead output error, the data model parameters can be updated online, thus enabling the algorithm to possess good adaptability and robustness. On the other hand, sliding-mode control (SMC), as an effective algorithm for handling nonlinear system problems, exhibits control behavior independent of system state and parameters, and still demonstrates good robustness in the presence of external disturbances and model uncertainties. However, sliding-mode control still relies on the mathematical model of the system for implementation and suffers from significant chattering issues.

[0004] In recent years, many studies have focused on combining the advantages of MFAC and SMC. The literature [Ye Jinxin, Xie Lirong, Wang Hongwei. Model-free adaptive quasi-sliding mode control for non-uniform sampling systems [J]. Electro-Optics and Control, 2023, 30(01):35-41.] addresses the problem of external disturbances in non-uniform sampling nonlinear systems by proposing a model-free adaptive quasi-sliding mode control algorithm. This algorithm treats the pseudo-Jacobi matrix (PPD) parameter estimation error as an external disturbance and uses sliding mode control to suppress it. The literature [Li Wei, Wang Yuanhao, Zheng Wei. Research on a non-series DVR composite control strategy with MFAC compensation [J]. Industrial Instrumentation and Automation Devices, 2014, (06):7-10.] introduces MFAC into DVR inverter control to enhance the performance of PI control. Reference [Chang Debiao, Cao Rongmin, Hou Zhongsheng. Model-free adaptive frictional compensating contour control of two-dimensional linear motors [J]. Control Engineering, 2025, 32(03): 459-468.] addresses the nonlinear friction and uncertain internal and external disturbances in linear motors by combining MFAC with the extended state observer (ESO) in Active Disturbance Rejection Control (ADRC) to form a model-free adaptive observer. Simultaneously, a model-free ADRC cross-coupling control scheme is proposed based on single-axis control. Reference [Zhu Ze, Zhu Zhanxia. Quasi-sliding mode model-free adaptive control of a class of second-order nonlinear systems [J]. Control and Decision, 2024, 39(08): 2663-2670.] designs a quasi-sliding mode model-free adaptive control algorithm (SM-MFAC) using a free-floating space manipulator as an example. It decomposes the model into two discrete subsystems in series and designs the controller using the output data of the two subsystems. However, this design is specific to a particular type of second-order nonlinear system, and its universality may require further investigation. The literature [Jiang Haobin, Feng Zhangqi, Hong Yangke, et al. Model-free adaptive sliding mode predictive control method applied to vehicle longitudinal control [J]. Automotive Engineering, 2022, 44(03):319-329.] combines model-free adaptive control algorithm, sliding mode control algorithm and model predictive control (MPC) algorithm to solve the problems of strong nonlinearity, time-varying uncertainty of parameters and multiple external disturbances and hysteresis that are common in vehicle longitudinal control systems. Hysteresis switching logic is designed to achieve smooth switching between braking and drive control algorithms. However, the controller integrates multiple algorithms, which may lead to increased structural complexity and difficulty in parameter tuning.

[0005] Patent CN119472250A proposes a control method for an intelligent electric molten magnesium furnace based on neural network adaptive PID. It employs a neural network adaptive PID control strategy, using a radial basis function neural network (RBFNN) to self-tune the PID parameters. However, it still relies on the linear model framework of PID control. The neural network is only used to optimize the PID gain, resulting in a delayed response to rapid time-varying disturbances such as arc fluctuations and power grid interference. Under complex operating conditions, parameter fluctuations can easily lead to decreased control accuracy, and it cannot achieve finite-time convergence. Patent CN105571340A proposes an automatic control device and electrode current control method for a rotary electric molten magnesium furnace. It uses traditional PID and improved filtering control, performing PID adjustment after filtering the current signal by limiting the average value. This method relies entirely on a fixed-structure PID algorithm, lacking adaptive capability and unable to cope with time-varying characteristics such as changes in raw material composition and fluctuations in molten pool height. The filtering only suppresses high-frequency noise and has no compensation effect on systemic disturbances (such as power grid voltage fluctuations). Patent CN107526293A proposes a PID control method for electrode current switching in an electric fused magnesium furnace based on a compensation signal. It employs a switching PID and an unmodeled dynamic compensator, using a conventional PID during stable conditions and switching to a PID with a compensator during large disturbances. Its switching mechanism relies on error threshold judgment, easily generating switching jitter and affecting control smoothness; the compensator is based on the unmodeled dynamic of the previous step, resulting in weak tracking and compensation capabilities for real-time disturbances. Patent CN1316070A proposes a model-free adaptive controller and control system for industrial processes, employing a linear dynamic neural network structure. This system exhibits poor stability under external disturbances; the pseudo-gradient estimation uses fixed gain updates without a damping adjustment mechanism, making it susceptible to oscillations caused by high-frequency disturbances; it lacks a dedicated disturbance observer, relying solely on feedforward compensation for measurable disturbances, resulting in poor suppression of unmeasurable disturbances such as arc fluctuations. Patent CN101765257A proposes an intelligent control method for electrode lifting in an electric fused magnesium furnace, based on intelligent control using preset rules, adjusting electrode lifting through fixed logic rules. Its control rules rely on human experience and are not adaptive, making it unable to cope with dynamic changes such as raw material composition and smelting conditions; it also lacks adaptive algorithms and disturbance compensation mechanisms, resulting in low control accuracy.

[0006] In summary, traditional PID controllers have a fixed structure and rely on linear models, resulting in limited disturbance rejection capabilities and large current fluctuations in strongly coupled, rapidly changing systems. Improved PID controllers (such as neural network adaptive PID and switching PID) still adhere to the linear model framework, are insufficient for adapting to strong nonlinearities, and may suffer from problems such as switching jitter and disturbance compensation lag. Simple model-free adaptive control (MFAC), while not requiring a mechanistic model, lacks robustness, and pseudo-gradient estimation is susceptible to disturbances, leading to high-frequency chattering. Rule-based intelligent control lacks adaptability and cannot cope with dynamically changing operating conditions. Summary of the Invention

[0007] To address the shortcomings of existing technologies, such as reliance on linear models, insufficient robustness, susceptibility to chattering due to pseudo-gradient estimation, and weak anti-interference capability, the present invention aims to provide a model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems. This method integrates a composite strategy of model-free adaptive control and sliding mode control to achieve high-precision, robust, and low-chattering control of the electrode current in fused magnesium furnaces.

[0008] The technical solution of this invention is:

[0009] A model-free sliding mode adaptive control and damping estimation enhancement method for an electrode system includes the following steps:

[0010] (1) Constructing a partial scheme dynamic linearization (PFDL) data model for the electrode system of an electric fused magnesium furnace: ,in Let be the system output increment vector of the three-phase electrode current of the fused magnesium furnace at time k+1. It is a time-varying pseudo-gradient matrix. For sliding time windows [ The control input increment vector for the rotation direction and frequency of the three-phase electrode motor within [k], Δu(k)=u(k)-u(k-1) is the control input increment; This represents the lumped disturbance of the electrode system of the fused magnesium furnace at time k;

[0011] (2) Design a model-free sliding mode (PFDL-SMC) controller. The controller adopts a terminal sliding mode surface containing an exponentially decaying integral term and a double power-law approaching law, and suppresses chattering through a saturation function.

[0012] (3) A sliding mode damped pseudo-gradient estimation method is adopted, and the pseudo-gradient update gain is dynamically adjusted based on the sliding surface amplitude to suppress high-frequency oscillations during the estimation process;

[0013] (4) Construct a sliding mode disturbance observer (SMDO) to estimate lumped disturbances in real time. It also feeds forward compensation to the control law, outputs control signals to adjust the rise and fall of the three-phase electrodes of the fused magnesium furnace, and realizes the tracking control of the electrode current to the set value.

[0014] In the aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system, step (2) defines the expression for the terminal sliding surface as follows: ,in Let k be the terminal sliding surface vector. Let k be the system output error vector at time k. For the system The desired output signal at time t, with a scaling factor λ>0, terminal attractor parameter δ∈(0,1), and μ>0 representing the integral gain. γ is the exponential decay integral term, and γ∈(0,1] is the forgetting factor.

[0015] In the aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system, the expression for the reaching law in step (2) is: Where η>0 represents the linear convergence coefficient, κ>0 represents the nonlinear convergence coefficient, T represents the discrete-time sampling period, and q∈(0,1) represents the double power convergence factor. For a saturated function, satisfying: when hour ,when hour θ>0 is the function boundary layer value.

[0016] In the aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system, step (3) of the expression for the update law of the sliding mode damping pseudo-gradient estimator is: ,in = The increment matrix representing the pseudo gradient estimate; update step size ; Penalty factor; sliding mode damping factor ; s(k) represents the terminal sliding surface vector at time k; This represents the output prediction error vector at time k. Δy(k) = y(k) - y(k-1) represents the pseudo gradient estimation matrix at time k-1, T represents the transpose of the matrix / vector, Δy(k) = y(k) - y(k-1) represents the system output increment vector of the three-phase electrode current of the fused magnesium furnace at time k, and ΔU(k-1) = U(k-1) - U(k-2) represents the control input increment vector at time k-1.

[0017] In the aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system, step (4) defines the expression for the sliding mode disturbance observer as follows: ,in This represents the vector of perturbation estimates at time k+1. Let represent the vector of perturbation estimates at time k, and η>0 be the perturbation observer gain. For observation error, It is a boundary saturation function.

[0018] In the electrode system model-free sliding mode adaptive control and damping estimation enhancement method, in step (1), the length L of the sliding time window is a positive integer, L∈[2,5].

[0019] The aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system, lumped disturbance This includes unmodeled dynamics, power grid disturbances, and arc fluctuations.

[0020] The aforementioned model-free sliding mode adaptive control and damping estimation enhancement method for the electrode system constructs a joint Lyapunov function. Verify system stability, including , , It is proved that the sliding surface converges in finite time, and the perturbation estimation error and the pseudo-gradient estimation error converge asymptotically; among them, The Lyapunov function representing the terminal sliding surface. For the terminal sliding surface; The Lyapunov function representing the perturbation observer. = This represents the perturbation estimation error vector at time k. The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the disturbance at time k. The vector representing the estimated perturbation values ​​at time k; The Lyapunov function representing pseudo-gradient estimation, = This represents the pseudo-gradient estimation error matrix at time k.

[0021] The design concept of this invention is:

[0022] To address the practical requirements of controlling fused magnesium furnaces, this invention proposes a model-free sliding mode adaptive control (PFDL-SMC) method. Based on the PFDL-MFAC framework, this method introduces a sliding mode disturbance observer (SMDO) and a sliding mode damped pseudo-gradient estimator to further enhance system performance, as detailed below:

[0023] (1) Based on the Partially Phenomenon Dynamic Linearization (PFDL) data model, this invention designs a model-free sliding mode controller that incorporates an improved reaching law. A model-free data-driven model is established based on PFDL, avoiding reliance on complex furnace mechanism models.

[0024] (2) The model-free sliding mode controller of the present invention adopts a terminal sliding surface containing an exponentially decaying integral term and a double power-law approaching law, which aims to improve the convergence speed and suppress chattering. By designing a terminal sliding surface containing an exponentially decaying integral term and a double power-law approaching law, and combining it with a saturation function to suppress chattering, finite-time convergence is achieved.

[0025] (3) This invention proposes a sliding mode damped pseudo-gradient estimation method, which dynamically adjusts and updates the gain based on the amplitude of the sliding mode surface, effectively suppressing high-frequency oscillations in the estimation process.

[0026] (4) This invention introduces a sliding mode disturbance observer (SMDO) for online estimation and feedforward compensation of unknown disturbances, which significantly improves the anti-interference capability of the system.

[0027] The advantages and beneficial effects of this invention are:

[0028] 1. Significantly improved tracking accuracy: The sliding mode disturbance observer has high estimation accuracy for complex disturbances, significantly enhanced anti-interference ability, and smaller current tracking error. It does not need to rely on a precise mechanism model and can achieve control based on input and output data, adapting to the strong nonlinearity and time-varying parameters of fused magnesium furnaces.

[0029] 2. Enhanced anti-interference capability: Under the combined disturbance of “1000cos(kπ / 20) + 1000cos(kπ / 15)”, the current is stable near the set value (15000A) without periodic deviation.

[0030] 3. Smoothness optimization: Sliding mode damped pseudo-gradient estimation effectively suppresses high-frequency chattering, reduces the amplitude of input chattering, improves smoothness, avoids mechanical impact on the electrode lifting mechanism, and extends equipment life.

[0031] 4. Faster convergence speed: It can enter a stable tracking state in the initial stage (0~10s), and the terminal sliding surface and the double power-law converge in finite time, which greatly improves the convergence speed compared with the traditional linear sliding mode.

[0032] 5. Reduced energy consumption: The control energy is reasonable and there is no excessive impact, which extends the life of the electrode lifting mechanism. The energy consumption per ton is lower than that of traditional PID control, which balances control performance and energy saving requirements. Attached Figure Description

[0033] Figure 1 The curves show the comparison of tracking performance of the baseline discrete nonlinear system under the desired output trajectory.

[0034] Figure 2 The current response curve of the A-phase electrode in an electric fused magnesium furnace under combined disturbance is shown.

[0035] Figure 3 This is the control input curve for the A-phase electrode of the fused magnesium furnace. A negative value indicates the direction of electrode descent.

[0036] Figure 4 The curves show the estimation performance of the sliding mode disturbance observer (SMDO) for the composite disturbance. Detailed Implementation

[0037] In practical implementation, the electrode system of an electric fused magnesium furnace faces problems such as arc fluctuations, power grid interference, and strong nonlinearity during operation, making it difficult for traditional control methods relying on precise mathematical models to achieve high-performance control. To address this, this invention proposes a model-free adaptive control (MFAC) method based on Partially Pseudo-Scheme Dynamic Linearization (PFDL), introducing a sliding mode control (SMC) mechanism to enhance system robustness. The process includes data acquisition → PFDL modeling → sliding surface calculation → pseudo-gradient estimation → SMDO disturbance observation → control output → feedback correction. To address the issue of pseudo-gradient estimation under sliding mode control being susceptible to high-frequency chattering, a sliding mode damped pseudo-gradient estimation method is further proposed. By dynamically adjusting the estimation gain, the stability and accuracy of the estimation process are effectively improved. Furthermore, this invention constructs a sliding mode disturbance observer (SMDO) for real-time estimation and compensation of unknown disturbances, thereby significantly enhancing the system's anti-interference capability. Simulation results show that this composite control strategy can significantly improve the tracking accuracy of the electrode current, control smoothness, and anti-interference robustness.

[0038] 1. Description of the control problem

[0039] The smelting process of fused magnesia uses the rotation direction and frequency of a three-phase motor as input and the current of a three-phase electrode as output. It adopts the submerged arc method, that is, the three-phase electrodes are buried in the raw ore, and the material is added while melting. It has a complex process mechanism, key parameters cannot be measured, it is affected by uncertain factors such as raw material composition and production conditions, and the dynamic changes in the smelting process.

[0040] For the smelting process of fused magnesia, the literature [Fan Yunpeng. Modeling of the smelting process of fused magnesia furnace [D]. Northeastern University, 2011.] established the following dynamic model of electrode current based on the law of conservation of energy:

[0041]

[0042] in, Representing the three-phase electrodes A, B, and C respectively, the input variables are... For the first Phase motor rotation direction and frequency (Hz), output variables For the first Phase electrode current (A), F i (•) represents a nonlinear function (1 / (Ω•m)) that varies slowly with the characteristics of the raw materials, the height of the molten pool, etc., Q i (•) represents a nonlinear function that varies slowly with motor characteristics, transmission mechanism parameters, etc. (1 / (Hz•A)) 2 ), and The function is a nonlinear time-varying function, and the parameters and physical meaning of the function are shown in Table 1.

[0043] The dynamic model of electrode current is discretized using the Euler method, because and It changes slowly over time, so it is assumed to be constant. The resulting modeling error is determined by the rate of change of the current. To compensate, the discretized model of the electrode current dynamic model can be expressed as:

[0044]

[0045] Where k represents the discrete time step, k=0,1,2,…; y i (k) represents the current (A) of the i-th phase electrode at step k; F i Q i This represents the value of F within a sampling period. i (•) and Q i (•) Equivalent parameters obtained by approximating constants; Δy is the sampling time; i (k) represents the modeling error compensation term (A); , The shift operator is used for units.

[0046] The objective of this invention is to design a saturated-constrained one-step optimal controller for the discretized model of the electrode current dynamic model in the fused magnesia smelting process, so that the electrode current tracks its set value. The saturation-constrained one-step optimal control strategy is executed by an industrial PLC (such as the Siemens S7-400 series) to ensure the real-time performance and reliability of control commands.

[0047] Table 1. Physical meaning of electrode current model parameters and functions in fused magnesia furnaces.

[0048]

[0049] The randomness of electric arc and furnace operation and the instability of the smelting process make it extremely difficult to establish an accurate model of the fused magnesium furnace. An inaccurate model leads to errors in controller design. This invention constructs a model-free adaptive control framework based on partial scheme dynamic linearization (PFDL), introducing an improved sliding surface, a sliding mode disturbance observer, and a sliding mode damped pseudo-gradient estimation method. This achieves a composite control strategy that combines strong robustness, high accuracy, and low chattering, ensuring high-performance tracking control of the electrode current to the desired current.

[0050] 2. PFDL-SMC Algorithm

[0051] 2.1 PFDL Algorithm Design

[0052] Based on the model-free adaptive control literature [Fan Yunpeng. Modeling of the smelting process in an electric fused magnesium furnace [D]. Northeastern University, 2011.], the electrode adjustment system for electric fused magnesium can be represented in the following general form:

[0053]

[0054] In the formula: (Three-dimensional vector), representing the three-phase electrode current output vector (A) of the system at time k; (Three-dimensional vector), representing the system in The three-phase electrode control input vector (Hz) at time t; y(k) and u(k) correspond to the actual melting current value of the three-phase fused magnesium furnace and the three-phase electrode speed of the fused magnesium electrode adjustment system, respectively; This represents the output lag order, which is an unknown positive integer. This represents the input lag order, which is an unknown positive integer. The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the external disturbance and the unmodeled dynamics.

[0055] definition For a sliding time window The vector consisting of all control input signals within is as follows:

[0056]

[0057] Satisfy when Sometimes, Where u(k) represents the control input vector (Hz) at time k; integers To control the input linearization length constant (LLC), L∈[2,5] positive integers; It is a dimension of The zero vector; k represents the discrete time step, k=0,1,2,….

[0058] This invention makes the following two basic assumptions about system (1):

[0059] Except for finite points in time, Regarding the first The variable to the first The partial derivatives of the variables exist and are continuous.

[0060] The system satisfies the generalized Lipschitz condition, that is, for any , and From time to time .

[0061] in, The nonlinear discrete state function representing system (1); Represents the output lag order, the first... The variable to the first Each variable corresponds to Inputting , ,…, These L control input variables; , 0, Represents the control input increment vector; This represents the input lag order; This represents the linearization length constant of the control input. Assumption i is a typical constraint on nonlinear systems in control system design, and assumption ii is a constraint on the upper bound of the system output rate of change. The fused magnesium electrode regulation system clearly satisfies both assumptions.

[0062] For a system that satisfies two assumptions (1), given ,when At that time, there must exist a time-varying parameter vector called PG. System (1) can be transformed into the following PFDL data model:

[0063]

[0064] Where Δy(k+1) represents the system output increment (A); The time-varying pseudo-gradient (PG) matrix represents a bounded matrix. It is the constant for controlling the linearization length of the input, L∈[2,5] positive integers; The vector represents the control input increment, and Δu(k) = u(k) - u(k-1) is the control input increment (Hz). The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the external disturbance and the unmodeled dynamics.

[0065] Design the following control input criterion function:

[0066]

[0067] Among them, weighting factors ; y(k+1) represents the expected output signal (A) at time k+1; y(k+1) represents the system output at time k+1 predicted by equation (3); u(k) represents the control input vector (Hz) at time k.

[0068] Substitute equation (3) into the criterion function equation (4) and perform... Differentiating, we get:

[0069]

[0070] in, Represents the control input increment vector; This represents the step size factor, used to adjust the convergence speed of the pseudo-gradient estimation. Step size factor The introduction of this feature makes the control algorithm design more flexible; Φ1(k) represents the pseudo-gradient Φ p,L The first sub-block of (k) (corresponding to the coefficient of Δu(k); Φ i (k) represents the pseudo gradient Φ p,L The i-th sub-block of (k) (corresponding to the coefficient of Δu(k-i+1)), i=2,…,L; d represents the constant offset term, which is used to compensate for model error, and is usually taken as d=0; weight factor α>0.

[0071] In order to implement the control algorithm of equation (5), the value of PG needs to be known. Since the mathematical model of the system is unknown and the value of PG is time-varying, the input and output data of the control system need to be used to estimate the value of PG.

[0072] The following pseudo-gradient (PG) estimation criterion function is proposed:

[0073]

[0074] Where Δy(k) = y(k) - y(k-1) represents the system output increment (A); Φ p,L (k) represents the true time-varying pseudo-gradient (PG) matrix, and L is the control input linearization length constant; Represents pseudo gradient The estimated value; ΔU L (k-1)=[Δu(k-1),…,Δu(kL)] T The control input increment vector represents the rotation direction and frequency of the three-phase electrode motor within the sliding window. Δu(k) = u(k) - u(k-1) is the control input increment (Hz), and T represents the transpose of the matrix / vector. The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the disturbance, including external disturbances and unmodeled dynamics; weighting factor .

[0075] Based on the optimal conditions, equation (6) is related to... By finding the extreme values ​​and using the matrix inversion lemma, the PG estimation algorithm can be obtained as follows:

[0076]

[0077] Among them, step size factor The introduction of the penalty factor is to make the control algorithm more flexible; Ensure that the denominator is not zero. , Represents pseudo gradient The estimated value.

[0078] Based on the pseudo gradient estimation algorithm (7) and control algorithm (5) above, the MFAC control algorithm can be obtained:

[0079]

[0080] Among them, u mfac (k) represents the MFAC control input vector (Hz) at time k; u mfac (k-1) represents the MFAC control input vector (Hz) at time k-1; ρ1∈(0,1) represents the step size factor, which is used to adjust the convergence speed of the pseudo gradient estimation. Represents pseudo gradient The estimated value; Φ1(k) represents the pseudo gradient Φ p,L The first sub-block of (k); Represents the pseudo gradient Φ p,L The estimated value of the i-th sub-block of (k), i=2,…,L; y(k) represents the expected output signal (A) at time k+1; y(k) represents the actual output vector (A) at time k; Δu(k-i+1)=u(k-i+1)-u(ki) represents the control input increment (Hz); L represents the linearization length constant of the control input, which is a positive integer and ranges from L∈[2,5]; weighting factor .

[0081] Compared to the scalar parameters in the commonly used CFDL-MFAC The unknown parameters of the data model of the present invention It is A time-varying vector of dimension makes it easier to design parameter estimation algorithms to capture the complex behavior of the system. Furthermore, it considers the... Output changes at time intervals and within a fixed-length sliding time window The relationship between changes in control inputs within the system, rather than CFDL compressing all factors into... This reduces the complexity of the estimation algorithm.

[0082] 2.2 PFDL-TSMC Algorithm

[0083] The PFDL model-free control algorithm designed above does not require a model of the control system, only the system's input and output data. However, its performance is poor against external disturbances. Therefore, this section combines the model-free control algorithm (MFAC) with an improved terminal sliding mode control (TSMC), enabling the system state to converge to the equilibrium point in a finite time (while traditional linear sliding mode can only converge asymptotically), achieving high-precision tracking. The combined MFAC-TSMC has a faster response than traditional MFAC and a smoother response than traditional SMC, enhancing the system's disturbance rejection capability without requiring a system model.

[0084] Define the system output error:

[0085]

[0086] Where e(k) represents the system output error vector (A) at time k. Representative System The actual output vector (A) at time t. For the system The desired output signal (A) at time t.

[0087] The terminal sliding surface is designed as follows:

[0088]

[0089] in, Let be the terminal sliding surface vector at time k, and be the scaling factor. Integral gain , The exponential decay integral term is defined by equation (11).

[0090]

[0091] Among them, the forgetting factor This represents the memory strength of historical errors. The closer γ is to 1, the higher the weight of historical errors. A typical value for γ is 0.9. That is, the initial integral term is 0. By performing an exponentially weighted summation of historical errors, recent errors have a higher weight, while long-term errors have a lower weight. attenuation.

[0092] Design a discrete convergence law:

[0093]

[0094]

[0095] Wherein, s(k+1) represents the predicted value of the sliding surface at time k+1; η>0 represents the linear convergence coefficient, which adjusts the linear convergence speed of the sliding surface, with a typical value of η∈(0,10); κ>0 represents the nonlinear convergence coefficient, which adjusts the nonlinear convergence speed of the sliding surface, with a typical value of κ∈(0,10); q∈(0,1) represents the double power convergence factor, which balances the convergence speed and chattering suppression, with a typical value of q=0.5; T represents the discrete time sampling period (s); sat(s(k)) represents the saturation function, defined by equation (13), used to suppress sliding chattering.

[0096] Using saturation function Replacement symbol function It can suppress chattering caused by sliding surface switching to a certain extent.

[0097]

[0098] In the formula, the function boundary layer value Choosing a suitable value ensures the smoothness of the saturation function. The discrete-time sampling period (s); When the sliding surface value is outside the boundary layer, a sign function is used; When the sliding surface value is within the boundary layer, a linear function is used.

[0099] Substituting the PFDL system model (Equation 3) into the error formula e(k+1):

[0100]

[0101] Where e(k+1) represents the system output error vector (A) at time k+1. y(k+1) represents the expected output signal (A) of the system at time k+1; y(k+1) represents the actual output vector (A) of the system at time k+1; y(k) represents the actual output vector (A) of the system at time k. ΔU represents the time-varying pseudo-gradient (PG) matrix, which is solved using the PG estimation algorithm of equation (7), where T represents the transpose of the matrix / vector; L (k)=[Δu(k),…,Δu(k-L+1)] T The control input increment vector represents the rotation direction and frequency of the three-phase electrode motor within the sliding window, and Δu(k) = u(k) - u(k-1) is the control input increment (Hz). The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the external disturbance and the unmodeled dynamics.

[0102] Substitution The final control law is:

[0103]

[0104] Where Δu(k) = u(k) - u(k-1) is the control input increment (Hz); ; Represents the pseudo gradient Φ p,L (k) The estimated value of the first sub-block; Represents the pseudo gradient Φ p,L The estimated value of the i-th sub-block of (k), i=2,…,L; y(k) represents the expected output signal of the system at time k+1 (A); y(k) represents the actual output vector of the system at time k (A); e(k) represents the output error vector of the system at time k (A); μ>0 represents the integral gain; γ∈(0,1] represents the forgetting factor; I(k) represents the exponential decay integral term, defined by equation (11); λ>0 represents the proportional coefficient; δ∈(0,1) represents the terminal attractor parameter; η>0 represents the linear convergence coefficient; T represents the discrete-time sampling period (s); sat(s(k)) represents the saturation function, defined by equation (13); κ>0 represents the nonlinear convergence coefficient; q∈(0,1) represents the double power convergence factor; s(k) represents the terminal sliding surface vector at time k, defined by equation (10); L represents the linearization length constant of the control input, which is a positive integer with a value range of L∈[2,5].

[0105] 2.3 Sliding Mode Damped Pseudo-Gradient Estimation Method

[0106] In the control model based on Partial Pseudo-Scheme Dynamic Linearization (PFDL) discussed earlier, the pseudo-gradient estimator was used to approximate the mapping relationship between the incremental input and output of the system. However, under the excitation of the strongly nonlinear reaching law of the sliding mode controller, the traditional pseudo-gradient estimator is highly susceptible to high-frequency oscillations and chattering, leading to instability in the estimation process and even causing the control law to diverge.

[0107] To enhance the robustness of the pseudo-gradient estimator under sliding mode conditions, this invention introduces a sliding mode damped pseudo-gradient estimation method. The estimator update gain is dynamically adjusted by the sliding surface amplitude, thereby effectively suppressing oscillations while maintaining estimation convergence.

[0108] Under the PFDL modeling approach, the system incremental model is as follows:

[0109]

[0110] Where Δy(k)=y(k)-y(k-1) represents the system output increment vector (A) of the three-phase electrode current of the fused magnesium furnace at time k; Φ(k-1) represents the real time-varying pseudo-gradient (PG) matrix at time k-1; ΔU(k-1)=U(k-1)-U(k-2) represents the control input increment vector (Hz) at time k-1; and T represents the transpose of the matrix / vector.

[0111] Define pseudo-gradient estimation error:

[0112]

[0113] in, The pseudo-gradient estimation error matrix at time k; The true time-varying pseudo-gradient (PG) matrix represents time k; This represents the pseudo-gradient estimate matrix at time k.

[0114] The output prediction error is:

[0115]

[0116] in, This represents the output prediction error vector at time k. This represents the pseudo-gradient estimate matrix at time k-1.

[0117] The estimator update law, after introducing the sliding mode damping term, becomes:

[0118]

[0119] in, = The increment matrix representing the pseudo gradient estimate; update step size ; To prevent the denominator from being zero; sliding mode damping factor ; s(k) represents the terminal sliding surface vector at time k, defined by equation (10).

[0120] Substituting equation (18) into equation (17), we get:

[0121]

[0122] Where α(k)∈(0,1) represents the damping adjustment factor, defined by equation (21); Represents a positive semidefinite matrix; the damping adjustment factor is:

[0123]

[0124] because It is a positive semi-definite matrix, and This indicates that the product matrix has stable compressibility. Therefore:

[0125]

[0126] in, =Φ(k)- This represents the pseudo-gradient estimation error matrix; Representing a bounded space, it indicates that the error of the pseudo-gradient estimation is always bounded; and when , The damping term approaches 1, indicating gradual convergence. Therefore, this pseudo-gradient estimation method exhibits good boundedness and asymptotic convergence capability.

[0127] When the sliding surface amplitude is large, that is Damping term This suppresses rapid updates of the estimator and prevents oscillations from amplifying errors; when the sliding mode approaches the stable range, i.e. When the damping term approaches 1, the estimator resumes its normal update rate, improving estimation accuracy.

[0128] 2.4 Sliding Mode Disturbance Observer Design

[0129] In the electrode current control system, there are unknown disturbance terms such as unmodeled dynamics, power grid disturbances, and electrode melting fluctuations. To improve system robustness, this invention introduces a sliding mode disturbance observer into the control structure to estimate the disturbance online and use it as a feedforward term to compensate the control law, thereby effectively reducing error accumulation and control oscillations.

[0130] The core idea of ​​the observer is to estimate the disturbance term in real time using the system's input-output model while maintaining a simple model structure. The disturbance model in the control system of this invention is as follows:

[0131]

[0132] in, = The system output increment vector (A) represents the three-phase electrode current of the fused magnesium furnace at time k+1; Φ p,L (k) represents the time-varying pseudo-gradient (PG) matrix, which is obtained by solving the PG estimation algorithm of equation (7); =[Δu(k),…,Δu(k-L+1)] T The control input increment vector represents the rotation direction and frequency of the three-phase electrode motor within the sliding window. Δu(k) = u(k) - u(k-1) is the control input increment (Hz), and T represents the transpose of the matrix / vector. The lumped disturbance of the fused magnesium furnace electrode system at time k includes external disturbances and unmodeled dynamics, with the disturbance term... for:

[0133]

[0134] The disturbance observation error is defined as:

[0135]

[0136] in, The vector representing the perturbation observation error at time k; The actual output vector (A) of the system at time k; The vector (A) represents the estimated output values ​​of the system at time k.

[0137] because Since it can only be obtained in the future, it cannot be directly estimated using equation (23). This invention employs the sliding mode concept to construct the following disturbance observer:

[0138]

[0139] in, The vector representing the perturbation estimate at time k+1; The vector representing the estimated perturbation values ​​at time k; For observation error, The observer gain is the disturbance, η∈[100,500]; This is the boundary saturation function used above. This perturbation observer structure depends only on the input, output, and pseudo-gradient estimation.

[0140] 2.5 Stability Analysis

[0141] To verify the stability of the sliding mode enhanced model-free adaptive controller proposed in this invention, a joint Lyapunov function of the controller, disturbance observer, and pseudo-gradient estimator is constructed to address the nonlinear characteristics and multi-source uncertainties of the system, and the error convergence and boundedness are derived.

[0142] To ensure the theoretical validity of the controller and pseudo-gradient estimator, the following reasonable assumptions are introduced:

[0143] For any given expected output There always exists a bounded control input. This allows the system's output, driven by this input signal, to approach... .

[0144] There exists a constant , so that at any time If the input direction remains unchanged, then the pseudo gradient vector Each component does not undergo a sign mutation within consecutive time intervals.

[0145] External disturbances It is a slowly varying signal and has an upper bound. .

[0146] The rate of change of the pseudo-gradient is bounded, i.e. .

[0147] Assumption i is a fundamental condition ensuring the controllability of the system; Assumption ii restricts drastic changes in the pseudo-gradient direction, ensuring the smoothness of the estimation; Assumptions iii and iv respectively specify the boundedness of external disturbances and the rate of change of the pseudo-gradient, which are standard prerequisites for stability analysis. The stability proof is as follows:

[0148] 2.5.1 Stability analysis using Lyapunov

[0149] Define the following three types of errors:

[0150] Tracking error:

[0151] Perturbation estimation error:

[0152] Pseudo-gradient estimation error:

[0153] Define three Lyapunov function components respectively.

[0154] (1) Lyapunov function of sliding surface

[0155]

[0156] in, The Lyapunov function representing the terminal sliding surface is used to analyze the stability of the sliding mode approach process; For the terminal sliding surface.

[0157] (2) Perturbation observer Lyapunov function

[0158]

[0159] in, The Lyapunov function representing the perturbation observer is used to analyze the convergence of the perturbation estimation error; = This represents the perturbation estimation error vector at time k. The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the disturbance at time k. This represents the vector of perturbation estimates at time k.

[0160] (3) Pseudo-gradient estimation of Lyapunov function

[0161]

[0162] in, The Lyapunov function representing pseudo-gradient estimation is used to analyze the convergence of pseudo-gradient estimation errors. = This represents the pseudo-gradient estimation error matrix at time k.

[0163] Construct the joint Lyapunov function as follows

[0164]

[0165] in, This represents the joint Lyapunov function, used to prove the global stability of the entire closed-loop system (including sliding mode control, disturbance observer, and pseudo-gradient estimation);

[0166] (1) Sliding mode approach error difference:

[0167] Under the saturation approach law:

[0168]

[0169] in, The terminal sliding surface vector at time k+1; >0 represents the saturation approach rate parameter, which adjusts the approach speed of the sliding surface in the boundary layer; >0 represents the linear approach rate parameter, which adjusts the linear approach speed of the sliding surface; The saturation function is defined by equation (13) and is used to suppress sliding mode chattering.

[0170] when At that time, the Lyapunov function difference of the sliding surface is:

[0171]

[0172] when At that time, the Lyapunov function difference of the sliding surface is:

[0173]

[0174] in, = The difference representing the Lyapunov function of the terminal sliding surface is used to analyze the stability of the sliding mode approach process; The terminal sliding surface vector at time k is defined by equation (10); >0 represents the boundary layer value of the saturation function, as defined by equation (13); , representing the Lyapunov difference upper bound coefficient, guarantees the sliding surface's finite-time convergence outside the boundary layer; , representing the Lyapunov difference upper bound coefficient, indicates that the system converges to the vicinity of the boundary layer in a finite time, and then becomes exponentially stable.

[0175] (2) Error analysis of disturbance observer

[0176] The disturbance observer structure is as follows:

[0177]

[0178] in, The vector representing the perturbation estimate at time k+1; η represents the perturbation estimate vector at time k; η>0 represents the perturbation observer gain, used to adjust the convergence speed of the perturbation estimate, with typical values ​​η∈[100,500]. The saturation function, defined by equation (13), is used to suppress observer chattering.

[0179] The disturbance error is updated as follows:

[0180]

[0181] in, This represents the perturbation estimation error vector at time k. This represents the change in disturbance.

[0182] The Lyapunov function difference of the perturbation observer is:

[0183]

[0184] in, = The difference representing the Lyapunov function of the perturbation observer is used to analyze the convergence of the perturbation estimation error; Let η represent the perturbation estimation error vector at time k, and η>0 represent the perturbation observer gain, used to adjust the convergence speed of the perturbation estimation. Typical values ​​are... ∈[100,500]; The saturation function is defined by equation (13) and is used to suppress sliding mode chattering. = Represents the disturbance increment vector; >0 represents the final upper bound of the perturbation estimation error, which is determined by... The boundedness and choice of η are determined.

[0185] when Bounded and When chosen appropriately, the asymptotic convergence of the disturbance estimation error can be guaranteed, that is:

[0186]

[0187] (3) Error analysis of sliding mode damped pseudo-gradient estimation

[0188] The pseudo-gradient update law is:

[0189]

[0190] in, The matrix representing the pseudo-gradient estimates at time k+1; The matrix representing the pseudo-gradient estimates at time k; >0 represents the pseudo-gradient estimation step size factor, used to adjust the convergence speed of pseudo-gradient estimation; λ(s(k))>0 represents the sliding mode damping function, which is related to the sliding surface s(k) and is used to suppress chattering in pseudo-gradient estimation; Φ(k) represents the true pseudo-gradient matrix at time k.

[0191] Error updated to:

[0192]

[0193] in, = The pseudo-gradient estimation error matrix at time k; Represents the pseudo-gradient increment matrix; the pseudo-gradient estimation Lyapunov function difference is:

[0194]

[0195] in, = The difference represents the pseudo-gradient estimation Lyapunov function, used to analyze the convergence of the pseudo-gradient estimation error; >0 represents the final upper bound of the pseudo-gradient estimation error, which is determined by... Boundedness and The choice determines;

[0196] According to the assumption ,like If it is bounded, then

[0197]

[0198] in, = The pseudo-gradient estimation error matrix at time k; This represents the pseudo-gradient increment matrix; it shows that the pseudo-gradient estimation error converges to a small neighborhood under damped suppression.

[0199] 2.5.2 Joint Convergence Conclusion

[0200] The total Lyapunov difference is:

[0201]

[0202] in, = The difference representing the total Lyapunov function is used to analyze the joint stability of the closed-loop system; The Lyapunov function representing the terminal sliding surface; The Lyapunov function representing the perturbation observer; The Lyapunov function representing pseudo-gradient estimation; >0 represents the sliding surface convergence coefficient; >0 represents the convergence coefficient of the disturbance observation error; >0 represents the convergence coefficient of pseudo-gradient estimation error; The terminal sliding surface vector at time k; The vector representing the perturbation estimation error at time k; This represents the pseudo-gradient estimation error matrix. >0 represents the disturbance boundary term. As long as the control parameters meet the design conditions and the disturbance variation is bounded, then:

[0203] Sliding surface Converging to the boundary layer within a finite time; pseudo-gradient estimation error. asymptotic convergence to a small neighborhood; perturbation observation error It converges to the boundary layer; the joint system is globally uniformly bounded and possesses asymptotic stability.

[0204] 3. Simulation Analysis

[0205] To verify the advantages of the proposed PFDL-SMC control method under strong disturbances and time-varying nonlinear conditions in terms of tracking accuracy, robustness, and control smoothness, and to demonstrate the effectiveness of sliding mode disturbance observation and damped pseudo-gradient estimation in controller design by comparing its performance with comparative methods, the following simulation experiment was conducted to verify the effectiveness of the proposed PFDL-SMC method:

[0206]

[0207] in, Represents the output of a scalar system, used to simulate single-electrode current; Represents scalar control inputs used for simulating single-electrode motor control; time-varying parameters. This system has strong nonlinear and time-varying characteristics, which is quite suitable for fused magnesium systems.

[0208] The system expects the following output:

[0209]

[0210] in, This represents the desired output signal (A) at time k+1.

[0211] like Figure 1 As shown in the simulation graph, the tracking performance of the electrode current to the desired current is demonstrated. The red curve converges rapidly to the desired current within a short time, maintaining minimal fluctuations and no overshoot throughout the control process; while the comparative method exhibits significant oscillations and steady-state deviations under disturbances. It can be seen that the method of this invention significantly improves tracking accuracy, convergence speed, and dynamic stability. This result verifies the effective compensation effect of the sliding mode disturbance observer and the ability of damped pseudo-gradient estimation to suppress chattering.

[0212] In the smelting process of fused magnesia, the power supply system supplies power to the three-phase electrodes A, B, and C, generating electric arcs at their ends to release heat and melt the raw ore, forming a molten pool. Due to the high melting temperature, the three-phase electrodes are embedded in the raw ore during the smelting process, using a submerged arc method to melt and add material simultaneously. The current control system controls the motor to move the three-phase electrodes up and down, ensuring the electrode current tracks its set value. The fused magnesia furnace smelting system takes the rotation direction and frequency of the three-phase motor as input and the three-phase electrode current as output. Using the fused magnesia model described above, taking the A-phase electrode as an example, the discretized model of the electrode current in the fused magnesia smelting process is as follows:

[0213]

[0214] in: for Electrode current (A) at time t; for The direction and frequency (Hz) of the motor rotation at any given time; For modeling error compensation; and It is a nonlinear function and changes slowly with time. Therefore, it is assumed to be a constant. and The following equation is obtained:

[0215]

[0216] in: Represents the current self-growth coefficient. This represents the control input coupling coefficient. The reference input for the electrode current is... A.

[0217] The smelting process of fused magnesia is affected by uncertainties such as impurity composition and external conditions, as well as unpredictable external disturbances, which degrades the transient performance of the system. To simulate real working conditions, a composite disturbance is injected into equation (43). The system was controlled using the control method of this invention, with a sampling period of 0.01s and a simulation time of 500s. The simulation results are as follows. Figure 2 As shown.

[0218] The main parameters of the algorithm in this invention are selected based on the following criteria:

[0219] (1) PFDL-SMC method: , , , , , , , , , , .

[0220] (2) Traditional PFDL-SMC method: , , , , , , .

[0221] like Figure 2 As shown, the output curve of the A-phase electrode current in the fused magnesia smelting process using the control method of this invention is as follows: The traditional PFDL-SMC control exhibits significant fluctuations in the output current (orange) under disturbances, with amplitudes ranging from 14000A to 18000A, and deviating considerably from the target value of 15000A. This indicates that its pseudo-gradient learning and input update oscillate under disturbances, making it highly sensitive to external disturbances. In contrast, the PFDL-SMC method of this invention (blue) stably tracks the desired current curve, remaining essentially around 15000A with minimal fluctuations, demonstrating the good disturbance suppression effect of the sliding mode control element in this invention. Comparing the two output curves reveals that the traditional PFDL-SMC cannot maintain stable output under continuous disturbances, and its control performance exhibits a periodic trend with the disturbance. However, the PFDL-SMC control method (blue) with the addition of a disturbance observer and a double power-law approach shows stable and reliable output throughout the 500s control process, stably tracking the desired current (red), demonstrating stronger robustness and anti-interference capabilities.

[0222] Meanwhile, the PFDL-SMC control curve response exhibits no overshoot, rapidly converges in the initial stage, and maintains the sliding mode, ensuring that the tracking error remains within a very small neighborhood. This demonstrates that the sliding surface design guarantees the system's rapid entry into the sliding mode state. Introducing a double power-law approach law effectively reduces chattering while maintaining convergence speed.

[0223] like Figure 3 As shown, by control input The changes show that the control input fluctuates somewhat in the initial stage, then oscillates within a small, bounded range. After using damped pseudo-gradient estimation, the chattering of the control input is significantly reduced, and the changes become smoother. The control energy remains at a reasonable level throughout the process, without excessive shocks, providing good protection for the actuator. These results demonstrate that the proposed method not only ensures rapid and stable current control but also balances control energy consumption and engineering feasibility.

[0224] like Figure 4 As shown in the disturbance observation effect curves, the blue curve represents the disturbance value estimated by the sliding mode disturbance observer. The red dashed line represents the actual disturbance injected into the system. It can be observed that during the entire 0~500s simulation process, after several adjustments, there is almost no observation error, indicating that the designed disturbance observer has extremely high estimation accuracy and dynamic tracking capability.

[0225] In the initial stage, the estimator quickly approaches the true value of the perturbation within 0-10 seconds, indicating that the sliding mode observer has strong convergence capability. However, as the perturbation frequency and amplitude gradually increase, Still able to closely follow The changes showed virtually no lag, further validating SMDO's ability to identify rapid time-varying disturbances in real time.

[0226] As can be seen from the current tracking results in this figure, it is precisely because the disturbance observer accurately estimates and compensates for external disturbances in real time that the current output can always maintain precise tracking of the desired current. When the disturbance observer is removed, the control system will exhibit significant fluctuations (see...). Figure 4 The orange curve further illustrates the necessity and effectiveness of the sliding mode disturbance observer proposed in this invention in the controller.

[0227] The implementation results show that, addressing the problems of strong nonlinearity, model uncertainty, and complex external disturbances in the electrode current control system of an electric molten magnesium furnace, this invention studies a composite control strategy integrating model-free adaptive control (MFAC) and sliding mode control (SMC). Its main contents and conclusions are as follows:

[0228] 1. The data-driven framework based on PFDL effectively avoids dependence on complex furnace mechanism models and improves the engineering feasibility of the algorithm;

[0229] 2. By superimposing an improved terminal sliding mode term into the MFAC control law and employing a double power-law approaching law and a saturated boundary layer design, finite-time convergence can be achieved and chattering can be suppressed.

[0230] 3. The proposed SMDO can accurately estimate and compensate for external disturbances in real time, significantly enhancing the robustness of the system; the sliding mode damped pseudo-gradient estimation effectively suppresses high-frequency chattering during the estimation process by dynamically adjusting the update gain.

[0231] 4. Based on joint Lyapunov analysis, the stability and error convergence of the controller-observer-estimator coupled system are proved; simulation results further verify the effectiveness of the method and its potential for engineering applications.

Claims

1. A model-free sliding mode adaptive control and damping estimation enhancement method for an electrode system, characterized in that, Includes the following steps: (1) Constructing a partial scheme dynamic linearization (PFDL) data model for the electrode system of an electric fused magnesium furnace: ,in Let be the system output increment vector of the three-phase electrode current of the fused magnesium furnace at time k+1. It is a time-varying pseudo-gradient matrix. For sliding time windows [ The control input increment vector for the rotation direction and frequency of the three-phase electrode motor within [k], Δu(k)=u(k)-u(k-1) is the control input increment; This represents the lumped disturbance of the electrode system of the fused magnesium furnace at time k; (2) Design a model-free sliding mode (PFDL-SMC) controller. The controller adopts a terminal sliding mode surface containing an exponentially decaying integral term and a double power-law approaching law, and suppresses chattering through a saturation function. (3) A sliding mode damped pseudo-gradient estimation method is adopted, and the pseudo-gradient update gain is dynamically adjusted based on the sliding surface amplitude to suppress high-frequency oscillations during the estimation process; (4) Construct a sliding mode disturbance observer (SMDO) to estimate lumped disturbances in real time. It also feeds forward compensation to the control law, outputs control signals to adjust the rise and fall of the three-phase electrodes of the fused magnesium furnace, and realizes the tracking control of the electrode current to the set value.

2. The electrode system model-free sliding mode adaptive control and damping estimation enhancement method according to claim 1, characterized in that, In step (2), the expression for the terminal sliding surface is: ,in Let k be the terminal sliding surface vector. Let k be the system output error vector at time k. For the system The desired output signal at time t, with a scaling factor λ>0, terminal attractor parameter δ∈(0,1), and μ>0 representing the integral gain. γ is the exponential decay integral term, and γ∈(0,1] is the forgetting factor.

3. The electrode system model-free sliding mode adaptive control and damping estimation enhancement method according to claim 2, characterized in that, In step (2), the expression for the reaching law is: Where η>0 represents the linear convergence coefficient, κ>0 represents the nonlinear convergence coefficient, T represents the discrete-time sampling period, and q∈(0,1) represents the double power convergence factor. For a saturated function, satisfying: when hour ,when hour θ>0 is the function boundary layer value.

4. The model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems according to claim 1, characterized in that, In step (3), the expression for the update law of the sliding mode damped pseudo-gradient estimator is: ,in = The increment matrix representing the pseudo gradient estimate; update step size ; Penalty factor; sliding mode damping factor ; s(k) represents the terminal sliding surface vector at time k; This represents the output prediction error vector at time k. Δy(k) = y(k) - y(k-1) represents the pseudo gradient estimation matrix at time k-1, T represents the transpose of the matrix / vector, Δy(k) = y(k) - y(k-1) represents the system output increment vector of the three-phase electrode current of the fused magnesium furnace at time k, and ΔU(k-1) = U(k-1) - U(k-2) represents the control input increment vector at time k-1.

5. The model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems according to claim 1, characterized in that, In step (4), the expression for the sliding mode disturbance observer is: ,in This represents the vector of perturbation estimates at time k+1. Let represent the vector of perturbation estimates at time k, and η>0 be the perturbation observer gain. For observation error, It is a boundary saturation function.

6. The model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems according to claim 1, characterized in that, In step (1), the length L of the sliding time window is a positive integer, L∈[2,5].

7. The model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems according to claim 1, characterized in that, Lumped disturbance This includes unmodeled dynamics, power grid disturbances, and arc fluctuations.

8. The model-free sliding mode adaptive control and damping estimation enhancement method for electrode systems according to claim 1, characterized in that, By constructing joint Lyapunov functions Verify system stability, including , , It is proved that the sliding surface converges in finite time, and the perturbation estimation error and the pseudo-gradient estimation error converge asymptotically; among them, The Lyapunov function representing the terminal sliding surface. For the terminal sliding surface; The Lyapunov function representing the perturbation observer. = This represents the perturbation estimation error vector at time k. The lumped disturbance of the electrode system of the fused magnesium furnace at time k represents the disturbance at time k. The vector representing the estimated perturbation values ​​at time k; The Lyapunov function representing pseudo-gradient estimation, = This represents the pseudo-gradient estimation error matrix at time k.

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