A method for model predictive control of kraft pulp kappa number in a batch cooking process

Abstract

The application provides a model predictive control method for pulp kappa number in batch cooking process, and belongs to the technical field of pulping and papermaking. The method comprises the following steps: first, collecting multi-dimensional measurable state variables related to delignification in the cooking process to form an original state vector; then, using a Koopman operator and extended dynamic mode decomposition, the original state vector is mapped to a high-dimensional linear space to obtain an upgraded state vector; based on historical data offline training, a Koopman global linear prediction model is established, and a free liquid temperature setting sequence in a future control time domain is optimized by a model predictive controller; after each optimization, the first control amount of the temperature sequence is applied to the cooking process in real time, and the rolling optimization is repeatedly executed in combination with feedback correction in the next sampling period. The method overcomes the end-point kappa number steady-state deviation problem existing in batch cooking applications, and improves the adaptability to model mismatch and the robustness of the control system.

Description

Technical Field

[0001] This invention belongs to the field of pulp and paper technology, and specifically relates to a method for predicting and controlling the kappa value of pulp in an intermittent cooking process. Background Technology

[0002] The core task of batch pulping cooking is to dissolve lignin and retain cellulose to the maximum extent by reacting with the chemical solution in a closed container through heating. The degree of delignification is often measured by the kappa number of the pulp. In recent years, to address the production demands of multiple raw materials, small batches, and frequent changes in process parameters, researchers have introduced the Koopman operator into this field. By mapping low-dimensional nonlinear states such as temperature and solid-liquid phase concentration to a high-dimensional linear space, a global linear prediction model is constructed. Based on this, a model predictive control method is combined to continuously optimize the free liquid phase temperature, achieving accurate tracking of the kappa number setpoint. This method does not require local linearization of the reaction mechanism and can obtain global prediction capabilities with limited computational resources, making it considered an effective solution that balances control accuracy and system real-time performance. However, current practices typically remain at the level of "offline model training and direct online application": once there are fluctuations in wood species or active alkali formulation, model mismatch is reflected in the endpoint deviation; predictive control loops only use the current state as the initial value, lacking memory of historical deviations, resulting in residual steady-state error even after the system reaches steady state; in addition, constraints such as saturation of field actuators and measurement drift are not included in the state variables, and the optimization results often give unattainable temperature commands, forcing operators to revert to manual intervention. A more prominent contradiction is that traditional solutions simplify the "controller" to a piece of algorithm code, ignoring the coupling design with the cooker body, measurement system, and regulating valve. When the sensor experiences zero-point drift or the valve linearity deteriorates, the algorithm cannot detect it and is difficult to adaptively compensate, ultimately requiring the extension of cooking time or the addition of chemical solution to "pull back" the indicators, which increases energy consumption and amplifies fluctuations. Summary of the Invention

[0003] In view of this, the present invention provides a model prediction and control method for pulp kappa value in a batch cooking process, which solves the problems of steady-state deviation of the endpoint kappa value and model mismatch adaptability caused by the lack of integrated design of error memory and execution constraints in traditional batch cooking control methods (such as endpoint control based on H factor).

[0004] This invention is implemented as follows:

[0005] This invention provides a method for predictive control of kappa value in pulp during intermittent cooking processes, comprising:

[0006] S10: In the batch digester body and the field-level measurement system connected to it, collect multidimensional measurable state variables related to delignification in the batch digester as the original state vector, including but not limited to solid phase component concentration, liquid phase component concentration, free liquid phase temperature and solid-liquid composite phase temperature, to form the original state vector.

[0007] S20: In the control station of edge computing or distributed control system, the extended dynamic mode decomposition method is used to map the original state vector to a high-dimensional linear space to obtain the corresponding up-dimensional state vector, thereby realizing the global linearization of nonlinear dynamics.

[0008] S30: In the same control station, a Koopman linear prediction model describing the process dynamics is established through offline training based on historical operating data; the Koopman linear prediction model uses the free liquid phase temperature as the control input and the pulp kappa value as the control output.

[0009] S40: At each sampling time, the model prediction control program in the control station uses the Koopman linear prediction model to continuously optimize the free liquid phase temperature sequence in the future control time domain, so that the Kapoor value tracks to the set value and meets the upper and lower temperature limits given by the process.

[0010] S50: After each optimization, the first control quantity of the temperature sequence is applied to the cooking process in real time, and the rolling optimization is repeated in the next sampling cycle in combination with feedback correction to form a closed-loop control, so that the kappa value is finally tracked to the set value.

[0011] Koopman operator theory has opened up new perspectives for the study of complex nonlinear dynamic systems. This paper combines the data-driven approach of the Koopman operator with model predictive control algorithms. By matching appropriate observation functions, a more accurate Koopman model is established as the predictive model. Taking into account factors such as control objectives, performance optimization, and input constraints, the Kappa value of a batch pulping and cooking machine is precisely controlled. Simulation studies in MATLAB demonstrate the effectiveness of the proposed method. Furthermore, the robustness of the designed KMPC system is verified under perturbations of parameters such as effective alkali concentration, solid lignin content, and temperature. The proposed method has the following main advantages:

[0012] The Koopman operator, combined with extended dynamic mode decomposition and using the Gaussian function as the observation function, constructs an accurate linear prediction model in the finite-dimensional state space through data-driven global linearization. This method fundamentally eliminates the local linearization error and model mismatch that are prevalent in traditional nonlinear control methods.

[0013] By utilizing the Koopman operator, complex nonlinear control problems are transformed into convex quadratic programming solutions. Compared with traditional nonlinear MPC schemes, this significantly improves computational efficiency and real-time feasibility.

[0014] For strongly coupled, nonlinear batch cooking processes, a proposed model predictive control strategy based on the Koopman operator achieves precise control of the kappa value. The control system exhibits significant robustness when faced with parameter step perturbations, consistently maintaining stable kappa value tracking. This research provides an advanced solution for real-time optimal control of batch cooking processes, combining theoretical depth with engineering feasibility.

[0015] Upgraded state refers to the mapping of low-dimensional and highly nonlinear measurable state variables (such as temperature and concentrations of phase components) during batch cooking to a high-dimensional linear space using a set of predefined Gaussian functions, thus constructing a new state vector. This vector follows an approximately linear dynamic evolution law in the upgraded space, transforming the nonlinear dynamics of the original system into a linear relationship. Based on this linear representation, a corresponding predictive model can be established to describe the future behavior of the system, thereby providing a theoretical basis for implementing model predictive control.

[0016] Specifically, the original state vector consists of the data of 19 system state variables collected in the data acquisition section.

[0017] Based on the above technical solution, the method for predictive control of pulp kappa value in an intermittent cooking process can be further improved as follows:

[0018] The center point and bandwidth of the Gaussian function are determined through offline cross-validation to ensure that the relative error of the Koopman model in predicting the Kabbal value is no greater than 1.2%.

[0019] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: by using the Gaussian function selected through cross-validation for dimensionality mapping, the model error can be reduced to an extremely low level in the offline stage. After going online, the controller does not need to be adjusted again and the training results can be reused directly, which significantly shortens the on-site debugging cycle.

[0020] Furthermore, the historical operating data was obtained by randomly changing the initial conditions within the free liquid phase temperature range of 415K to 430K and performing multiple open-loop simulations on the mechanism model, covering the dynamic characteristics of the entire process of intermittent pulping and cooking.

[0021] The beneficial effects of adopting the above-mentioned improved scheme are as follows: Within the temperature range of 415–430K, batch simulations are conducted through a systematic random sampling design. A high-quality historical sample dataset covering the entire dynamic range of the system is generated in one go, ensuring that the subsequent Koopman model has seen various operating conditions, including "cold and hot start-up" and "high and low alkali". Based on this complete dataset, the model trained for online application, even when faced with fluctuations in raw material properties, can quickly and reliably provide accurate predictions thanks to its learned dynamic memory, without relying on additional online data accumulation or model correction.

[0022] Furthermore, the prediction time domain for rolling optimization is set to 15 sampling periods, the control time domain is set to 8 sampling periods, and the temperature constraint is set to a lower limit of 410K and an upper limit of 432K.

[0023] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: setting the prediction time domain to 15 minutes, the control time domain to 8 minutes, and the temperature to 410–432K not only provides the optimizer with sufficient foresight but also avoids computing power overflow. Traditional distributed control systems can also complete the solution, and there is no need to replace high-performance hardware for engineering implementation.

[0024] Furthermore, a Kapper value error integral term is introduced before rolling optimization to eliminate steady-state tracking bias.

[0025] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: when a new state is added, a "Kab value error accumulator" will be added to the upgraded state. When the optimizer calculates the temperature sequence each time, it will also clear the historical deviation to zero. Even if there is a slight mismatch or valve position drift in the model, the integral will continue to push the temperature until the measured Kab value completely coincides with the set value, thus completely getting rid of the steady-state error.

[0026] Furthermore, the process of introducing a cabochon value error integral term before rolling optimization includes the following steps:

[0027] A Kapoor value error accumulator is added to the controller to accumulate the difference between the set Kapoor value and the measured Kapoor value in real time. The controller is a computing unit or software module that embeds the Koopman model predictive control algorithm.

[0028] After each sampling is completed, the current difference is multiplied by a fixed ratio and accumulated into the accumulator to form an integral that grows over time.

[0029] The integral is added as a new state variable and fed into the Koopman linear prediction model along with the original upgraded state vector, so that the optimizer considers historical error accumulation information when rolling the calculation of future temperature sequences.

[0030] The optimization objective includes adding a penalty weight to the integral quantity, which forces the controller to actively offset the accumulated error in subsequent adjustments;

[0031] Once the system reaches steady state, the integral remains constant, and the difference between the measured Kab value and the set value is gradually approached to zero, thus achieving zero steady-state error tracking.

[0032] If a continuous disturbance causes the integral to be too large, the accumulator output can be clamped by limiting or anti-saturation logic to prevent the temperature command from exceeding the process allowable range.

[0033] "System" refers to a complete intermittent cooking closed-loop control system consisting of the cooker body, measuring or actuating mechanisms, and control algorithms; only this whole system can be said to "enter steady state and have errors approaching zero".

[0034] Furthermore, when the effective base or active hydride concentration experiences a +5% step disturbance, the controller still maintains the steady-state error of the kappa value within ±0.5.

[0035] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: when the laboratory reports that "the effective base or hydride is 5% higher", the error accumulator immediately senses the accelerated reaction and the drop in the kappa value, and automatically lowers the temperature. There is no need to modify the process card throughout the process. The system can rebalance by online integration, ensuring that the endpoint kappa value does not deviate from the target.

[0036] Furthermore, when the concentration of solid-phase highly reactive lignin, low reactive lignin, or their total concentration experiences a +5% step disturbance, the controller adaptively adjusts the temperature to ensure that the kappa value overshoot does not exceed 2%.

[0037] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: once the quality of wood chips changes and the high and low activity lignin increase at the same time, the integral state will bring the "reaction is difficult to chew" information to the optimizer, and the instruction temperature will be actively raised, avoiding the operator from manually adding cooking time, and the kappa value will still be steadily sent into the qualified area, and the overshoot will be locked within the allowable bandwidth.

[0038] Furthermore, when the measured temperature of the solid-liquid composite phase is 3-5K higher than expected, the controller adjusts the free liquid phase temperature accordingly using a feedforward compensation method to maintain the Kapper value tracking error within ±1.

[0039] The beneficial effects of adopting the above-mentioned improvement scheme are as follows: when the thermocouple aging reading is 3-5K higher, the error integration window will treat the "false high" as a continuous deviation, and in turn reduce the steam valve opening to offset the risk of overcooking caused by measurement drift. This ensures that the actual kappa value still follows the set curve, and extending the probe calibration cycle does not affect the pulp quality.

[0040] Furthermore, the Koopman linear prediction model is fixed at 27 dimensions, and once the model parameters are trained offline, they are not modified during the entire cooking batch operation to reduce the amount of online computation.

[0041] The beneficial effects of adopting the above-mentioned improved scheme are as follows: the 27-dimensional model is fixed for life after one training, the DCS only needs to save three constant tables, and only matrix multiplication and QP solution are performed in the online stage. The memory usage and CPU load are comparable to those of traditional PID.

[0042] Compared with existing technologies, the beneficial effects of the kappa value model predictive control method for pulp in intermittent cooking processes provided by this invention are as follows: This invention treats the algorithm, measurement, and execution as a whole, embeds the Koopman extended model into the distributed control system or edge computing node, and introduces the kappa value error integral into the state vector, making historical deviations a predictable and penalized "visible state," thereby directly incorporating the steady-state error elimination mechanism into the optimization objective. When the effective alkali concentration, lignin activity, or temperature reference undergoes slow drift, the integral state will continuously accumulate deviation information, forcing the optimizer to actively adjust the temperature curve in subsequent cycles, pulling the endpoint back to the target area without manual modification of the process card. By unifying the error integral, measurement drift, and valve saturation into an augmented state model, the controller can avoid infeasible regions in advance during the solution process, avoiding "accurate calculations but inability to execute" commands, reducing frequent saturation and rebound of the actuator, improving valve life, and reducing steam waste. The solution employs offline cross-validation to determine Gaussian function parameters. Once deployed, the model's dimension and coefficients are fixed. In the online phase, only matrix multiplication and standard quadratic programming are performed, requiring hardware computing power comparable to traditional PID control. Upgrading older equipment requires no replacement of high-performance CPUs or additional servers, enabling a "plug-and-play" transformation. Due to the introduction of integral states, the system's tolerance to model-object mismatch is significantly improved. Even with changes in timber origin, fluctuations in active alkali, or aging temperature probes, closed-loop memory can automatically compensate, maintaining a stable Kaplan value curve and reducing the additional costs associated with re-cooking, downgrading, and adding bleach. The entire method can achieve millisecond-level solutions on general-purpose PLCs or embedded controllers, synchronized with existing DCS scan cycles. Field personnel only need to replace the original temperature control loop, without altering the digester's structure or adding complex instruments, thus completing the transition from "experience-based endpoint judgment" to "model-predictive endpoint control." This provides an engineering path for intermittent cooking that balances investment protection and performance improvement. Attached Figure Description

[0043] Figure 1 This is a schematic diagram of the intermittent pulping and cooking process;

[0044] Figure 2 The temperature change curves of the free liquid phase and the solid-liquid composite phase are shown.

[0045] Figure 3 A graph showing the concentration changes of the components in the retained liquid phase and the free liquid phase;

[0046] Figure 4 This is a graph showing the change in mass concentration of the solid phase component.

[0047] Figure 5 This is a graph showing the change in kappa number of pulp during intermittent cooking.

[0048] Figure 6 A comparison of the Kappa values ​​of the three basis function Koopman linear models with the original nonlinear models;

[0049] Figure 7 A comparison of the solid-liquid composite phase temperature in the Koopman linear model with the original nonlinear model using three basis functions;

[0050] Figure 8 Plots showing the total relative root mean square error of the Koopman model with different dimensions of the Gaussian function;

[0051] Figure 9 The free liquid temperature control curve for Koopman-MPC is shown below.

[0052] Figure 10 A graph showing the Kappa value tracking curve for Koopman-MPC;

[0053] Figure 11 The graph shows the free liquid temperature control curves under perturbation of the parameters of the active effective base and the active hydride.

[0054] Figure 12 The Kappa value tracking curve is provided for the perturbation of the parameters of the active effective base and the active hydrosulfide.

[0055] Figure 13 Free liquid temperature control curves under perturbation of solid-phase lignin component parameters;

[0056] Figure 14 Kappa value tracking curves when solid-phase lignin component parameters are perturbed;

[0057] Figure 15 A graph showing the free liquid temperature control curve under temperature parameter perturbation of the solid-liquid composite phase.

[0058] Figure 16 Kappa value tracking curve for solid-liquid composite phase under temperature parameter perturbation;

[0059] Figure 17 This is a flowchart of the present invention. Detailed Implementation

[0060] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0061] like Figure 1The diagram shown is a flowchart of a predictive control method for kappa value model of pulp in an intermittent cooking process provided by the present invention, which includes the following:

[0062] Kinetic model of intermittent pulping and cooking process:

[0063] Mechanistic models of pulping processes are all based on the physical and chemical mechanisms of the cooking process. Based on the laws of conservation of matter / energy / momentum, the principle of phase equilibrium, or the principles of reaction / mass transfer kinetics, idealized functional relationships between operable variables and controllable target variables are derived. This paper, based on the extended Purdue model established by Wisnewski and Doyle in the literature, conducts a high-dimensional Koopman linear prediction model study on the chemical batch cooking pulping process.

[0064] The extended Purdue model assumes that the pulping dynamics of a batch digester consist of three phases: a solid phase, a retentate liquid phase, and a free liquid phase. The solid phase specifically refers to the pulping raw materials; the retentate liquid phase refers to the liquid existing in the pores of the raw materials; and the free liquid phase refers to the free liquid surrounding the raw materials. In the sulfate pulping process, pretreated raw materials and a cooking liquor (mainly composed of sodium hydroxide and sodium sulfide, called "white liquor") are fed into the digester from the feed inlet and thoroughly mixed. Under closed conditions, steam is introduced to gradually increase the temperature and pressure to the cooking conditions. The cooking liquor reacts chemically with the raw materials, dissolving and separating components such as lignin while retaining cellulose to form pulp. The cooking endpoint is determined by extracting a black liquor sample and measuring its residual lignin concentration to confirm that the predetermined delignification effect has been achieved. After cooking, the pulp and black liquor are discharged from the bottom outlet of the digester. The pulp enters subsequent washing and bleaching processes to ultimately form high-quality pulp; the black liquor is treated or residual alkali is recovered to achieve energy conservation, emission reduction, and clean green production. A schematic diagram of the intermittent pulping and cooking process is shown below. Figure 1 As shown.

[0065] The pulping and cooking process mainly involves two irreversible chemical reactions: delignification and cellulose degradation. The corresponding material transformations are shown below.

[0066] ;

[0067] The kinetic behavior of each component in the solid, trapped liquid, and free liquid phases is dominated by mass and energy transfer during the cooking process, and is influenced by the dissolution and diffusion mechanisms of the materials within the cooker. The extended Purdue model establishes the reaction kinetic differential equations for each component based on the principles of mass and energy balance.

[0068] Assuming the wood chips are homogeneous and all extractable substances have been removed, and that they are fully impregnated upon entering the digester with no trapped air, the mass conservation equation for the batch pulping and digesting process is shown below.

[0069] ;

[0070] In the formula =1,2,…,5; =1,2,…,6.

[0071] The formulas for calculating the parameters in the mass conservation equation are shown below.

[0072] ;

[0073] Assume the temperature of the solid-liquid complex phase in the batch digester system ( ), free liquid phase temperature ( Heat transfer occurs between the solid-liquid composite phase and the free liquid phase. The energy conservation of the system includes both the energy conservation of the solid-liquid composite phase and the energy conservation of the free liquid. The energy conservation equation for the intermittent pulping and cooking process is shown below.

[0074] ;

[0075] The kappa number is a chemical indicator characterizing the residual lignin content in pulp, and also relatively represents the degree of delignification of the pulping raw materials during cooking. In the extended Purdue model, the calculation model for the pulp kappa number is shown below.

[0076] ;

[0077] The meanings of the parameters in the extended Pudu model for the intermittent pulping and cooking process are shown in Table 1.

[0078] Table 1. Meaning of the symbols for each parameter in the Extended Purdue Model

[0079]

[0080] A model for predicting the kappa value in batch cooking processes based on the Koopman operator:

[0081] Koopman operator theory:

[0082] Consider the following nonlinear dynamic system that includes control input:

[0083] ;

[0084] In the formula, For state space On Maintain system status. Let be the discrete time step of the system. For the first The system state at each time step. To control space On Dimensional control input, For the first Time step control input, It is a state space The next state evolution function. The observation function is defined. The set of all observed functions constitutes an infinite-dimensional vector space. Define the Koopman operator. It is an action on the observation function The linear operators on are shown below.

[0085] ;

[0086] Nonlinear systems in finite-dimensional space China and Israel Nonlinear evolution, based on observation function Transform the nonlinear system to an infinite-dimensional space. China and Israel Linear evolution.

[0087] Finite-dimensional approximation of the Koopman operator:

[0088] The infinite-dimensional nature of the Koopman operator hinders its direct application. To obtain a finite-dimensional approximation model of the Koopman operator, the Extended Dynamic Mode Decomposition (EDMD) algorithm can be used. The essence of the EDMD method is to extract the system's state evolution information from historical or simulation data, combine it with a predefined basis function space, and then approximate the Koopman operator using a finite-dimensional matrix. This transforms an infinite-dimensional problem into a finite-dimensional problem, making it suitable for high-dimensional and nonlinear systems.

[0089] Based on the EDMD method, select a set of basis functions Describing a nonlinear dynamical system, the lift function is defined as:

[0090] ;

[0091] in:

[0092] ;

[0093] ;

[0094] The promotion state is defined as:

[0095] ;

[0096] In the formula To enhance the system dimension.

[0097] Based on the properties of the Koopman operator, the high-dimensional linear model of the Koopman operator is constructed as follows:

[0098] ;

[0099] In the formula, The state of the original nonlinear system The estimate, For high-dimensional systems The system state at the time step. , , These are all linear time-invariant matrices of the system, which can be obtained by solving an optimization problem:

[0100] ;

[0101] A high-dimensional linear Koopman prediction model for kappa values ​​in batch cooking processes was established:

[0102] This describes a batch cooking process for pulping, and the state vector is defined as follows: ,in - The first The time step includes the concentrations of five components in the solid phase, the concentrations of six components in the retained liquid phase, the concentrations of six components in the free liquid phase, and the temperature variable of the solid-liquid composite phase. Control inputs. This indicates the first step in the intermittent pulping and cooking process. Free liquid temperature at time step.

[0103] To construct a high-dimensional space for observable functions, basis functions can typically be selected from thin-plate spline radial basis functions, Gaussian functions, and multiharmonic basis functions. Thin-plate spline radial basis functions are shown below:

[0104] ;

[0105] The Gausky function formula is shown below:

[0106] ;

[0107] The formulas for multiple harmonic basis functions are shown below:

[0108] ;

[0109] This paper selects the Gaussian function as the basis function, where, For the first The center point of each basis function The bandwidth is Gaussian. If the center point is chosen as... The promotion function is as follows.

[0110] ;

[0111] Based on the original nonlinear system, i.e., the dynamic system, discrete-time observation data of the system's state vector and input vector are collected through numerical simulation to establish the following dataset:

[0112] ;

[0113] Based on the state promotion function, the original state dataset and After being elevated to a higher dimension, they are respectively and :

[0114] ;

[0115] According to the EDMD method, the system matrix , , This can be derived by solving the following optimization problem:

[0116] ;

[0117] The analytical solution is:

[0118] ;

[0119] in, It is a Moore-Penrose pseudo-inverse.

[0120] The above method transforms the original nonlinear system into a high-dimensional global linear model based on the Koopman operator. This model can effectively predict the system's future dynamic behavior, providing a theoretical basis for the design of subsequent controllers.

[0121] Design of a predictive controller for the kappa value in a batch cooking process based on the Koopman operator:

[0122] Model predictive control (MPC) is an advanced control strategy based on rolling time-domain optimization. Its core lies in using a mathematical model to predict the future dynamics of the system, solving a constrained optimization problem within a finite prediction time domain, and implementing only the optimal control input at the current time step. The optimization window is then pushed forward in a rolling time-domain manner, and this process is repeated at each time step. However, traditional MPC faces challenges of high computational complexity and poor real-time performance when dealing with strongly nonlinear systems. To address this, this paper introduces Koopman operator theory, constructing an efficient linear predictive model by globally mapping the nonlinear system to a high-dimensional linear space. This combined strategy retains the advantages of MPC in handling constraints while significantly improving the control performance and computational efficiency of nonlinear systems.

[0123] Based on the established high-dimensional linear prediction model using the Koopman operator, then from the... The state of time step improvement The initial evolution of the future state can be described as follows:

[0124] ;

[0125]

[0126] ;

[0127]

[0128] ;

[0129] In the formula, To control the time domain, To predict the time domain, and They represent Predicted state and control input at any given time.

[0130] Based on the estimated input of the batch pulping and cooking system (i.e. ). This is the estimated output of the batch pulping and cooking system, assuming... equal , It can be represented as:

[0131] ;

[0132] The system's prediction equation can be reformulated as:

[0133] ;

[0134] in:

[0135]

[0136]

[0137]

[0138]

[0139] Setting the reference trajectory,

[0140] Reference trajectories are typically represented as time series, with each time step containing the expected value of the system state or output. A reference trajectory can be represented as:

[0141] ;

[0142] in, It is the first The expected state or output value at each time step.

[0143] The objective function in MPC based on the Koopman operator is:

[0144] ;

[0145] ;

[0146] ;

[0147] ;

[0148] ;

[0149] in, and These are the state weighting matrix and control weighting matrix of the nonlinear system, respectively, both of which are positive definite weighting matrices; To predict the output, For controlling input.

[0150] To improve the efficiency of the solution, formula (24) can be converted into the following expression:

[0151] ;

[0152] In the formula:

[0153] ;

[0154] ;

[0155] ;

[0156] ;

[0157] This problem can be solved as a standard linear quadratic programming problem, thus quickly obtaining the global optimum. In practical applications, adjustments are needed. and The matrix enables the system to achieve high-precision trajectory tracking within specified constraints.

[0158] Simulation results and analysis:

[0159] The experiments in this paper were implemented in the same simulation environment (Intel(R) Core(TM) i5-10210U CPU (1.60GHz), RAM (16GB), MATLAB R2018b). The model parameters of the pulping and cooking system are shown in Table 2. The nonlinear dynamic model of the batch pulping and cooking process was discretized using the Euler method, and the open-loop simulation results are as follows: Figures 2-5 As shown.

[0160] Table 2 Initial values ​​of model parameters for the intermittent pulping and cooking process

[0161]

[0162] Figure 2 This is a curve showing the temperature change over time of the free liquid phase and the solid-liquid composite phase during intermittent cooking. Figure 2 It can be seen that the temperature of the free liquid phase initially decreases and then increases during the cooking process, while the temperature of the solid-liquid composite phase consistently increases, with both eventually remaining around 429 K. This temperature change trend is consistent with the heat transfer characteristics between the free liquid phase and the solid-liquid composite phase. The initial temperatures of the free liquid phase and the solid-liquid composite phase are 423 K and 383 K, respectively, with a significant temperature difference between them. The larger the temperature difference, the faster the heat transfer rate. Therefore, in the initial stage of cooking, the temperature of the solid-liquid composite phase rises rapidly, while the temperature of the free liquid phase initially decreases and then increases. In the middle and later stages of cooking, the temperature difference between the two phases decreases, and their temperatures essentially reach equilibrium. The temperature of the solid-liquid composite phase then changes with the temperature of the free liquid phase.

[0163] Figure 3 The curves show the changes in the concentrations of six components in the retained liquid phase and the free liquid phase over time during intermittent cooking. Figure 3 It can be seen that in the retentate phase during the initial stage of cooking, the main reactants for delignification are the active effective base and the active hydrogen sulfide. and The concentration increases rapidly from 0, while in the free liquid phase its ( and The concentration of lignin decreases rapidly from its initial value, and the resulting concentration difference is the main driving force for the transfer of the same substance between the two phases to form liquid-liquid equilibrium. In the later stages of cooking, most of the lignin is removed, and a large amount of active effective alkali and active hydrogen sulfides are consumed. The concentrations of both gradually decrease in the retentate phase and the free liquid phase, respectively forming interphase equilibrations; simultaneously, the concentration of lignin in the retentate phase (…) decreases rapidly from its initial value, and the resulting concentration difference is the main driving force for the transfer of the same substance between the two phases to form liquid-liquid equilibrium. and ) and free liquid phase ( and The concentrations of the main delignification reaction products, low-activity effective base and low-activity hydrogen sulfide, gradually increase from 0 and form phase equilibrium respectively.

[0164] Partially dissolved lignin and dissolved carbohydrates are products of the delignification reaction, and both are present in the retentate phase. and The concentration of all components gradually increases from 0, while in the free liquid phase, the concentration of all components (…) increases gradually. and The initial value shows a slight decrease followed by a gradual increase, eventually reaching a phase equilibrium. The cooking liquor is prepared by mixing white liquor and black liquor, and the residual lignin in the black liquor affects the free liquid phase... and It has a certain initial concentration; in addition, due to the low temperature inside the digester during the initial stage of cooking, dissolved lignin and dissolved carbohydrates easily polymerize and adhere to the wood chips, making... and The concentration decreased slightly. In summary, the trends in the concentrations of the retained liquid phase and the free liquid phase are completely consistent with the actual pulping process, and can well reflect the kinetic characteristics of the lignin removal reaction and the interphase diffusion characteristics of substances in the batch pulping process.

[0165] Figure 4 The curves show the changes in the mass concentration of the five solid components over time during intermittent cooking. Figure 4 It can be seen that as the cooking process deepens, the highly reactive lignin ( ) and low-reactivity lignin ( The concentration of active cellulose (cellulose) decreases continuously due to the delignification reaction, but it does not reach zero. Similarly, the concentration of active cellulose (cellulose) decreases continuously due to the delignification reaction, but it does not reach zero. ), active xylan ( ), galactoglucan ( Under alkaline conditions, the mass concentration of all molecules continuously decreases due to cellulose degradation. It is easily degraded under high temperature and alkaline conditions; Due to its low initial concentration and poor chemical stability, it rapidly degrades during the initial stages of cooking; while Under alkaline conditions, only partial decomposition occurs, and the component still maintains a high mass concentration at the end of cooking. The dynamic changes in the mass concentration of the above components are consistent with the reaction kinetics of the actual batch cooking process.

[0166] Figure 5 This is a curve showing the change in kappa number of pulp over time during intermittent cooking. Figure 5It can be seen that in the initial stage of cooking, due to the low temperature of the solid-liquid composite phase, the delignification chemical reaction proceeds slowly, and the dissolved lignin in the cooking liquor easily polymerizes and adheres to the wood chips, resulting in a slight upward trend in the pulp kappa number. As the cooking temperature increases, the lignin in the wood chips is rapidly degraded, and the cooking process enters a stage of significant delignification, causing the pulp kappa number to decrease rapidly. In the later stage of cooking, most of the lignin in the wood chips has been removed, entering the stage of residual lignin removal, where the delignification reaction becomes slower, and the pulp kappa number tends to stabilize. In summary, the trend of pulp kappa number changes can fully reflect the dynamic characteristics of lignin degradation during intermittent cooking.

[0167] Model Tracking Performance Analysis: To construct the high-dimensional feature space of the system, dynamic system state data of the batch cooking process were collected through numerical simulation. The sampling time interval was set to 1 minute, and 1000 sets of differentiated initial control input conditions were systematically selected. To ensure that the collected data more comprehensively describes the system characteristics, the initial free liquid temperatures were randomly generated within the range [415K, 430K]. Starting from each initial point, system state variable response data were continuously collected for 300 minutes, obtaining state trajectories containing 300 consecutive sampling points. The final dataset contains 1000 independent system trajectories, each representing the evolution of the model within a 300-minute running cycle. The constructed high-dimensional Koopman model of the batch cooking process was validated based on the initial values ​​of the parameters shown in Table 2.

[0168] The fit of the Koopman linear model is evaluated by the relative root mean square error (RRMSE), which reflects the fitting error between the state trajectory of the high-dimensional linear model and the actual state trajectory of the nonlinear system. It is often used to measure the accuracy of model identification. The calculation formula is shown below.

[0169] ;

[0170] In the formula, and These are the actual system values ​​and the values ​​from the Koopman high-dimensional linear model, respectively. This represents the number of data points.

[0171] Model Basis Function Selection: A Comparison of Key State Variables Kappa Values ​​and Solid-Liquid Composite Phase Temperature in the Intermittent Pulping and Cooking Process Constructed Using Thin-Plate Spline Radial Basis Functions, Multiharmonic Basis Functions, and Gaussian Functions. Figure 6 , Figure 7 As shown. By Figure 6 and Figure 7It is evident that the high-dimensional linear Koopman model of the batch cooking process established using the Gaussian function better fits the original nonlinear model throughout the entire time range, fully describing the dynamic behavior of the batch cooking system. In contrast, the Koopman models obtained using thin-plate spline radial basis functions and multiharmonic basis functions deviate from the original nonlinear model and cannot accurately characterize the dynamic and static performance of the batch cooking system. The relative root mean square errors of the Koopman linear models established using the three basis functions and the extended Purdue model are shown in Table 3. To achieve optimal dynamic characteristics and overall fitting accuracy, a Koopman model based on the Gaussian function is used to linearly describe the batch cooking process.

[0172] Table 3. Relative root mean square error between the Koopman linear model and the original nonlinear model with different basis functions.

[0173]

[0174] Model Dimension Selection: Based on the selection of the Gaussian function as the basis function, this study further explored Koopman linear models constructed using the Gaussian function in different dimensions. Since the key state variable Kappa value has different dimensions from the solid-liquid composite phase temperature, the sum of the relative root mean square error (RRMSE) of the Kappa value and the solid-liquid composite phase temperature was used to evaluate the fitting accuracy of the Koopman linear models in different dimensions. The overall RRMSE trend of the 20-dimensional to 40-dimensional Koopman linear models is shown below. Figure 8 As shown in the figure. The results indicate that the 27-dimensional and 39-dimensional Koopman models perform similarly in terms of prediction accuracy for key state variables, and both can effectively characterize the key dynamic characteristics of the system. However, the 39-dimensional Koopman model has higher structural complexity and computational cost than the 27-dimensional model. Therefore, the 27-dimensional Koopman model exhibits the best accuracy-complexity balance, with relative root mean square errors of 1.16% and 0.0019% for the Kappa value and the temperature of the solid-liquid composite phase, respectively. Considering both model complexity and accuracy, a 27-dimensional Koopman model based on the Gaussian function is finally established to linearize the description of the batch cooking process.

[0175] MPC Tracking Performance Analysis Based on Koopman Operator: To verify the control performance of the proposed model predictive control strategy based on the Koopman operator in the batch pulping and cooking process, we conducted numerical simulations under typical operating conditions for 300 minutes. The control objective was to achieve accurate tracking of the pulp Kappa value while strictly constraining the control input (free liquid temperature) within the process-allowed range of 410K to 432K. The controller parameters are shown in Table 4. Simulation results are as follows: Figure 9 and Figure 10As shown, the simulation results of the tracking performance of the control input free liquid temperature curve and the controlled output Kappa value are presented respectively.

[0176] Table 4 MPC Controller Design Parameters

[0177]

[0178] Figure 9 The temporal variation of the closed-loop control input (free liquid temperature) in the cooking process is described. In the initial stage, the controller rapidly increases the temperature to the constraint limit of 432K to accelerate the reaction and shorten the response time using high temperature. Subsequently, the temperature is smoothly adjusted according to the dynamic change of the Kappa value to avoid overcooking. In the final stage, the temperature is slightly increased again to offset the deviation caused by the decrease in reaction rate. The entire input trajectory is smooth and without frequent chattering, demonstrating the high-precision linearization capability of the Koopman operator for nonlinear cooking dynamics. Finally, the dynamic adjustment of the free liquid temperature successfully drives the Kappa value to the setpoint, verifying the feasibility and superiority of the proposed strategy under input constraints.

[0179] Figure 10 The tracking performance of the Kappa value throughout the cooking process was demonstrated. In the initial stage, the Kappa value rapidly approached the setpoint from its initial state, exhibiting good dynamic response characteristics. In the subsequent process, the Kappa value changed smoothly and gradually approached the setpoint without significant overshoot or oscillation. Ultimately, the Kappa value was driven to the setpoint, essentially coinciding with it, demonstrating the controller's excellent tracking capability. Throughout the control period, the system output remained smooth and continuous without significant fluctuations, verifying the Koopman-MPC's robustness and control accuracy in handling nonlinear processes.

[0180] Robustness Analysis under Parameter Perturbation: To systematically verify the robustness performance of the Koopman-MPC strategy constructed in this paper under uncertainties in model parameter perturbation, this study conducted a comprehensive evaluation through a series of closed-loop simulation experiments. Step-type parameter perturbations were applied to the concentration of key components in the free liquid phase, the lignin composition in the solid phase, and the temperature of the solid-liquid composite phase, respectively, comprehensively revealing the adaptability and robustness of the strategy under various real-world uncertainties. The specific simulation experiment settings are as follows:

[0181] The concentrations of the key components in the free liquid phase—active effective base (EA) and active hydrosulfide (HS)—are subject to a +5% parameter perturbation.

[0182] Highly reactive lignin in the solid phase ( ), low reactive lignin ( The parameters of both and their total concentrations were perturbed by +5%.

[0183] The solid-liquid composite phase temperature Tc was subjected to parameter perturbations of +3K and +5K, respectively.

[0184] Based on the simulation conditions of the above-mentioned intermittent pulping and cooking process parameter perturbation, the closed-loop test results of the proposed Koopman-MPC strategy are as follows: Figures 11 to 16 As shown.

[0185] Figure 11 The control input (free liquid temperature) adjustment curves were compared under parameter perturbations of EA and HS concentrations by +5%. Under both perturbation scenarios, the controller effectively tracked the Kappa value by adjusting the temperature, and the control input consistently met the constraints throughout the process. Compared to the nominal case, the temperature was lower under the EA concentration +5% scenario than under the HS perturbation scenario, reflecting the controller's differentiated response strategy to perturbations of different chemical components. The controller compensated for the reaction rate changes caused by the increase in EA and HS concentrations by adjusting the input; the temperature curves remained smooth under both perturbations without drastic fluctuations, indicating that the Koopman-MPC maintains good stability when dealing with parameter uncertainties.

[0186] Figure 12 The tracking performance of the system output Kappa value is demonstrated under parameter perturbations of EA and HS concentrations +5%. When the EA concentration increases, the Kappa value deviates from the setpoint by a greater margin throughout the response than under the HS perturbation scenario, reflecting that the process dynamics are more sensitive to changes in EA concentration. The steady-state error of the Kappa value is small under both parameter perturbations, indicating good tracking of the setpoint and demonstrating the controller's excellent adaptability to model parameter perturbations.

[0187] Figure 13 Demonstrates solid components , and The dynamic response characteristics of the control input (free liquid temperature) are analyzed when the concentration is subjected to a +5% parameter perturbation. Simulation results show that under all perturbation conditions, the controller can effectively track the Kappa value by adjusting the temperature, and the control input strictly satisfies the constraints. The temperature response under different perturbations exhibits significant differences: The overall control temperature is highest during combined disturbances. Individual perturbations are the next best option. The temperature is lowest under individual perturbation. This response characteristic conforms to the reaction kinetic mechanism—the system compensates for the highly reactive lignin ( The changes in reaction kinetics caused by the increase of ) necessitate a moderate increase in temperature to maintain the reaction rate; As a low-reactivity component, a higher temperature is required to maintain the reaction rate, while The synergistic effect generated by the combined increase further inhibits the reaction process, resulting in the most significant temperature requirement. All temperature curves remain smooth without drastic fluctuations, demonstrating the good robustness and stability of the Koopman-MPC control strategy under solid-state parameter perturbations.

[0188] Figure 14 Shown in , and The system's Kappa value tracking performance was analyzed when the concentration was subjected to a +5% parameter perturbation. Simulation results show that the Kappa value exhibits good setpoint tracking capability under different types of solid component perturbations. The system remains stable under all operating conditions, and the Kappa value eventually converges to the setpoint neighborhood, with steady-state deviations controlled within the allowable range. This result verifies the robustness and control effectiveness of the Koopman-MPC control strategy in the face of solid component parameter perturbations.

[0189] Figure 15 The dynamic response characteristics of the control input free liquid temperature were compared when the solid-liquid composite phase temperature was perturbed by +3K and +5K parameters, respectively. Simulation results show that the controller can effectively control the free liquid temperature under both perturbation conditions by adjusting the input parameters. The system effectively maintains thermodynamic stability, and all control inputs strictly meet the constraints. Analysis of the temperature response curve shows that, compared to... +3K operating conditions The free liquid temperature is generally higher under a +5K disturbance. This response characteristic reflects the feedforward compensation mechanism embedded in the control strategy: when When a positive step change occurs, the controller increases accordingly to counteract the resulting thermal effect and maintain a stable reaction process. The operating setpoint, and the increase is positively correlated with the disturbance intensity. With When the disturbance increases from +3K to +5K, the system reaches equilibrium at a higher temperature level. The required compensation is relatively reduced, and the temperature drop is also reduced accordingly, reflecting the controller's accurate perception of the disturbance intensity and its adaptive adjustment. During the simulation, all temperature response curves remained smooth and stable, verifying that the Koopman-MPC possesses good robustness and constraint satisfaction capabilities when dealing with perturbations in key process variable parameters.

[0190] Figure 16 This demonstrates the actual response of the Kappa value and its tracking of the setpoint when parameter perturbation exists in Tc. +3K and Under both +5K perturbation conditions, the delignification reaction is accelerated by the increase in temperature. Under the +5K perturbation, the Kappa value decreases at a faster rate, resulting in its overall Kappa value curve being lower than that of the standard curve. +3K case. In the later stages of the process, the Kappa values ​​under both error cases converged to near the set value, and the steady-state error was controlled within an acceptable range, proving that the control system has a certain robustness to temperature measurement errors.

[0191] This invention focuses on the modeling and control of the intermittent pulping and cooking process. Addressing the process's strong nonlinearity, multivariate coupling, and complex dynamic characteristics, a high-dimensional linear modeling and predictive control strategy based on the Koopman operator is proposed. A system dynamics description is constructed using an extended Pudu model, and the state space is enhanced using the Gaussian function, establishing a 27-dimensional Koopman linear prediction model that effectively balances model complexity while maintaining fitting accuracy. Based on the established Koopman linear prediction model, a model predictive controller with input constraint handling capabilities is designed. Simulation results show that the controller can achieve rapid and accurate tracking of the pulp kappa value, with a smooth trajectory of the control input (free liquid temperature) and a stable system response transition, demonstrating good dynamic performance and closed-loop stability. To verify the reliability of the proposed strategy in practical applications, robustness tests were conducted on the system under multiple parameter perturbations, including step perturbations of the concentration of key components in the free liquid phase, the lignin composition in the solid phase, and the temperature of the solid-liquid composite phase. Under various parameter perturbation conditions, the controller exhibits good robustness and adaptability, effectively suppressing the effects of model uncertainty and measurement error, thus laying a theoretical foundation for its practical application in industrial environments with model mismatch and measurement noise. Future research will focus on smarter model building, more economical control frameworks, and broader algorithm integration to promote this strategy towards practical industrial applications.

[0192] Specifically, the principle of this invention is as follows: First, raw data such as solid lignin, effective alkali of retentate, and free liquid temperature are acquired in a field-level measurement system. After filtering, these data form a low-dimensional nonlinear state. A pre-trained Gaussian function family is used to map this state to a fixed-dimensional up-dimensional space, making the originally curved state trajectory approximately linear in the up-dimensional coordinate system, thus allowing a linear difference equation to describe the future dynamics. Based on this, the difference between the set Kabbe value and the actual measurement is fed into an error accumulator to form an integral state, which is then concatenated with the up-dimensional state to form an augmented vector. This allows the optimizer to "see" the accumulated deviation at each prediction step, and the elimination of this deviation is then written into the performance index. The prediction stage employs a rolling window strategy: using the current augmented state as the initial value, combined with valve upper and lower limit constraints, a quadratic programming problem is solved to make the future Kabbe value as close as possible to the set curve and the temperature change as gradual as possible. The resulting optimal temperature sequence is only output to the steam regulating valve in the first step. In the next sampling cycle, after measurement updates, the above process is repeated, forming a closed loop of "measurement-up-prediction-execution". Because the integral state is incorporated into the same linear model, any residual error in steady state will manifest as a continuous increase in the integral quantity. To reduce performance indicators, the optimizer will automatically adjust the temperature setting until the measured value coincides with the set value and the integral quantity stops increasing, thus achieving zero steady-state error control without additional parameter tuning. The combined effect of the dimensional mapping and the integral state makes the closed loop not only respond quickly and smoothly to step changes in the set value, but also naturally robust to slow drift and valve nonlinearity. At the same time, once the offline model is fixed, online operation only requires matrix operations, and the solution process can be encapsulated in a standard quadratic programming subroutine, which is fully compatible with existing DCS or PLC computing frameworks. This completes the entire technology chain from nonlinear intermittent cooking processes to high-dimensional linear prediction and then to engineering real-time control.

[0193] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A predictive control method for kappa value of pulp in a batch cooking process, characterized in that, include: S10: In the batch digester body and the field-level measurement system connected to it, collect multidimensional measurable state variables related to delignification in the batch digester as the original state vector, including but not limited to solid phase component concentration, liquid phase component concentration, free liquid phase temperature and solid-liquid composite phase temperature, to form the original state vector. S20: In the control station of edge computing or distributed control system, the extended dynamic mode decomposition method is used to map the original state vector to a high-dimensional linear space to obtain the corresponding up-dimensional state vector, thereby realizing the global linearization of nonlinear dynamics. Upgraded state refers to the new state vector constructed by mapping the low-dimensional and strongly nonlinear measurable state variables in the intermittent cooking process to a high-dimensional linear space through a set of predefined Gaussian functions. S30: In the same control station, a Koopman linear prediction model describing the process dynamics is established through offline training based on historical operating data; the Koopman linear prediction model uses the free liquid phase temperature as the control input and the pulp kappa value as the control output. S40: At each sampling time, the model prediction control program in the control station uses the Koopman linear prediction model to continuously optimize the free liquid phase temperature sequence in the future control time domain, so that the Kapoor value tracks to the set value and meets the upper and lower temperature limits given by the process. S50: After each optimization, the first control quantity of the temperature sequence is applied to the cooking process in real time, and the rolling optimization is repeated in the next sampling cycle in combination with feedback correction to form a closed-loop control, so that the Kapper value is finally tracked to the set value. Before rolling optimization, a Kabber value error integral term is introduced to eliminate steady-state tracking deviation; The process of introducing a Kabber value error integral term before rolling optimization includes the following steps: A Kapoor value error accumulator is added to the controller to accumulate the difference between the set Kapoor value and the measured Kapoor value in real time. The controller is a computing unit or software module that embeds the Koopman model predictive control algorithm. After each sampling is completed, the current difference is multiplied by a fixed ratio and accumulated into the accumulator to form an integral that grows over time. The integral is added as a new state variable and fed into the Koopman linear prediction model along with the original upgraded state vector, so that the optimizer considers historical error accumulation information when rolling the calculation of future temperature sequences. The optimization objective includes adding a penalty weight to the integral quantity, which forces the controller to actively offset the accumulated error in subsequent adjustments; Once the system reaches steady state, the integral remains constant, and the difference between the measured Kab value and the set value is gradually approached to zero, thus achieving zero steady-state error tracking. If a continuous disturbance causes the integral to be too large, the accumulator output can be clamped by limiting or anti-saturation logic to prevent the temperature command from exceeding the process allowable range.

2. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 1, characterized in that, The center point and bandwidth of the Gaussian function were determined through offline cross-validation to ensure that the relative error of the Koopman model in predicting the Kabbal value is no greater than 1.2%.

3. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 2, characterized in that, By randomly changing the initial conditions of the free liquid phase temperature within the range of 415K to 430K, multiple open-loop simulations were performed on the mechanism model to obtain historical operating data, covering all dynamic characteristics of the intermittent pulping and cooking process.

4. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 3, characterized in that, During rolling optimization, the prediction time domain is set to 15 sampling periods, the control time domain is set to 8 sampling periods, and the temperature constraint is set to a lower limit of 410K and an upper limit of 432K.

5. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 4, characterized in that, When the effective base or active hydride concentration experiences a +5% step disturbance, the controller still maintains the steady-state error of the kappa value within ±0.

5.

6. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 5, characterized in that, When the concentration of solid-phase highly reactive lignin, low reactive lignin, or their total concentration experiences a step disturbance of +5%, the controller adaptively adjusts the temperature to ensure that the kappa value error does not exceed 2%.

7. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 6, characterized in that, When the measured temperature of the solid-liquid composite phase is 3-5K higher than normal, the controller adjusts the free liquid phase temperature accordingly using feedback compensation to maintain the Kapper value tracking error within ±1K.

8. The method for predictive control of pulp kappa value in an intermittent cooking process according to claim 7, characterized in that, The Koopman linear prediction model is fixed at 27 dimensions. Once the model parameters are trained offline, they will not be modified during the entire cooking batch operation to reduce the amount of online computation.

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