Three-node thermal model for high temperature fff hot end temperature prediction control

Abstract

This invention relates to the field of temperature control technology for additive manufacturing equipment, providing a three-node thermal model predictive temperature control method for high-temperature FFF hot ends. This method constructs a three-node thermal network model that explicitly retains the fourth-order nonlinear term of high-temperature radiative heat dissipation. Model parameters are accurately calibrated through equivalent parameterization and a two-stage identification strategy. An extended Kalman filter is employed, using the temperature of the only measurable sensor as the observation, to estimate the unmeasurable internal states of the heating core and heating block online. Then, using the estimated complete state vector as initial conditions, the NMPC algorithm is used to perform rolling optimization to solve for the optimal heating power sequence that satisfies the constraints. The optimization solution combines the projected gradient descent method with the analytical gradient of the adjoint variables. This method achieves accurate, rapid, and stable control of the nozzle temperature, significantly improving the molding quality of high-temperature consumables.

Description

Technical Field

[0001] This invention belongs to the field of temperature control technology for additive manufacturing equipment, and particularly relates to a three-node thermal model predictive temperature control method for the hot end of a high-temperature FFF. Background Technology

[0002] Fused Filament Fabrication (FFF) technology is widely used in additive manufacturing due to its flexible structural design, high material utilization, and low equipment cost. With the increasing demand for high-performance thermoplastic components in aerospace, biomedical, and high-end equipment manufacturing industries, high-performance polymers such as polyetheretherketone (PEEK) and polyetherimide (PEI) are increasingly used in FFF processes. These materials typically require stable melting and precise extrusion at high temperatures of 350-450 °C, which places extremely high demands on the accuracy of hot-end temperature control and dynamic response performance.

[0003] Currently, the nozzle temperature regulation of existing FFF equipment still mainly relies on proportional-integral-derivative (PID) control. Although this method is simple in structure and mature in engineering implementation, it is difficult to simultaneously balance dynamic response speed and steady-state tracking accuracy when facing the inherent nonlinearity, large thermal inertia, and input amplitude constraints of high-temperature hot-end systems. This can easily lead to problems such as temperature overshoot and long-term oscillation.

[0004] In terms of thermal modeling at the hot end, existing nozzles mostly employ a two-node thermal model that includes a heating rod and a heating block. This type of model fails to characterize the internal heat transfer hysteresis effect between the heating core and the heating block, resulting in an inaccurate portrayal of the dynamic characteristics during the rapid heating phase at high temperatures. Furthermore, existing models generally neglect the heat capacity, heat conduction, and heat dissipation characteristics of the temperature sensor itself, leading to a non-negligible deviation between the sensor-acquired temperature and the actual temperature of the heating block. This limits the model's prediction accuracy and consequently affects the control performance.

[0005] In terms of control algorithms, Model Predictive Control (MPC) is based on predicting future states using models and solving finite-time optimization problems online, which has significant advantages in handling constrained thermal process control. However, existing research mostly uses linear MPC or highly simplified predictive models. For high-temperature FFF hot ends, a systematic technical solution that simultaneously considers low-order identifiable thermal models, online estimation of internal states, and embedded real-time nonlinear MPC implementation is still insufficient. Summary of the Invention

[0006] The purpose of this invention is to provide a three-node thermal model predictive temperature control method for high-temperature FFF hot ends, aiming to solve the problems mentioned in the background art.

[0007] The present invention is implemented as follows: a three-node thermal model predictive temperature control method for high-temperature FFF hot ends includes the following steps:

[0008] Step 1: System parameter calibration. Measure the heat capacity, thermal conductivity, and heat dissipation coefficient of the heating core, heating block, and temperature sensor. Establish a three-node thermal network temperature dissipation model that includes the heating core, heating block, and temperature sensor. The model explicitly retains the fourth-order nonlinear term of high-temperature radiative heat dissipation.

[0009] Step 2: Set the target control temperature of the nozzle according to the consumable type, and complete the configuration of the constraint parameters, prediction time domain, and control time domain of the embedded MPC controller;

[0010] Step 3: The embedded main controller collects the measured temperature data from the nozzle temperature sensor in real time according to a fixed sampling period;

[0011] Step 4: Input the measured temperature data into the three-node thermal network temperature dissipation model, and perform online state estimation through extended Kalman filtering to obtain the complete state vectors of heating core temperature, heating block temperature and sensor temperature;

[0012] Step 5: Based on the estimated complete state vector, the MPC controller predicts the subsequent temperature change trend in the prediction time domain. With the optimization objectives of minimizing temperature deviation, minimizing terminal error, and minimizing power fluctuation, the optimal heating control quantity is solved by combining the projected gradient descent method with the analytical gradient of the adjoint variable.

[0013] Step 6: Adjust the heating rod power in real time according to the solution results to achieve closed-loop control of nozzle temperature;

[0014] Step 7: Execute the data acquisition, state estimation, prediction optimization and power adjustment process in a loop to achieve stable control of nozzle temperature throughout the process.

[0015] A further technical solution is that the three-node thermal network temperature dissipation model is constructed using the lumped parameter method, and includes three physical nodes: a heating core, a heating block, and a temperature sensor. Its heat transfer equations are as follows:

[0016]

[0017]

[0018]

[0019] In the formula: , and These are the heat capacities of the heating core, heating block, and temperature sensor, respectively. , and These are the temperatures of the heating core, heating block, and temperature sensor, respectively. Input thermal power to the heating rod; The equivalent heat transfer coefficient from the heating core to the heating block; The equivalent heat transfer coefficient from the heating block to the temperature sensor; The total heat dissipation power of the heating block includes natural convection, structural heat conduction, and thermal radiation. This refers to the heat absorption power of consumables.

[0020] A further technical solution is that the total heat dissipation of the heating block consists of natural convection, structural heat conduction, and thermal radiation:

[0021]

[0022] In the formula, The ambient temperature; Temperature of the larynx; and These are the equivalent heat transfer coefficients for natural convection and structural conduction, respectively. Let be the radiation coefficient, and , For surface emissivity, For effective radiation area, It is the Stefan-Boltzmann constant;

[0023] The extrusion heat consumption term is expressed as:

[0024]

[0025] In the formula, For the extrusion speed of consumables, The heat capacity of the consumable cross section, This is the initial temperature of the consumable.

[0026] A further technical solution involves reparameterizing the model in step 1, defining equivalent dynamic parameters:

[0027]

[0028] in, For equivalent dynamic parameters, ;

[0029] The system state equations are rewritten after the replacement as follows:

[0030]

[0031]

[0032] .

[0033] A further technical solution involves using a weighted mean square error objective function for system identification and introducing... Regularization terms prevent parameter overfitting:

[0034]

[0035] In the formula, To optimize the objective function, Indicates different experimental stages, For the corresponding stage weights, This represents the number of sampling points in this phase. and These are the measured and model-predicted values ​​of the sensor temperature, respectively. For regularization weights; parameter optimization first uses the differential evolution algorithm for global search, and then uses the L-BFGS-B algorithm for local fine optimization.

[0036] In a further technical solution, in step 5, MPC solves the following finite-time optimization problem at each sampling time:

[0037]

[0038]

[0039] In the formula, The predicted sensor temperature at time k. For the first Predicting sensor temperature at any given time. For reference target temperature, , and These are the tracking error weight, the terminal error weight, and the control rate of change weight, respectively. To predict the length of the time domain, To control the length of the time domain, To control the increment, and , This is the input at time k.

[0040] The three-node thermal model predictive temperature control method for high-temperature FFF hot ends provided in this invention has the following beneficial effects:

[0041] (1) High-precision thermal modeling: The three-node thermal model introduces heating core nodes and sensor heat capacity on the basis of the traditional two-node model to characterize internal heat transfer hysteresis and sensor dynamics, and retains the fourth-order nonlinear term of high-temperature radiation, which significantly improves the model accuracy.

[0042] (2) Accurate state estimation: Extended Kalman filter realizes online estimation of the unmeasurable temperature of heating core and heating block, providing complete and accurate state information for model predictive control, enabling the controller to predict the internal temperature change trend.

[0043] (3) Excellent control performance: Nonlinear model predictive control achieves advance adjustment and overcomes the lag of PID control.

[0044] (4) Embedded real-time implementation: Through the calculation of the adjoint variable gradient and the projection gradient descent method, as well as the lightweight processing of constraints and matrices, the algorithm can run in real time on the embedded main control chip with a single calculation cycle of 20ms and a control cycle of 0.3s, which has high practical value.

[0045] (5) Improved molding quality: Stable nozzle temperature ensures the uniformity of high-temperature consumable melting, reduces defects such as stringing, material shortage, and delamination deformation, and significantly improves the molding accuracy and mechanical properties of printed parts. Attached Figure Description

[0046] Figure 1 This is a schematic diagram of a high-temperature water-cooled hot end structure and heat transfer path.

[0047] Figure 2 This is an overall framework diagram of a three-node thermal model predictive temperature control method for high-temperature FFF hot ends provided in an embodiment of the present invention.

[0048] Figure 3 The results of the experimental platform and system identification;

[0049] Figure 4 A comparison of the temperature response of the three controllers under different target temperatures;

[0050] Figure 5 A comparison of the tracking errors of the three controllers at different target temperatures;

[0051] Figure 6 A comparison of the comprehensive performance indicators of the three controllers at different target temperatures;

[0052] Figure 7 A comparison of the PWM output of three controllers. Detailed Implementation

[0053] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0054] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0055] like Figure 1 and Figure 2 As shown, a three-node thermal model predictive temperature control method for the hot end of a high-temperature FFF (Free-Flying FFF) is provided in one embodiment of the present invention, comprising the following steps:

[0056] Step 1: System parameter calibration. Measure the heat capacity, thermal conductivity, and heat dissipation coefficient of the heating core, heating block, and temperature sensor. Establish a three-node thermal network temperature dissipation model that includes the heating core, heating block, and temperature sensor. The model explicitly retains the fourth-order nonlinear term of high-temperature radiative heat dissipation.

[0057] Step 2: Set the target control temperature of the nozzle according to the consumable type, and complete the configuration of the constraint parameters, prediction time domain, and control time domain of the embedded MPC controller;

[0058] Step 3: The embedded main controller collects the measured temperature data from the nozzle temperature sensor in real time according to a fixed sampling period;

[0059] Step 4: Input the measured temperature data into the three-node thermal network temperature dissipation model, and perform online state estimation through extended Kalman filtering to obtain the complete state vectors of heating core temperature, heating block temperature and sensor temperature;

[0060] Step 5: Based on the estimated complete state vector, the MPC controller predicts the subsequent temperature change trend in the prediction time domain. With the optimization objectives of minimizing temperature deviation, minimizing terminal error, and minimizing power fluctuation, the optimal heating control quantity is solved by combining the projected gradient descent method with the analytical gradient of the adjoint variable.

[0061] Step 6: Adjust the heating rod power in real time according to the solution results to achieve closed-loop control of nozzle temperature;

[0062] Step 7: Execute the data acquisition, state estimation, prediction optimization and power adjustment process in a loop to achieve stable control of nozzle temperature throughout the process.

[0063] In a preferred embodiment of the present invention, the three-node thermal network temperature dissipation model is constructed using the lumped parameter method and includes three physical nodes: a heating core, a heating block, and a temperature sensor. Its heat transfer equations are as follows:

[0064]

[0065]

[0066]

[0067] In the formula: , and These are the heat capacities of the heating core, heating block, and temperature sensor, respectively, in J / K. , and These are the temperatures of the heating core, heating block, and temperature sensor, respectively, in °C. The heating rod is supplied with thermal power, measured in W. The equivalent heat transfer coefficient from the heating core to the heating block is expressed in W / K. The equivalent heat transfer coefficient from the heating block to the temperature sensor is expressed in W / K. The total heat dissipation power of the heating block includes natural convection, structural heat conduction, and thermal radiation, and is expressed in W. This represents the heat absorption power of the consumables, expressed in W.

[0068] The total heat dissipation of the heating block consists of natural convection, structural heat conduction, and thermal radiation.

[0069]

[0070] In the formula, The ambient temperature; Temperature of the larynx; and These are the equivalent heat transfer coefficients for natural convection and structural conduction, respectively. Let be the radiation coefficient, and , For surface emissivity, For effective radiation area, It is the Stefan-Boltzmann constant, and W / (m 2 ·K 4 ).

[0071] The extrusion heat consumption term is expressed as:

[0072]

[0073] In the formula, For the extrusion speed of consumables, The heat capacity of the consumable cross section, This is the initial temperature of the consumable.

[0074] To address the challenge of directly identifying the original physical parameters, which appear in coupled form in the state equations, the model undergoes reparameterization, defining equivalent dynamic parameters:

[0075]

[0076] in, For equivalent dynamic parameters, .

[0077] The system state equations are rewritten after the replacement as follows:

[0078]

[0079]

[0080]

[0081] Through the aforementioned reparameterization process, the model retains the interpretability of the physical structure while reducing the coupling of parameters in the state equations, thus improving the system's identifiability. System identification employs a weighted mean square error objective function and introduces... Regularization terms prevent parameter overfitting:

[0082]

[0083] In the formula, To optimize the objective function, Indicates different experimental stages, For the corresponding stage weights, This represents the number of sampling points in this phase. and These are the measured and model-predicted values ​​of the sensor temperature, respectively. The weights are used for regularization. Parameter optimization first employs a differential evolution algorithm for global search, followed by L-BFGS-B algorithm for local fine-tuning.

[0084] A lightweight nonlinear model predictive control (NMPC) logic is employed: using the current temperature state estimated by extended Kalman filter as the initial state, the nozzle temperature change trajectory is predicted over multiple control cycles within a preset prediction time domain through a three-node heat dissipation model. The optimal control sequence is solved under upper and lower limits of heating power constraints, with the optimization objectives being to minimize temperature tracking error, terminal error, and heating power fluctuation. Only the optimal control quantity for the current cycle is output and applied to the heating rod. The next cycle involves resampling and continuous optimization, achieving proactive prediction and dynamic correction to avoid temperature overshoot and lag. Specifically, at each sampling time, MPC solves the following finite-time optimization problem:

[0085]

[0086]

[0087] In the formula, The predicted sensor temperature at time k. For the first Predicting sensor temperature at any given time. For reference target temperature, , and These are the tracking error weight, the terminal error weight, and the control rate of change weight, respectively. To predict the length of the time domain, To control the length of the time domain, To control the increment, and , This is the input at time k. The optimization problem is solved online using the projected gradient descent method, and the gradient is calculated analytically using the adjoint variable method. The gradient calculation for the entire control sequence can be completed with only one forward state prediction and one backward recursion, reducing the computational load and meeting the real-time operation requirements of embedded hardware.

[0088] As a preferred embodiment of the present invention, the MPC algorithm, after constraint simplification and matrix dimensionality reduction, can run independently in embedded microcontrollers and ARM main control chips, with a single control calculation cycle of approximately 20 ms, meeting the millisecond-level real-time response requirements. During the control process, the ambient temperature parameter of the closed-loop cavity is used as an external disturbance variable in the model calculation to counteract the interference of hot air from the cavity on the printhead temperature. For printing start / stop, filament retraction, and filament switching conditions, the controller automatically updates the model boundary conditions and dynamically matches the temperature control logic under different conditions.

[0089] After the device is powered on, the embedded main control unit first completes the parameter calibration. First, a three-node thermal network model is established: based on the traditional two-node model of heating rod and heating block, a heating core physical node is added to explicitly characterize the internal heat transfer hysteresis effect from the heating core to the heating block, while retaining the fourth-order nonlinear term of high-temperature radiative heat dissipation. Through an equivalent parameterization method, the original coupled physical parameters are transformed into a combination of nine identifiable dynamic parameters.

[0090] To accurately calibrate the model parameters, four types of identification experiments were designed, including open-loop step excitation, pseudo-random binary sequence excitation, steady-state step excitation, and natural cooling. A weighted mean square error objective function was constructed and introduced. For the regularization term, a differential evolution algorithm is used for global search and an L-BFGS-B algorithm is used for local fine-tuning optimization to complete parameter identification.

[0091] Based on the target temperature set by the printing consumables, the system completes the initialization of MPC control parameters and sets the prediction time domain. (Approximately 9 seconds), control time domain (Approximately 4.5 s) and heating power constraints. During the printing process, the embedded chip periodically reads the signal collected by the printhead temperature sensor at a fixed cycle of 0.3 s (consistent with Klipper's default temperature control cycle).

[0092] Within each control cycle, the acquired sensor temperatures are first input into an extended Kalman filter. A one-step state prediction is then performed based on a discrete three-node thermal model, followed by correction using the current measured values ​​to obtain complete state estimates for the heating core temperature, heating block temperature, and sensor temperature. This process involves setting the process noise covariance matrix. Observation noise covariance matrix This achieves a recursive fusion of model predictions and sensor observations, whereby... It is an identity matrix.

[0093] The MPC controller estimates the complete temperature state (sensor temperature) using EKF (Extended Kalman Filter). heating block temperature heating rod temperature Using the initial state as an example, the subsequent temperature change trend is continuously predicted within the prediction time domain. The optimization objectives are to minimize temperature tracking deviation, terminal error, and power fluctuation, thus solving for the current optimal heating output power. The optimization problem is solved online using the projected gradient descent method. The gradient is calculated analytically using the adjoint variable method, and the step size is determined through backtracking search (initial step size). Attenuation factor Armijo conditional parameters The solution result is output to the heating rod drive circuit in real time.

[0094] When faced with conditions such as hot air interference in the cavity circulation, filament retraction, and printing start-stop, the model simultaneously incorporates external disturbances and operating condition switching conditions, dynamically adjusts and optimizes the target and constraint boundaries, and independently completes real-time calculation and closed-loop control on the embedded end to stabilize the nozzle temperature within the set target range, ensuring uniform filament extrusion and stable molding quality.

[0095] Experimental results show that, at three target temperatures of 400 °C, 425 °C, and 450 °C, the coefficient of determination of the identified three-node model on the independent validation dataset reaches 0.9997, and the root mean square error is 1.81 °C. Compared with traditional PID control, the proposed control method reduces the average ITAE by about 8.8%, the average overshoot by about 74.8%, and the average rise time by about 3.4%, while keeping the steady-state error within 1%. The calculation time for a single online optimization is about 20 ms.

[0096] like Figure 3As shown, on 12 independent validation datasets covering four types of experiments—open-loop step, pseudo-random binary sequence, steady-state step, and natural cooling—the identified three-node model achieved an average determination coefficient of 0.9997 and an average root mean square error (RMSE) of 1.81 °C, a reduction of approximately 80.93% compared to the traditional two-node model's RMSE (9.60 °C). The model's predicted values ​​closely match the measured values. The equivalent parameterization identification method transforms the coupled physical parameters into a combination of nine identifiable dynamic parameters. Combined with differential evolution and the L-BFGS-B two-stage optimization strategy, the accuracy and stability of parameter identification are improved. Extended Kalman filtering enables online estimation of the two unmeasurable internal states of the heating core and heating block, allowing the controller to distinguish the dynamic relationships between the heating core, heating block, and sensor nodes. This allows for input adjustment before approaching the setpoint, and predictive control is no longer limited to single-sensor temperature feedback.

[0097] like Figure 4 As shown, at three target temperatures of 400 °C, 425 °C, and 450 °C, the proposed nonlinear model predictive control (NMPC) exhibits a smoother temperature response curve compared to traditional PID control. The temperature rises steadily to near the target value and then converges rapidly without significant overshoot or oscillation. Figure 6 As shown, the overshoot of NMPC at the three target temperatures was 0.36%, 0.61%, and 0.19%, respectively, all significantly lower than that of PID (1.49%, 1.98%, and 1.18%), with an average overshoot reduction of approximately 74.8%. The ITAE indices were 464,219, 585,152, and 664,865, respectively, representing reductions of approximately 8.8%, 7.5%, and 10.0% compared to PID, with an average reduction of approximately 8.8%. The average rise time was shortened by approximately 3.4% compared to PID. Figure 5 As shown, the temperature tracking error of NMPC remains lower than that of PID during the heating process, the error convergence speed is faster, and the steady-state error is kept within 1%.

[0098] like Figure 7 As shown, the NMPC's PWM output exhibits characteristics of proactively increasing power in the early stage, a smooth transition in the middle stage, and fine adjustment in the later stage, demonstrating the MPC's proactive adjustment capability based on model prediction. In contrast, the PID's PWM output exhibits a passive oscillating adjustment mode with significant power fluctuations. These results indicate that the NMPC effectively avoids the lag problem of PID control by predicting future temperature change trends and optimizing the control output in advance.

[0099] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A three-node thermal model predictive temperature control method for the hot end of a high-temperature FFF (Fryer-Filler) circuit, characterized in that, Includes the following steps: Step 1: System parameter calibration. Measure the heat capacity, thermal conductivity, and heat dissipation coefficient of the heating core, heating block, and temperature sensor. Establish a three-node thermal network temperature dissipation model that includes the heating core, heating block, and temperature sensor. The model explicitly retains the fourth-order nonlinear term of high-temperature radiative heat dissipation. Step 2: Set the target control temperature of the nozzle according to the consumable type, and complete the configuration of the constraint parameters, prediction time domain, and control time domain of the embedded MPC controller; Step 3: The embedded main controller collects the measured temperature data from the nozzle temperature sensor in real time according to a fixed sampling period; Step 4: Input the measured temperature data into the three-node thermal network temperature dissipation model, and perform online state estimation through extended Kalman filtering to obtain the complete state vectors of heating core temperature, heating block temperature and sensor temperature; Step 5: Based on the estimated complete state vector, the MPC controller predicts the subsequent temperature change trend in the prediction time domain. With the optimization objectives of minimizing temperature deviation, minimizing terminal error, and minimizing power fluctuation, the optimal heating control quantity is solved by combining the projected gradient descent method with the analytical gradient of the adjoint variable. Step 6: Adjust the heating rod power in real time according to the solution results to achieve closed-loop control of nozzle temperature; Step 7: Execute the data acquisition, state estimation, prediction optimization and power adjustment process in a loop to achieve stable control of nozzle temperature throughout the process.

2. The three-node thermal model predictive temperature control method for high-temperature FFF hot ends according to claim 1, characterized in that, The three-node thermal network temperature dissipation model is constructed using the lumped parameter method and includes three physical nodes: a heating core, a heating block, and a temperature sensor. Its heat transfer equations are as follows: ； ； ； In the formula: , and These are the heat capacities of the heating core, heating block, and temperature sensor, respectively. , and These are the temperatures of the heating core, heating block, and temperature sensor, respectively. Input thermal power to the heating rod; The equivalent heat transfer coefficient from the heating core to the heating block; The equivalent heat transfer coefficient from the heating block to the temperature sensor; The total heat dissipation power of the heating block includes natural convection, structural heat conduction, and thermal radiation. This refers to the heat absorption power of consumables.

3. The three-node thermal model predictive temperature control method for high-temperature FFF hot ends according to claim 2, characterized in that, The total heat dissipation of the heating block consists of natural convection, structural heat conduction, and thermal radiation. ； In the formula, The ambient temperature; Temperature of the larynx; and These are the equivalent heat transfer coefficients for natural convection and structural conduction, respectively. Let be the radiation coefficient, and , For surface emissivity, For effective radiation area, It is the Stefan-Boltzmann constant; The extrusion heat consumption term is expressed as: ； In the formula, For the extrusion speed of consumables, The heat capacity of the consumable cross section, This is the initial temperature of the consumable.

4. The three-node thermal model predictive temperature control method for high-temperature FFF hot ends according to claim 3, characterized in that, In step 1, the model is reparameterized, and equivalent dynamic parameters are defined: ； in, For equivalent dynamic parameters, ; The system state equations are rewritten after the replacement as follows: ； ； 。 5. The three-node thermal model predictive temperature control method for high-temperature FFF hot ends according to claim 4, characterized in that, The system identification uses a weighted mean square error objective function and introduces... Regularization terms prevent parameter overfitting: ； In the formula, To optimize the objective function, Indicates different experimental stages, For the corresponding stage weights, This represents the number of sampling points in this phase. and These are the measured and model-predicted values ​​of the sensor temperature, respectively. For regularization weights; parameter optimization first uses the differential evolution algorithm for global search, and then uses the L-BFGS-B algorithm for local fine optimization.

6. The three-node thermal model predictive temperature control method for high-temperature FFF hot ends according to claim 5, characterized in that, In step 5, MPC solves the following finite-time optimization problem at each sampling time: ； ； In the formula, The predicted sensor temperature at time k. For the first Predicting sensor temperature at any given time. For reference target temperature, , and These are the tracking error weight, the terminal error weight, and the control rate of change weight, respectively. To predict the length of the time domain, To control the length of the time domain, To control the increment, and , This is the input at time k.

Images