Underactuated electrothermal system based on available energy temperature tracking control method

Abstract

This invention relates to the interdisciplinary fields of electrothermal engineering, thermal system modeling, and intelligent control, and provides a temperature tracking control method for underactuated electrothermal systems based on available energy. This method establishes a dynamic model including the temperature states of the quartz lamp and the test specimen, as well as their coupled heat transfer relationship. It constructs a Lyapunov function of available energy based on the temperature deviation of the test specimen, determines the system's natural dissipation terms and the energy injection channel by analyzing the function's derivative, and then constructs a virtual energy control target and error adjustment process. The dynamic constraint relationship of the energy injection channel is derived and mapped to the actual control input of each quartz lamp, adjusting the energy transfer between the quartz lamp and the test specimen to ensure the test specimen temperature stably tracks the reference value. This invention, without directly controlling the test specimen temperature, constructs a unified control framework based on the system's energy transfer and dissipation mechanisms, effectively improving the stability and robustness of the control system.

Description

Technical Field

[0001] This invention relates to the interdisciplinary fields of electrothermal engineering, thermal system modeling and intelligent control, and in particular to a temperature tracking control method for an underdriven electrothermal system based on available energy. Background Technology

[0002] In high-end industrial and scientific research fields such as thermal testing of aerospace materials, performance testing of high-temperature structural components, and simulation of complex thermal environments, extremely stringent requirements are placed on the temperature control accuracy, dynamic response speed, and overall energy utilization efficiency of the heated object (usually referred to as the test piece). Multi-quartz lamp electrothermal systems, with their high power density, rapid heating, and flexible array arrangement, have become key equipment for realizing the aforementioned radiative heating and thermal environment simulation processes. In this type of system, the quartz lamp array serves as the primary heat source, transferring energy to one or more test pieces through various heat exchange methods such as radiation, convection, and conduction. The temperature state of the test piece is not directly controlled by independent actuators, but rather indirectly determined by the temperature distribution of the quartz lamps and the complex heat exchange process between them and the test piece. This structural characteristic leads to a typical underactuated property in the system, meaning that the number of control inputs (such as the driving power of each quartz lamp) is less than the number of outputs to be precisely controlled (i.e., the temperature of each test piece), or the control inputs cannot directly act on the controlled outputs.

[0003] Furthermore, the energy transfer relationships within the system are highly complex and nonlinearly coupled. Specifically, there is strong, temperature-dependent radiative heat transfer between heat sources (quartz lamps), between heat sources and the heated object (test specimen), and between the heated objects themselves, all superimposed with convection and conduction effects, forming a tightly coupled thermodynamic network of multiple physics fields. This combination of strong nonlinearity and underdriven characteristics poses a fundamental challenge to traditional temperature control strategies.

[0004] Currently, most temperature control methods applied to such electrothermal systems are based on empirical models or significantly simplified linear models, commonly employing algorithms such as proportional-integral-derivative (PID) control, fuzzy logic control, or model predictive control (MPC), adjusting the quartz lamp power to track the set temperature curve. However, these methods have several inherent limitations: First, their control law design typically uses temperature error as feedback directly, failing to explicitly integrate and utilize the system's inherent energy conservation and dissipation mechanisms, resulting in a disconnect between the controller design process and the system's physical nature. Second, when system parameters are perturbed, external operating conditions interfere, or multiple test pieces need to be heated collaboratively, the performance and stability of controllers based on simplified models are insufficient. Crucially, some existing methods, to handle underactuated structures, tend to introduce additional virtual control quantities or perform mathematical equivalent transformations at the theoretical level, attempting to convert the system into a formally fully driven system before designing the controller. While this approach facilitates theoretical analysis, in engineering practice it often leads to complex control algorithm structures, a lack of physical basis for energy distribution strategies among heat sources, and difficulty in conducting a unified and rigorous demonstration of the global stability of the closed-loop system from the fundamental perspective of energy generation, transfer, and dissipation. Summary of the Invention

[0005] Therefore, it is necessary to address the shortcomings of existing solutions in terms of control performance, stability, and energy utilization rationality by providing a temperature tracking control method for an underdriven electrothermal system based on available energy, which ensures that the temperature of the test specimen can efficiently, stably, and accurately track the target value by rationally adjusting the energy output of the heat source without directly controlling the temperature of the test specimen.

[0006] This invention provides a temperature tracking control method for an underdriven electrothermal system based on available energy, applied to an electrothermal system comprising multiple quartz lamps and multiple test specimens. The control input acts directly only on the quartz lamps, and the temperature of the test specimens is indirectly regulated through heat transfer between the quartz lamps and the test specimens. The method includes: A dynamic model of the electrothermal system is established, which simultaneously includes the temperature state of each quartz lamp and the temperature state of each test piece, and characterizes the coupled heat transfer relationships formed between quartz lamps, between quartz lamps and test pieces, and between test pieces through radiation, convection and heat conduction. Based on the temperature dynamic model of the test specimen and the preset temperature reference value of the test specimen, a usable energy Lyapunov function based on the temperature deviation of the test specimen is constructed. The available energy Lyapunov function is differentiated over time, and the temperature dynamic model of the test piece is substituted into it. The terms formed by energy exchange between the test piece and the test piece, between the test piece and the environment, and between the test piece and the base are identified as the natural dissipation terms of the system. At the same time, the terms formed by energy exchange between the quartz lamp and the test piece are extracted as the aggregated energy injection channel related to the temperature state of the quartz lamp. Based on the polymer energy injection channel and the temperature deviation of the test specimen, a virtual energy control target is constructed, such that under the virtual energy control target, the time derivative of the available energy Lyapunov function is negatively definite or semi-negatively definite with respect to the temperature deviation of the test specimen. Based on the deviation between the virtual energy control target and the aggregated energy injection channel, an error adjustment process is constructed, and the dynamic constraint relationship of the aggregated energy injection channel is derived. The dynamic constraint relationship of the energy injection channel is mapped to the actual control input of each quartz lamp. By solving and allocating the control input of each quartz lamp, the energy transfer between the quartz lamp and the test piece is adjusted so that the temperature of the test piece stably tracks its reference value.

[0007] In one embodiment, establishing the dynamic model of the electrothermal system specifically includes: Establish a dynamic temperature model for the quartz lamp, for the first The temperature dynamic equation for a quartz lamp is: in, , For the number of quartz lamps, , , The first The temperature of the quartz lamp, the derivative of temperature with respect to time, and the control input; The equivalent heat capacity of a single quartz lamp. The efficiency coefficient for converting electrical energy into heat energy. The emissivity of the quartz lamp surface. The Stefan-Boltzmann constant, The effective heat exchange area of ​​the quartz lamp. The radiation viewing angle factor between the quartz lamp and the test specimen. The convective heat transfer coefficient between the quartz lamp and the test specimen is given. The equivalent thermal resistance between the quartz lamp and the test piece. The radiation angle factor between quartz lamps. The convective heat transfer coefficient between quartz lamps. The equivalent thermal resistance between quartz lamps. The radiation angle factor between the quartz lamp and the environment. The convective heat transfer coefficient between the quartz lamp and the environment. For ambient temperature, The temperature of the quartz lamp base. The equivalent thermal resistance between the quartz lamp and the base. For the number of test pieces, For the first Temperature of each test piece; Establish a dynamic temperature model for the test specimen, for the first... The temperature dynamic equation for each test specimen is as follows: in, , Number of test specimens; For the first The derivative of temperature with respect to time for each test specimen; For the equivalent heat capacity of a single test specimen, The emissivity of the test specimen surface. The effective heat transfer area of ​​the test specimen. The radiation viewing angle factor between test specimens. The convective heat transfer coefficient between the test specimens is... The equivalent thermal resistance between the test pieces, The radiation perspective factor between the test specimen and the environment. The convective heat transfer coefficient between the test specimen and the environment. The temperature of the test specimen base. The equivalent thermal resistance between the test piece and the base.

[0008] In one embodiment, the construction based on the available energy Lyapunov function of the specimen temperature deviation is specifically as follows: in, This indicates that the available Lyapunov function is... Indicates the first The preset temperature reference value corresponding to each test piece.

[0009] In one embodiment, the time derivative of the available energy Lyapunov function is performed, and the temperature dynamic model of the test piece is substituted into it. The term formed by energy exchange between test pieces, between the test piece and the environment, and between the test piece and the base is identified as the system's natural dissipation term. Simultaneously, the term formed by energy exchange between the quartz lamp and the test piece is extracted as a polymeric energy injection channel related to the temperature state of the quartz lamp. Specifically, this includes: The time derivative of the available energy Lyapunov function Represented as: in, Characterizing the energy exchange between the quartz lamp and the test specimen. , , These characterize the energy exchange between test specimens, between test specimens and the environment, and between test specimens and the base, respectively. The sum of the last three terms is determined as the system's natural dissipation term: in, , , ; Simultaneously, terms related to the temperature state of the quartz lamp were extracted as a polymer energy injection channel. and its weight function : .

[0010] In one embodiment, constructing a virtual energy control target based on the polymer energy injection channel and the temperature deviation of the test piece specifically involves: The temperature deviation of the test piece is characterized as a weighting function. ; when At that time, the virtual energy control target is constructed as follows: ,when season ; in, Positive control gain, This is the residual energy term introduced by the inconsistency between the test specimen temperature reference value and the ambient or base temperature.

[0011] In one embodiment, the step of constructing an error adjustment process based on the deviation between the virtual energy control target and the aggregated energy injection channel, and deriving the dynamic constraint relationship on the aggregated energy injection channel, specifically includes: Introducing auxiliary state variables The following second-order error adjustment process is constructed: , in, For virtual energy control targets, To inject energy into the channel, deviation , , Positive control gain; Depend on and The dynamic constraint relationship of the polymerization energy injection channel is derived as follows: in, This represents the time derivative of the energy injection channel. This represents the time derivative of the virtual energy control target.

[0012] In one embodiment, the dynamic constraint relationship of the energy injection channel is mapped to the actual control input of each quartz lamp, specifically as follows: according to The definition and dynamic model of quartz lamp temperature transform the dynamic constraint relationship into a relationship with respect to the control input. The equation is: in, This is the known energy change term caused by the system's inherent heat exchange relationship.

[0013] In one embodiment, the energy transfer between the quartz lamp and the test piece is adjusted by solving and allocating the control inputs of each quartz lamp, specifically including: Define the aggregate quantity of the quartz lamp control input. and ; The following formula can be obtained by solving it. and : Based on the current temperature status of the quartz lamp Distribute the control inputs to each quartz lamp This ensures that the polymerization quantity relationship is satisfied.

[0014] In one embodiment, the method stabilizes the closed-loop system by selecting a control gain, specifically satisfying the following: and This ensures the temperature of the test piece. asymptotically converges to its reference value .

[0015] The aforementioned underdriven electrothermal system, based on a temperature tracking control method using available energy, establishes a dynamic model that simultaneously incorporates the temperature states of each quartz lamp and the test specimen, characterizing the multi-physics coupled heat transfer relationship between them. This provides a precise physical basis for analyzing the system from an energy perspective. Furthermore, by constructing a Lyapunov function of available energy based on the test specimen's temperature deviation, the temperature tracking problem is transformed into a problem of characterizing the energy contained when the test specimen deviates from its reference state. After differentiating this function and substituting it into the model, the natural dissipation term formed by energy exchange between the test specimens and with the environment and base, as well as the aggregated energy injection channel formed by energy exchange between the quartz lamps and the test specimens, are explicitly separated, thus unifying and integrating the energy transfer process in the control design. The system's inherent energy dissipation characteristics are analyzed, and the complex underactuated control problem involving multiple heat sources and multiple test specimens is abstracted into the regulation problem of a single aggregated energy channel. By constructing a virtual energy control target based on this channel and temperature deviation, the derivative of the Lyapunov function is ensured to be negatively definite with respect to temperature deviation, thus strictly guaranteeing the stability of the closed-loop system in an energy sense. Finally, by constructing an error adjustment process, the dynamic constraints on this aggregated channel are derived and mapped to the actual control inputs of each quartz lamp. Therefore, without directly controlling the temperature of the test specimen, the energy transfer between the test specimen and the heat source is indirectly and precisely regulated by coordinating the energy output of each quartz lamp, achieving stable tracking of the test specimen temperature to the target value. This method fundamentally overcomes the shortcomings of existing schemes, which, due to model simplification and the lack of explicit utilization of energy mechanisms, struggle to coordinate the underactuated structure and nonlinear coupling. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 This is an overall flowchart of the temperature tracking control method for an underdriven electric heating system based on available energy, provided in an embodiment of the present invention. Figure 2 The simulation result diagram of the target temperature of test piece 1 provided in the embodiment of the present invention; Figure 3 The simulation result diagram of the target temperature of test piece 2 provided in the embodiment of the present invention; Figure 4 The simulation result diagram of the target temperature tracking test piece 1 provided by the comparison method in the embodiment of the present invention is shown. Figure 5 The simulation results of the comparison method for tracking the target temperature of test piece 2 provided in this embodiment of the invention are shown in the figure. Figure 6This is an internal structural diagram of an electronic device according to an embodiment of the present invention. Detailed Implementation

[0018] 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. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] The following is combined Figures 1-6 The present invention describes a temperature tracking control method for an underdriven electric heating system based on available energy.

[0020] like Figure 1 As shown, in one embodiment, an underdriven electrothermal system is provided with a temperature tracking control method based on available energy. This method is applied to an electrothermal system comprising multiple quartz lamps and multiple test pieces, wherein the control input acts directly only on the quartz lamps, and the temperature of the test pieces is indirectly regulated through heat transfer between the quartz lamps and the test pieces.

[0021] Specifically, the electrothermal system addressed in this embodiment exhibits significant underactuated characteristics. In this system, the quartz lamp serves as the heat source, and its power or voltage can be directly adjusted by the controller, constituting the system's control input. However, the object being heated, i.e., the test piece, does not possess an independent heating or cooling actuator; its temperature change depends entirely on the heat flow transferred from the quartz lamp through radiation, convection, and conduction. This physical separation between the control input and the controlled object results in fewer control inputs than the number of outputs to be precisely controlled, or the control inputs cannot directly act on the controlled outputs, forming a typical underactuated structure. Furthermore, the energy transfer relationships within the system are highly complex and nonlinearly coupled. Specifically, there is strong, temperature-dependent radiative heat transfer between heat sources (quartz lamps), between heat sources and the heated object (test piece), and between the heated objects themselves, all superimposed with convection and conduction effects, forming a tightly coupled multi-physics thermodynamic network. This structure makes it impossible to control the temperature of the test specimen directly through simple single-input single-output feedback. Instead, the temperature of the test specimen must be indirectly affected by adjusting the state of the quartz lamp through a complex coupled thermal network, which greatly increases the difficulty of control design.

[0022] To address the temperature tracking problem in underactuated systems, this embodiment employs a control strategy based on available energy, transforming the traditional temperature error tracking problem into a regulation problem of energy injection and dissipation. The method specifically includes the following steps: Step S110: Establish a dynamic model of the electrothermal system.

[0023] The dynamic model simultaneously includes the temperature state of each quartz lamp and the temperature state of each test piece, and characterizes the coupled heat transfer relationships formed between quartz lamps, between quartz lamps and test pieces, and between test pieces through radiation, convection, and heat conduction.

[0024] Establishing an accurate dynamic model is fundamental to achieving high-performance control. Due to the complex physical field couplings within the electrothermal system, such as high-temperature radiation interference between quartz lamps, heat conduction between test pieces, and convective heat transfer between them and the environment, these factors significantly affect the temperature field distribution. By establishing a system of differential equations encompassing all these coupling relationships, the energy conservation relationships at each node of the system are fully described, providing an accurate mathematical framework for subsequent energy-based control design. It should be understood that although this embodiment focuses on the quartz lamps and test pieces, this modeling method is equally applicable to systems with more heat sources or more complex heat transfer structures in other embodiments.

[0025] Step S120: Construct an available energy Lyapunov function based on the temperature deviation of the test specimen, taking into account the temperature dynamic model of the test specimen and the preset temperature reference value of the test specimen.

[0026] Unlike traditional control methods that directly use a quadratic form of temperature error as an evaluation index, this embodiment constructs a usable energy function from a thermodynamic perspective. This function not only reflects the magnitude of the deviation between the current temperature of the test specimen and the reference temperature, but also profoundly characterizes the "work potential" when the system state deviates from equilibrium. In this way, the control objective is no longer merely to eliminate the temperature difference, but to drive the system's energy state to converge towards the energy level corresponding to the reference state, which facilitates the utilization of the system's inherent thermodynamic properties.

[0027] Step S130: The time derivative of the available energy Lyapunov function is calculated, and the temperature dynamic model of the test piece is substituted into it. The terms formed by energy exchange between the test piece and the test piece, between the test piece and the environment, and between the test piece and the base are identified as the natural dissipation terms of the system. At the same time, the terms formed by energy exchange between the quartz lamp and the test piece are extracted as the aggregated energy injection channels related to the temperature state of the quartz lamp.

[0028] By differentiating the Lyapunov function, it becomes clear that the rate of change of the system's energy consists of two parts: one part is energy dissipation caused by heat dissipation within the system and to the environment, which always tends to reduce the system's energy and is called the natural dissipation term, which is beneficial to the system's stability; the other part is the energy injected by the quartz lamp, called the aggregated energy injection channel, which is the only part that can be adjusted by the control input. Separating these two parts allows the control design to fully utilize the stabilizing effect of the natural dissipation term, and global stability can be achieved simply by reasonably adjusting the energy injection channel.

[0029] Step S140: Based on the energy injection channel and the temperature deviation of the test piece, construct a virtual energy control target such that, under the virtual energy control target, the time derivative of the available energy Lyapunov function is negatively definite or semi-negatively definite with respect to the temperature deviation of the test piece.

[0030] To ensure the temperature of the test specimen steadily converges to the reference value, a desired energy injection target needs to be designed. By constructing a virtual energy control target, an ideal energy injection trajectory is artificially defined. When the actual aggregated energy injection channel can track this virtual target, combined with the aforementioned natural dissipation term, the time derivative of the Lyapunov function can be guaranteed to be negative, thus mathematically ensuring the stability of the system. This is equivalent to designing an energy-level "virtual trajectory" for the underactuated system, guiding the system state to evolve towards the target.

[0031] Step S150: Based on the deviation between the virtual energy control target and the aggregated energy injection channel, construct an error adjustment process and derive the dynamic constraint relationship of the aggregated energy injection channel.

[0032] Due to the thermal inertia of quartz lamps and the complexity of system dynamics, the actual energy injection channel often cannot instantly reach the virtual target. Therefore, an error adjustment process is introduced, and dynamic constraints are designed to ensure that the energy injection channel can smoothly and quickly approach the virtual energy control target. This process essentially constructs an inner-loop control at the energy level, solving the dynamic matching problem between the control variable and the controlled variable in an underactuated system.

[0033] Step S160: Map the dynamic constraint relationship of the energy injection channel to the actual control input of each quartz lamp. By solving and allocating the control input of each quartz lamp, adjust the energy transfer between the quartz lamp and the test piece so that the temperature of the test piece can stably track its reference value.

[0034] The dynamic constraint relationship derived in the preceding steps is an abstract equation concerning energy injection. This step transforms it into specific quartz lamp control voltage or power commands. Since the system contains multiple quartz lamps, there is control redundancy. This embodiment uses a solution and allocation algorithm to calculate the specific input value for each quartz lamp, thereby achieving precise regulation of the energy transfer process at the physical level. Ultimately, this ensures that the temperature of the test piece can stably and accurately track the preset reference value. Through the above steps, this embodiment successfully transforms the complex underactuated temperature control problem into a clear energy flow regulation problem, not only ensuring system stability but also improving the physical intuitiveness and robustness of the control. In one embodiment, a dynamic model of the electrothermal system is established, specifically including establishing a dynamic model of the quartz lamp temperature and a dynamic model of the test specimen temperature.

[0035] For the The temperature dynamic equation for a quartz lamp is: in, , For the number of quartz lamps, , , The first The temperature of the quartz lamp, the derivative of temperature with respect to time, and the control input; The equivalent heat capacity of a single quartz lamp. The efficiency coefficient for converting electrical energy into heat energy. The emissivity of the quartz lamp surface. The Stefan-Boltzmann constant, The effective heat exchange area of ​​the quartz lamp. The radiation viewing angle factor between the quartz lamp and the test specimen. The convective heat transfer coefficient between the quartz lamp and the test specimen is given. The equivalent thermal resistance between the quartz lamp and the test piece. The radiation angle factor between quartz lamps. The convective heat transfer coefficient between quartz lamps. The equivalent thermal resistance between quartz lamps. The radiation angle factor between the quartz lamp and the environment. The convective heat transfer coefficient between the quartz lamp and the environment. For ambient temperature, The temperature of the quartz lamp base. The equivalent thermal resistance between the quartz lamp and the base. For the number of test pieces, For the first The temperature of the test specimen. The first term on the right-hand side of the equation. The first term represents the electrothermal conversion input power of the quartz lamp, which is the sole control input source of the system. The second and third terms characterize the radiative heat transfer and convective / conductive heat transfer between the quartz lamp and the test piece, respectively. The fourth term reflects the strong nonlinearity of high-temperature radiative heat transfer. The fourth and fifth terms characterize the radiative and convective / conductive heat coupling between the quartz lamps, reflecting the mutual interference between multiple heat sources. The sixth and seventh terms characterize the radiative and convective heat loss between the quartz lamp and the environment. The last term characterizes the heat loss of the quartz lamp through the base. It should be understood that although this embodiment provides a complete model including all the above heat transfer terms, in other embodiments, depending on the structural characteristics of the actual system, some heat transfer terms may be negligible. For example, when the distance between quartz lamps is large, the coupling terms between quartz lamps may be small, but this still falls within the scope of protection of this invention.

[0036] For the The temperature dynamic equation for each test specimen is as follows: in, , Number of test specimens; For the first The derivative of temperature with respect to time for each test specimen; For the equivalent heat capacity of a single test specimen, The emissivity of the test specimen surface. The effective heat transfer area of ​​the test specimen. The radiation viewing angle factor between test specimens. The convective heat transfer coefficient between the test specimens is... The equivalent thermal resistance between the test pieces, The radiation perspective factor between the test specimen and the environment. The convective heat transfer coefficient between the test specimen and the environment. The temperature of the test specimen base. The equivalent thermal resistance between the test specimen and the base is given by this equation. This equation clearly shows the energy source and destination of the temperature change in the test specimen. The first two terms characterize the radiation, convection, and conduction energy received by the test specimen from all the quartz lamps. This is the main driving force for the temperature rise of the test specimen and also reflects the indirect effect of the control input. The third and fourth terms characterize the radiation, convection, and conduction coupling between the test specimens, reflecting the thermal interaction between multiple test specimens. The last three terms characterize the radiative heat loss, convective heat loss, and conduction heat loss from the test specimen to the environment, respectively. It is worth noting that the dynamic equation for the test specimen temperature does not contain any direct control input terms. This mathematically characterizes the underactuated nature of the system, meaning that the test specimen temperature can only be indirectly regulated by the state of the quartz lamps. This model not only covers the main heat exchange channel between the quartz lamps and the test specimen but also fully preserves the coupling relationships between the quartz lamps, between the test specimens, and between the system and the environment, avoiding the decrease in control accuracy caused by oversimplification in traditional methods.

[0037] In one embodiment, the derivation process of the core logic of the control law is explained, which constitutes the theoretical foundation of this invention based on the available energy control strategy. This embodiment describes the construction of the Lyapunov function, the extraction of energy characteristics, the design of the virtual target, and the derivation of dynamic constraint relationships as a complete logical chain.

[0038] This embodiment first constructs a usable energy Lyapunov function based on the temperature deviation of the test specimen, taking into account the temperature dynamic model of the test specimen and the preset temperature reference value of the test specimen. Specifically: in, This indicates that the available Lyapunov function is... Indicates the first The preset temperature reference value corresponds to each test piece. It should be understood that this function form is not arbitrarily chosen, but rather stems from the concept of "available energy" in thermodynamics. In thermodynamics, the degree to which a system's state deviates from equilibrium can be measured by its work potential. The function constructed in this embodiment... It precisely depicts the current temperature of the test piece. Relative to reference temperature The "energy distance". If and only if At a given temperature deviation, the function reaches its minimum value of 0; however, for any other temperature deviation, the function remains positive and monotonically increases with increasing deviation. This mathematical property ensures that it can serve as a legitimate measure of the system's tracking error, laying the foundation for subsequent stability analysis.

[0039] Subsequently, in this embodiment, the available energy Lyapunov function is differentiated over time, and the temperature dynamic model of the test specimen is substituted into it. The terms formed by energy exchange between test specimens, between test specimens and the environment, and between test specimens and the base are identified as the natural dissipation terms of the system. At the same time, the terms formed by energy exchange between the quartz lamp and the test specimen are extracted as the aggregated energy injection channels related to the temperature state of the quartz lamp.

[0040] Specifically, the time derivative of the available Lyapunov function will be used. Represented as: in, Characterizing the energy exchange between the quartz lamp and the test specimen. , , The energy exchange between test specimens, between test specimens and the environment, and between test specimens and the base are characterized separately. The key to this step is to identify and separate the physical properties of the above energy terms.

[0041] In this embodiment, the sum of the last three terms is determined as the system's natural dissipation term: in, , , These terms are called "natural dissipative terms" because, according to the second law of thermodynamics, heat spontaneously flows from high to low temperatures. During the dynamic evolution of the system, these terms tend to consume the system's available energy, bringing it towards equilibrium. Mathematically, these terms typically exhibit non-positive or semi-negative constant characteristics, naturally acting as a "damper" on the system's stability. Utilizing this physical property, control design does not need to actively cancel these terms; instead, it can "go with the flow," using their dissipative characteristics to assist in stabilizing the system.

[0042] Simultaneously, terms related to the temperature state of the quartz lamp were extracted as a polymer energy injection channel. and its weight function : .

[0043] Converging energy injection channel This represents the total energy injection capability of all quartz lamps into the test specimen; it is the only adjustable channel for the control input. Weighting function. This reflects the sensitivity of the test specimen's temperature deviation to energy injection. Through this separation, the complex system dynamics are decoupled into an "uncontrollable but beneficial dissipation component" and a "controllable injection component," greatly simplifying the design of the controller.

[0044] Based on the above separation results, this embodiment constructs a virtual energy control target based on the polymerization energy injection channel and the temperature deviation of the test piece, so that under the virtual energy control target, the time derivative of the available energy Lyapunov function is negatively definite or semi-negatively definite with respect to the temperature deviation of the test piece.

[0045] Specifically, the temperature deviation of the test specimen is characterized as a weighting function. ;when At that time, the virtual energy control target is constructed as follows: ,when season .in, Positive control gain, This is the residual energy term introduced by the inconsistency between the test specimen temperature reference value and the ambient or base temperature.

[0046] This construction process embodies profound control principles. Substituting into the derivative expression of the Lyapunov function, we can obtain... .because Constantly non-negative, Naturally not positive, therefore It is guaranteed to be negative constant or semi-negative constant. This means that if the actual polymerization energy injection channel can be controlled... Accurately track this virtual target Therefore, the available energy of the system will decrease monotonically with time, eventually converging to zero, thus theoretically guaranteeing that the temperature of the test specimen converges to the reference value without error. Here, "virtual" refers to... It is an ideal value derived from the control target, which provides a clear "bullseye" for actual energy injection.

[0047] However, in actual physical systems, due to the thermal inertia of quartz lamps and the nonlinearity of system dynamics, the actual energy injection channel... Virtual goals are often not achieved instantly. Therefore, this embodiment constructs an error adjustment process based on the deviation between the virtual energy control target and the aggregated energy injection channel, and derives the dynamic constraint relationship on the aggregated energy injection channel.

[0048] Specifically, auxiliary state variables are introduced. The following second-order error adjustment process is constructed: , in, For virtual energy control targets, To inject energy into the channel, deviation , , The gain is positive. This second-order system is equivalent to designing a damped dynamic filter for the energy tracking process, which can smoothly adjust the error convergence process and avoid system oscillations caused by sudden changes in control input. and The dynamic constraint relationship of the polymerization energy injection channel is derived as follows: in, This represents the time derivative of the energy injection channel. This represents the derivative of the virtual energy control target with respect to time. This dynamic constraint reveals the dynamic conditions that the control input must satisfy: the rate of change of the aggregated energy injection channel must not only follow the rate of change of the virtual target. It should also include auxiliary states. The regulatory effect of this constraint relationship transforms the abstract stability requirement into a concrete dynamic equation, providing a direct mathematical basis for mapping the control law to the actual physical input in the next step. Through the above steps, this embodiment completes the entire process from energy function construction to dynamic constraint derivation, constructing a controller framework with clear physical meaning and rigorous mathematical logic.

[0049] In one embodiment, the method for mapping the dynamic constraints of the energy injection channel to the actual control inputs of each quartz lamp is explained, as well as how to regulate the energy transfer between the quartz lamp and the test specimen by solving and allocating the control inputs of each quartz lamp. This step is a crucial step in transforming the aforementioned abstract energy control theory into engineering-executable instructions.

[0050] Specifically, the dynamic constraints of the energy injection channel are mapped to the actual control inputs of each quartz lamp, as follows: according to The definition and dynamic model of quartz lamp temperature transform the dynamic constraint relationship into a relationship with respect to the control input. The equation is: in, This represents the known energy change caused by the system's inherent heat exchange relationships. The left side of this equation is the rate of change of the polymerization energy injection channel. The specific details reveal the control input How to determine the temperature state of a quartz lamp This affects the energy injection. It's worth noting that the left side contains two related... Aggregate item: one is and The sum of the products corresponds to the radiation heat transfer term, since the radiative heat flux is proportional to the fourth power of temperature, and its rate of change naturally involves the third power of temperature; the other is... The direct summation corresponds to the convection and conduction heat transfer terms, as these heat flows are linearly related to temperature. This structure clearly reflects the different dependencies of different heat transfer mechanisms on the control input.

[0051] To solve the above equations, this embodiment adjusts the energy transfer between the quartz lamps and the test specimen by solving and allocating the control inputs of each quartz lamp, specifically including: Define the aggregate quantity of the quartz lamp control input. and .

[0052] It should be understood that the introduction of polymerization amount and This is a sophisticated mathematical approach. Since the system contains multiple quartz lamps, directly solving for each... The problem arises from an insufficient number of equations (one equation constrains multiple variables). By defining two aggregate quantities, the complex multivariate coupled problem is reduced to a problem of solving a binary linear equation, greatly simplifying the computational complexity. The following equation can be obtained by solving... and : The equation is about and Given the values ​​of the terms on the right-hand side of a linear equation, a unique set of equations can be determined. Solution. This set of solutions represents the "total power" and "total radiative potential" that all quartz lamps need to provide in order to meet the energy tracking target.

[0053] The amount of polymerization is obtained by solving the problem. and Then, taking into account the current temperature of the quartz lamp. Distribute the control inputs to each quartz lamp This ensures that the polymerization quantity relationship is satisfied.

[0054] Specifically, the allocation process needs to solve the inverse mapping problem from aggregate quantity to individual quantity. Due to control redundancy (multiple lights), the above aggregation relationship must be satisfied. The power allocation scheme is not unique. This embodiment can employ various allocation strategies, such as average allocation, temperature-weighted allocation, or optimized energy consumption allocation. Taking temperature-weighted allocation as an example, power can be allocated based on the current temperature of each quartz lamp. Lamps with higher temperatures are allocated less power to avoid overheating, while lamps with lower temperatures are allocated more power for faster response. This balances the heat load of each lamp while meeting the total energy demand, extending the equipment's lifespan. Regardless of the specific allocation strategy used, as long as the control input of each lamp... satisfy and This ensures the injection of polymer energy into the channel. Accurately track virtual targets Furthermore, through the system's inherent energy dissipation mechanism, the temperature of the test specimen can be stably tracked to the reference value. This design approach of "aggregate solution + flexible allocation" not only ensures the rigid realization of the control objective but also provides sufficient flexibility for engineering implementation, perfectly adapting to the control characteristics of underactuated multi-input systems.

[0055] In one embodiment, the stability proof process of the control system and the selection conditions for key control gain parameters are described. This process rigorously proves theoretically that the control method proposed in this invention can ensure that the temperature of the test specimen asymptotically converges to its reference value, providing clear guidance for parameter tuning in engineering practice.

[0056] Specifically, in order to analyze the overall stability of the closed-loop system, this embodiment constructs a global Lyapunov function V, which has the following form: in, The available energy Lyapunov function constructed in the foregoing embodiments is used as an energy measure to characterize the temperature deviation of the test specimen. The deviation between the energy injection channel and the virtual energy control target; This is an auxiliary state variable introduced during the error adjustment process. The overall Lyapunov function integrates energy information from three aspects: system state deviation, energy tracking error, and the dynamic adjustment process, thus comprehensively characterizing the system's stability. It should be understood that although this embodiment provides the specific function form described above, in other embodiments, any positive definite function that can reflect the trend of system energy changes can be considered as a candidate Lyapunov function, and this still falls within the scope of protection of this invention.

[0057] Taking the time derivative of the overall Lyapunov function V and substituting it into the dynamic equation derived in the previous embodiment, we get: To simplify the analysis and ensure good convergence performance of the system, this embodiment selects... At this point, the cross term in the above derivative expression They are eliminated. Furthermore, by applying Young's inequality to the remaining cross terms, we can... The upper bound is represented as: Observing the above inequalities, we can find that in order to ensure To ensure system stability and possess negative definiteness, two key conditions must be met. First, in order to counteract the right-hand side... This requires utilizing the positive definiteness of the Lyapunov function V itself. Because... If selected Then there is This condition ensures that the energy tracking error term does not disrupt the overall energy dissipation trend. Secondly, to guarantee the auxiliary state... The coefficient of the quadratic term must be negative, which requires satisfying... This condition ensures that the error adjustment process has sufficient damping to quickly suppress dynamic fluctuations.

[0058] In summary, this embodiment stabilizes the closed-loop system by selecting a control gain, specifically satisfying the following: and This ensures the temperature of the test piece. asymptotically converges to its reference value .

[0059] The selection of this parameter has significant engineering implications. It not only provides sufficient conditions to ensure system stability but also reveals the coupling relationships between the various control gains. For example, when the coupling gain... Increasing the value of the test specimen temperature deviation means that its impact on the dynamic error is enhanced, thus requiring a corresponding increase in the value of the test specimen temperature deviation. To maintain system stability; similarly, increase While it can accelerate the convergence of temperature deviations, it also requires... The lower bound is adjusted accordingly. This explicit parameter boundary relationship allows engineers to be more targeted during debugging, avoiding the blindness of traditional trial-and-error methods and significantly improving the debugging efficiency and operational reliability of the control system. Through the above theoretical analysis, this embodiment rigorously proves the global asymptotic stability of the method of the present invention under underactuated and strongly nonlinear conditions, verifying the effectiveness of the available energy-based control strategy.

[0060] In one specific embodiment, to verify the effectiveness and superiority of the temperature tracking control method based on available energy for the underactuated electrothermal system provided by this invention, a specific simulation verification application scenario is constructed. This scenario includes two quartz lamps and two test pieces, constituting a typical multi-input multi-output underactuated electrothermal system. The quartz lamps are numbered as follows: The test piece number is The system thermodynamic model adopts the dynamic model established in the aforementioned embodiments. This model fully includes the radiation, convection, and thermal conduction coupling terms between the quartz lamp and the test piece, as well as the energy exchange terms between test pieces, between the test piece and the environment and the base. It can truly reflect the underactuated and strongly nonlinear characteristics of the system.

[0061] Specifically, the simulation parameters were set as follows: the reference temperature for both test pieces was set to 600K, the initial temperatures were set to 320K and 330K respectively, the initial temperatures of the quartz lamps were set to 350K and 360K respectively, the ambient temperature and the base temperature were both set to 300K, and the remaining thermophysical parameters were set as follows: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Under the same model parameters and initial conditions, control simulations were performed using the method of this invention and the proportional-integral comparison method based on temperature error, respectively, to compare the temperature tracking performance of different control strategies in an underdriven strongly nonlinear electrothermal system. Figures 2 to 4 Temperature response curves of test specimen 1 and test specimen 2 under the method of the present invention and the comparative method are given respectively.

[0062] Reference Figure 2 The figure shows the temperature tracking trajectory of test piece 1 under the method of this invention. As can be seen from the figure, under an initial temperature of approximately 320K, the temperature of test piece 1 rapidly rises to near the reference value of 600K within a very short time. The entire heating process is smooth and continuous, without significant oscillations or overshoot. Based on the curve, it can be estimated that the settling time reaches more than 98% of the reference value (i.e., above 588K) within approximately 20 seconds, and enters the ±1% error band (594K~606K) within approximately 30 seconds and remains stable, with a steady-state error close to 0K, essentially coinciding with the reference curve.

[0063] Reference Figure 3The temperature tracking trajectory of test specimen 2 under the method of this invention is shown. Test specimen 2, with an initial temperature of approximately 330K, also reached 98% of the reference value level in about 20 seconds and entered the ±1% error range within about 30 seconds. The heating rate and convergence time of the two test specimens were extremely similar, with a convergence time difference of less than 5% and a steady-state error of less than 0.5K, indicating that under multi-test specimen coupling conditions, the method of this invention can achieve consistent energy coordination control.

[0064] Reference Figure 4 The figure shows the temperature tracking trajectory of test piece 1 under the comparative method. As can be seen from the figure, under the same model parameters and initial conditions, although the temperature of test piece 1 eventually approaches 600K, the time it takes to reach 98% of the reference value is approximately 200s, and the time to enter the ±1% error band is close to 300s. Compared to the approximately 30s settling time of the method of this invention, the settling time is extended by about 8 to 10 times. A residual error of approximately 4 to 6K exists in the steady-state stage, and the temperature approximation process exhibits a gradual and slow convergence characteristic.

[0065] Reference Figure 5 The temperature tracking trajectory of test piece 2 under the comparison method is shown. Its dynamic response trend is similar to... Figure 3 The time to reach 98% of the reference value is approximately 220 seconds, and the time to enter the ±1% error band exceeds 300 seconds, which is significantly slower than the method of this invention. The steady-state error is also on the order of 5K, indicating that under underactuated and strongly nonlinear radiative coupling conditions, PI control relying solely on temperature error feedback is insufficient to achieve rapid energy redistribution.

[0066] In summary, under the same system model and parameter conditions, the temperature tracking control method based on available energy proposed in this invention reduces the settling time by more than 85% and the time to enter the ±1% steady-state error band by about 90% compared to the traditional error-driven PI control method, while reducing the steady-state error by about 50% to 80%. Simultaneously, it maintains a highly consistent dynamic response under multi-sample coupling conditions. The above quantitative comparison results demonstrate that the method of this invention has significant advantages in energy utilization efficiency, convergence speed, and steady-state accuracy, verifying the effectiveness of constructing a Lyapunov function and a converged energy channel regulation mechanism based on available energy.

[0067] Figure 6 This example illustrates a schematic diagram of the physical structure of an electronic device, which can be a smart terminal. Its internal structure diagram can be as follows: Figure 6As shown, the electronic device includes a processor, memory, and a network interface connected via a system bus. The processor provides computing and control capabilities. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The network interface is used to communicate with external terminals via a network connection. When the computer program is executed by the processor, it implements the temperature tracking control method based on available energy for the underdriven electric heating system according to any of the above embodiments.

[0068] Those skilled in the art will understand that Figure 6 The structure shown is merely a block diagram of a portion of the structure related to the present invention and does not constitute a limitation on the electronic device to which the present invention is applied. A specific electronic device may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0069] On the other hand, the present invention also provides a computer storage medium storing a computer program, which, when executed by a processor, implements the temperature tracking control method for the underdriven electric heating system based on available energy in any of the above embodiments.

[0070] In another aspect, a computer program product or computer program is provided, which includes computer instructions stored in a computer-readable storage medium. A processor of an electronic device reads the computer instructions from the computer-readable storage medium, and when the processor executes the computer instructions, it implements the temperature tracking control method for the underdriven heating system based on available energy according to any of the above embodiments.

[0071] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. This computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided by this invention can include non-volatile and / or volatile memory. Non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory.

[0072] By way of illustration and not limitation, RAM is available in a variety of forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0073] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0074] The above-described embodiments are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.

Claims

1. A temperature tracking control method for an underdriven electric heating system based on available energy, characterized in that, An electrothermal system comprising multiple quartz lamps and multiple test specimens, wherein a control input acts directly only on the quartz lamps, and the temperature of the test specimens is indirectly regulated through heat transfer between the quartz lamps and the test specimens, the method comprising: A dynamic model of the electrothermal system is established, which simultaneously includes the temperature state of each quartz lamp and the temperature state of each test piece, and characterizes the coupled heat transfer relationships formed between quartz lamps, between quartz lamps and test pieces, and between test pieces through radiation, convection and heat conduction. Based on the temperature dynamic model of the test specimen and the preset temperature reference value of the test specimen, a usable energy Lyapunov function based on the temperature deviation of the test specimen is constructed. The available energy Lyapunov function is differentiated over time, and the temperature dynamic model of the test piece is substituted into it. The terms formed by energy exchange between the test piece and the test piece, between the test piece and the environment, and between the test piece and the base are identified as the natural dissipation terms of the system. At the same time, the terms formed by energy exchange between the quartz lamp and the test piece are extracted as the aggregated energy injection channel related to the temperature state of the quartz lamp. Based on the polymer energy injection channel and the temperature deviation of the test specimen, a virtual energy control target is constructed, such that under the virtual energy control target, the time derivative of the available energy Lyapunov function is negatively definite or semi-negatively definite with respect to the temperature deviation of the test specimen. Based on the deviation between the virtual energy control target and the aggregated energy injection channel, an error adjustment process is constructed, and the dynamic constraint relationship of the aggregated energy injection channel is derived. The dynamic constraint relationship of the energy injection channel is mapped to the actual control input of each quartz lamp. By solving and allocating the control input of each quartz lamp, the energy transfer between the quartz lamp and the test piece is adjusted so that the temperature of the test piece stably tracks its reference value.

2. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 1, characterized in that, The establishment of the dynamic model of the electrothermal system specifically includes: Establish a dynamic temperature model for the quartz lamp, for the first The temperature dynamic equation for a quartz lamp is: ， in, , The number of quartz lamps. , , The first The temperature of the quartz lamp, the derivative of temperature with respect to time, and the control input; The equivalent heat capacity of a single quartz lamp. The efficiency coefficient for converting electrical energy into heat energy. The emissivity of the quartz lamp surface. The Stefan-Boltzmann constant is given. The effective heat exchange area of ​​the quartz lamp. The radiation viewing angle factor between the quartz lamp and the test specimen. The convective heat transfer coefficient between the quartz lamp and the test specimen is given. The equivalent thermal resistance between the quartz lamp and the test piece. The radiation angle factor between quartz lamps. The convective heat transfer coefficient between quartz lamps is... The equivalent thermal resistance between quartz lamps. The radiation angle factor between the quartz lamp and the environment. The convective heat transfer coefficient between the quartz lamp and the environment. For ambient temperature, Temperature of the quartz lamp base. The equivalent thermal resistance between the quartz lamp and the base. For the number of test pieces, For the first Temperature of each test piece; Establish a dynamic temperature model for the test specimen, for the first... The temperature dynamic equation for each test specimen is as follows: ； in, , Number of test specimens; For the first The derivative of temperature with respect to time for each test specimen; For the equivalent heat capacity of a single test specimen, The emissivity of the test specimen surface. The effective heat transfer area of ​​the test specimen. The radiation viewing angle factor between test specimens. The convective heat transfer coefficient between the test specimens is... The equivalent thermal resistance between the test pieces, The radiation perspective factor between the test specimen and the environment. The convective heat transfer coefficient between the test specimen and the environment. The temperature of the test specimen base. The equivalent thermal resistance between the test piece and the base.

3. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 1, characterized in that, The construction is based on the available energy Lyapunov function of the temperature deviation of the test specimen, specifically as follows: ， in, This indicates that the available Lyapunov function is... Indicates the first The preset temperature reference value corresponding to each test piece.

4. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 3, characterized in that, The process involves time-differentiating the available energy Lyapunov function and substituting it into the temperature dynamic model of the test specimen. The terms arising from energy exchange between test specimens, between the test specimen and the environment, and between the test specimen and the base are identified as the system's natural dissipation terms. Simultaneously, the term arising from energy exchange between the quartz lamp and the test specimen is extracted as a pooled energy injection channel related to the quartz lamp's temperature state. Specifically, this includes: The time derivative of the available energy Lyapunov function Represented as: ， in, Characterizing the energy exchange between the quartz lamp and the test specimen. , , These characterize the energy exchange between test specimens, between test specimens and the environment, and between test specimens and the base, respectively. The sum of the last three terms is determined as the system's natural dissipation term: ， in, , , ; Simultaneously, terms related to the temperature state of the quartz lamp were extracted as a polymer energy injection channel. and its weighting function : ， 。 5. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 4, characterized in that, The virtual energy control target is constructed based on the polymer energy injection channel and the temperature deviation of the test piece, specifically as follows: The temperature deviation of the test piece is characterized as a weighting function. ; when At that time, the virtual energy control target is constructed as follows: ,when season ; in, Positive control gain, This is the residual energy term introduced by the inconsistency between the test specimen temperature reference value and the ambient or base temperature.

6. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 5, characterized in that, The step of constructing an error adjustment process based on the deviation between the virtual energy control target and the aggregated energy injection channel, and deriving the dynamic constraint relationship of the aggregated energy injection channel, specifically includes: Introducing auxiliary state variables The following second-order error adjustment process is constructed: ， in, For virtual energy control targets, To inject energy into the channel, deviation , , Positive control gain; Depend on and The dynamic constraint relationship of the polymerization energy injection channel is derived as follows: ， in, This represents the time derivative of the energy injection channel. This represents the time derivative of the virtual energy control target.

7. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 6, characterized in that, The dynamic constraint relationship of the energy injection channel is mapped to the actual control input of each quartz lamp, specifically as follows: according to The definition and dynamic model of quartz lamp temperature transform the dynamic constraint relationship into a relationship with respect to the control input. The equation is: ， in, This is the known energy change term caused by the system's inherent heat exchange relationship.

8. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 7, characterized in that, By solving and allocating the control inputs of each quartz lamp, the energy transfer between the quartz lamp and the test piece is adjusted, specifically including: Define the aggregate quantity of the quartz lamp control input. and ; The following formula can be obtained by solving it. and : ； Based on the current temperature status of the quartz lamp Distribute the control inputs to each quartz lamp This ensures that the polymerization quantity relationship is satisfied.

9. The temperature tracking control method for an underdriven electric heating system based on available energy according to claim 6, characterized in that, The method stabilizes the closed-loop system by selecting a control gain, specifically satisfying the following: and This ensures the temperature of the test piece. asymptotically converges to its reference value .

10. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the temperature tracking control method for the underdriven electric heating system based on available energy as described in any one of claims 1 to 9.

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