A state coupling sliding mode control method and system for under-actuated electrothermal systems

Abstract

The application discloses a state coupling sliding mode control method and system of an underactuated electric heating system, relates to the cross technical fields of electric heating engineering, thermal system modeling and intelligent control, and comprises the following steps: constructing an electric heating system dynamic model comprising a quartz lamp temperature state, a test piece temperature state and a quartz lamp control input; based on a test piece temperature dynamic equation and a preset test piece temperature reference track, constructing a sliding mode function taking a test piece temperature tracking error and a time derivative as variables; deriving the constructed sliding mode function and combining the quartz lamp temperature dynamic equation and the test piece temperature dynamic equation to establish a direct mapping relationship between a sliding mode dynamic and a control input; designing a second-order sliding mode control law, introducing an auxiliary state and constructing a continuous second-order sliding mode reaching law, and inversely deriving the quartz lamp control input according to a linear algebra relationship. The method has clear control structure, clear physical meaning and strong engineering implementation, and is suitable for temperature control of a quartz lamp electric heating system under complex working conditions.

Description

Technical Field

[0001] This invention relates to the interdisciplinary fields of electrothermal engineering, thermal system modeling and intelligent control, specifically to a state-coupled sliding mode control method and system for an underactuated electrothermal system. Background Technology

[0002] Quartz lamp heating systems are widely used in materials heat treatment, aerospace component heating, advanced manufacturing test platforms, and various thermal experimental devices due to their advantages such as rapid heating, high power density, compact structure, and easy modular arrangement. In these systems, multiple quartz lamps are typically used as heating actuators, transferring heat to the test piece through radiation, convection, and conduction to achieve temperature regulation and control.

[0003] With the increasing complexity and sophistication of engineering applications, practical quartz lamp electrothermal systems often exhibit characteristics of multiple inputs and outputs, strong nonlinearity, and strong coupling. On the one hand, the heat transfer process between the quartz lamp and the test specimen is influenced by multiple physical mechanisms, including the fourth-order temperature term in radiative heat transfer, convective heat transfer, and contact conduction, resulting in significant nonlinear characteristics in the system dynamics. On the other hand, multiple quartz lamps and multiple test specimens form complex spatial coupling relationships through thermal radiation and conduction, causing mutual influence between channels and making simple decoupling control difficult. Furthermore, in practical systems, the quartz lamp is usually the only actuator that can directly apply control input, while the test specimen itself does not have an independent heating actuator; its temperature change depends entirely on the temperature of the quartz lamp and its heat exchange process with the surrounding environment. This structural characteristic makes the quartz lamp electrothermal system essentially an underactuated system, meaning that the dimension of the system's control input is smaller than the dimension of the controlled state, making traditional control methods based on the fully actuated assumption difficult to apply directly.

[0004] Existing control methods for quartz lamps or similar electrothermal systems mostly employ proportional-integral-derivative (PID) control, feedforward-feedback composite control, or control strategies based on linearized models. These methods typically rely on simplified linear models or linear approximations around specific operating points, making it difficult to fully characterize the strong nonlinear coupling relationship between the quartz lamp and the test specimen caused by radiative heat transfer. When system operating conditions change, parameters become uncertain, or external environmental disturbances are strong, control performance often degrades significantly, potentially leading to problems such as temperature overshoot, oscillations, or increased steady-state errors. On the other hand, some research has attempted to introduce nonlinear control methods such as sliding mode control and robust control into the field of thermal system control. However, most existing schemes still tend to treat the nonlinear terms in the system as uncertain disturbances for unified treatment, or indirectly achieve the control objective by introducing virtual control quantities and multi-layer backstepping designs. Although these methods have a certain degree of robustness in theory, they often suffer from complex control structures, difficult parameter design, unclear physical meaning, and significant practical implementation challenges in underactuated quartz lamp electrothermal systems. In particular, it is difficult to directly utilize the known physical coupling relationship between the quartz lamp temperature and the test specimen temperature to construct an efficient control law. Furthermore, in practical engineering applications, quartz lamp electrothermal systems are inevitably affected by factors such as parameter uncertainties, changes in thermal radiation coefficients, fluctuations in convective heat transfer conditions, and environmental temperature disturbances. This places higher demands on the stability and robustness of the control system. Currently, there is a lack of a control method that can construct a control law directly based on the system's physical model, utilizing the nonlinear coupling relationship between the quartz lamp and the test specimen, without introducing virtual control variables, and achieve accurate temperature tracking of the test specimen while ensuring system physical constraints.

[0005] Therefore, there is an urgent need to propose a new control method for the electrothermal system of an underactuated quartz lamp test specimen. This method can establish a direct mapping relationship between the quartz lamp control input and the temperature change of the test specimen based on a full characterization of the nonlinear thermal coupling characteristics of the system. It can also ensure the stability and robustness of the closed-loop system under the presence of parameter uncertainties and external disturbances, thereby meeting the requirements of complex engineering applications for the safety, accuracy and reliability of the electrothermal system. Summary of the Invention

[0006] In view of the above-mentioned problems, the present invention is proposed.

[0007] Therefore, the technical problem solved by this invention is that existing electrothermal systems composed of multiple quartz lamps and multiple test pieces have insufficient characterization of strong nonlinear coupling relationships, low control accuracy, and poor robustness, as well as the problem of how to directly establish input-output mapping without relying on virtual control quantities under underactuated constraints.

[0008] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a state-coupled sliding mode control method for an underactuated electrothermal system, comprising constructing a dynamic model of the electrothermal system including the temperature state of a quartz lamp, the temperature state of a test piece, and the control input of the quartz lamp, wherein the dynamic equation of the quartz lamp temperature includes an electrical power input term and a heat exchange nonlinear term, and the dynamic equation of the test piece temperature includes a nonlinear coupled heat transfer between the quartz lamp and the test piece temperature; based on the dynamic equation of the test piece temperature and a preset test piece temperature reference trajectory, constructing a model with the test piece temperature tracking error and time derivative as variables. A sliding mode function is constructed, and a state dependence coefficient related to the temperature state of the quartz lamp is introduced to reflect the nonlinear coupling relationship between the two. The derivative of the constructed sliding mode function is obtained, and combined with the temperature dynamic equations of the quartz lamp and the test specimen, a direct mapping relationship between the sliding mode dynamics and the control input is established. Based on the sliding mode function, a second-order sliding mode control law is designed, an auxiliary state is introduced, and a continuous second-order sliding mode arrival law is constructed to make the sliding mode variables and derivatives converge in a finite time. The control input of the quartz lamp is obtained by inverse calculation based on the linear algebraic relationship, so that the test specimen tracks the temperature reference trajectory under disturbance.

[0009] As a preferred embodiment of the state-coupled sliding mode control method for the underactuated electrothermal system described in this invention, the dynamic model of the electrothermal system includes an underactuated quartz lamp test specimen electrothermal system consisting of... A quartz lamp and The test specimens are composed of a temperature vector consisting of all the quartz lamp temperatures and the temperature vector consisting of all the test specimen temperatures. The control input in the electrothermal system is applied only to the quartz lamp channels, and the control input of all the quartz lamps is composed of an input vector. The dynamic model of the electrothermal system is constructed based on the temperature change mechanism jointly determined by electrothermal energy conversion and heat transfer, including multiple heat exchange mechanisms and corresponding physical coefficients.

[0010] As a preferred embodiment of the state-coupled sliding mode control method for the underdriven electrothermal system described in this invention, the dynamic equation for the quartz lamp temperature includes:

[0011] ;

[0012] in, Let be the first derivative of the quartz lamp temperature with respect to time. This refers to the temperature rise caused by the electrical power input of the quartz lamp. For electrothermal energy conversion efficiency, For equivalent heat capacity, For the first The control input for a quartz lamp, The nonlinear terms arising from heat exchange between quartz lamps, between the quartz lamp and the test specimen, and between the quartz lamp and the environment. Let be the temperature state vector of the quartz lamp. Let be the temperature state vector of the test specimen. For the first The total disturbance term of the temperature channel of a quartz lamp.

[0013] As a preferred embodiment of the state-coupled sliding mode control method for the underdriven electrothermal system described in this invention, the dynamic equation for the temperature of the test specimen includes:

[0014] ;

[0015] in, The first derivative of the specimen temperature with respect to time. This is a nonlinear coupled heat transfer term determined by both the temperature state of the quartz lamp and the temperature state of the test specimen. For the first The total disturbance term of the temperature channel of each test piece.

[0016] As a preferred embodiment of the state-coupled sliding mode control method for the underdriven electrothermal system described in this invention, the construction of the sliding mode function with the temperature tracking error and time derivative of the test piece as variables includes: setting a continuous and at least second-order differentiable temperature reference trajectory for each test piece; defining the temperature tracking error of each test piece based on the actual temperature state of the test piece and the temperature reference trajectory; obtaining the first derivative of the error by taking the time derivative of the temperature tracking error based on the dynamic equation of the test piece temperature; and constructing the corresponding sliding mode function for each test piece based on this, expressed as:

[0017] ;

[0018] in, For the first Sliding mode variables of each test piece, For the first The first derivative of the temperature tracking error of each test piece The sliding mode coefficient function is related to the system state. For the first Temperature tracking error of individual test specimens.

[0019] As a preferred embodiment of the state-coupled sliding mode control method for the underactuated electrothermal system described in this invention, the establishment of a direct mapping relationship between the sliding mode dynamics and the control input includes: taking the time derivative of the sliding mode function; using the partial derivative relationship of the test piece temperature dynamic equation with respect to the quartz lamp temperature, expressing the second derivative of the test piece temperature in a form that includes the rate of change of the quartz lamp temperature; substituting this into the time derivative of the sliding mode function to obtain the sliding mode dynamic expression that includes the rate of change of the quartz lamp temperature; and combining this with the quartz lamp temperature dynamic equation, converting the sliding mode dynamic expression into a linear algebraic relationship with respect to the quartz lamp control input, expressed as:

[0020] ;

[0021] in, The first derivative of the sliding mode variable. For the state coupling gain The coupling matrix formed The input vector consists of the control inputs of each quartz lamp. It is a vector term consisting of the system state, reference trajectory, and disturbance term.

[0022] As a preferred embodiment of the state-coupled sliding mode control method for the underdriven electrothermal system described in this invention, the second-order sliding mode control law includes: introducing auxiliary state variables corresponding to each sliding mode variable and forming an auxiliary state vector to construct a continuous second-order sliding mode arrival law for each test piece's corresponding sliding mode variable; the second-order sliding mode arrival law adopts a continuous form containing fractional terms and sets a control gain function related to the system state.

[0023] As a preferred embodiment of the state-coupled sliding mode control method for the underdriven electrothermal system described in this invention, the second-order sliding mode control law includes expressing the second-order sliding mode arrival law in vector form:

[0024] ;

[0025] ;

[0026] in, The first derivative of the sliding mode variable. It is a diagonal matrix composed of the corresponding gains of each test piece. It is a vector function composed of its components. For auxiliary state vectors, For the time derivative of the auxiliary state vector, It is the sliding mode vector composed of the sliding mode variables of each test piece.

[0027] Another objective of this invention is to provide a state-coupled sliding mode control system for an underactuated electrothermal system. This system can construct a sliding mode function with the temperature tracking error and time derivative of the test piece as variables based on the dynamic equation of the test piece temperature and the preset test piece temperature reference trajectory. It introduces a state dependence coefficient related to the temperature state of the quartz lamp to reflect the nonlinear coupling relationship between the two, thus solving the problem that current electrothermal systems composed of multiple quartz lamps and multiple test pieces cannot directly establish a mapping relationship.

[0028] As a preferred embodiment of the state-coupled sliding mode control system for the underactuated electrothermal system described in this invention, it includes: a dynamic modeling module, a sliding mode function construction module, a mapping relationship construction module, and a control law design and implementation module; the dynamic modeling module is used to construct a dynamic model of the electrothermal system including the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp, wherein the dynamic equation of the quartz lamp temperature includes an electrical power input term and a nonlinear term generated by heat exchange between quartz lamps, between the quartz lamp and the test piece, and between the quartz lamp and the environment, and the dynamic equation of the test piece temperature includes a nonlinear coupled heat transfer term determined by both the quartz lamp temperature and the test piece temperature; the sliding mode function construction module is used to construct a sliding mode function based on the dynamic equation of the test piece temperature and a preset test piece temperature reference trajectory, using the test piece temperature tracking error and its time derivative as... The sliding mode function of the variables incorporates a state dependence coefficient related to the temperature state of the quartz lamp, making the structure of the sliding mode function reflect the nonlinear coupling relationship between the quartz lamp temperature and the test piece temperature. The mapping relationship construction module is used to perform time differentiation on the sliding mode function, using the dependence of the test piece temperature dynamic equation on the quartz lamp temperature to obtain a sliding mode dynamic expression containing the rate of change of the quartz lamp temperature, and further combining it with the quartz lamp temperature dynamic equation to transform the sliding mode dynamic expression into a linear algebraic relationship with respect to the quartz lamp control input. The control law design and implementation module is used to design a second-order sliding mode control law based on the sliding mode dynamic equation. By introducing auxiliary states and constructing a continuous second-order sliding mode arrival law, the sliding mode variables and derivatives converge. At the same time, the quartz lamp control input is obtained by inverse calculation based on the linear algebraic relationship and applied to the electrothermal system.

[0029] Another object of the present invention is to provide a state-coupled sliding mode control storage medium for an underactuated heating system, wherein a computer program is stored thereon, and when the computer program is executed by a processor, the steps of a state-coupled sliding mode control method for an underactuated heating system are implemented.

[0030] The beneficial effects of this invention are:

[0031] The state-coupled sliding mode control method for underactuated electrothermal systems provided by this invention constructs a state-dependent sliding mode function that incorporates the temperature state of the quartz lamp, and combines it with a second-order sliding mode control strategy. Without introducing virtual control quantities, it achieves a direct mapping relationship between the quartz lamp control input and the temperature change of the test specimen in the underactuated electrothermal system, thereby effectively improving the coordination and accuracy of temperature regulation for multiple test specimens. Simultaneously, this invention can ensure the stability and robustness of the closed-loop system under conditions of system parameter uncertainty, external thermal disturbances, and changes in operating conditions. Under the premise of meeting control input and temperature safety constraints, it enables the test specimen temperature to quickly and reliably track a preset reference trajectory. It has advantages such as clear control structure, explicit physical meaning, and strong engineering feasibility, and is suitable for temperature control of quartz lamp electrothermal systems under complex operating conditions. This invention achieves better results in terms of test specimen temperature tracking accuracy, system dynamic process balance stability, and overall robustness. Attached Figure Description

[0032] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0033] Figure 1 The above is an overall flowchart of a state-coupled sliding mode control method for an underdriven electrothermal system provided in Embodiment 1 of the present invention.

[0034] Figure 2 The simulation results are shown in Embodiment 2 of the present invention for a state-coupled sliding mode control method for an underdriven electrothermal system.

[0035] Figure 3 The simulation results of the sliding mode comparison method for a state-coupled sliding mode control method for an underdriven electrothermal system provided in Embodiment 2 of the present invention are shown in the figure. Detailed Implementation

[0036] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present invention.

[0037] Example 1, referring to Figure 1 As an embodiment of the present invention, a state-coupled sliding mode control method for an underactuated electrothermal system is provided, comprising:

[0038] S1: Construct a dynamic model 100 of the electrothermal system that includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp. The dynamic equation of the quartz lamp temperature includes an electric power input term and a heat exchange nonlinear term, and the dynamic equation of the test piece temperature includes the nonlinear coupled heat transfer between the quartz lamp and the test piece temperature.

[0039] Specifically, a dynamic model 100 of the electrothermal system is established, which includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp. The dynamic equation of the quartz lamp temperature includes an electrical power input term and a nonlinear term generated by heat exchange between quartz lamps, between the quartz lamp and the test piece, and between the quartz lamp and the environment. The dynamic equation of the test piece temperature includes a nonlinear coupled heat transfer term determined by the temperature of the quartz lamp and the temperature of the test piece.

[0040] Furthermore, the dynamic model 100 of the electrothermal system includes the underdriven quartz lamp test specimen electrothermal system consisting of... A quartz lamp and The test specimens are composed of a temperature vector consisting of all the quartz lamp temperatures and the temperature vector consisting of all the test specimen temperatures. The control input in the electrothermal system is applied only to the quartz lamp channels, and the control input of all the quartz lamps is composed of an input vector. The dynamic model 100 of the electrothermal system is constructed based on the temperature change mechanism jointly determined by electrothermal energy conversion and heat transfer, including multiple heat exchange mechanisms and corresponding physical coefficients.

[0041] The underactuated quartz lamp test specimen's electrothermal system consists of... A quartz lamp and The test piece consists of several test pieces, of which the first one... The temperature of the quartz lamp is , ;No. The temperature of each test piece was , The time variable is Combine all quartz lamp temperatures into a temperature vector. Organize the temperatures of all test pieces into a temperature vector. The control input in the electrothermal system is applied only to the quartz lamp channel. The control input for a quartz lamp is All quartz lamp control inputs constitute the input vector. And each control input satisfies the physical limiting constraint. .

[0042] It should be noted that the temperature change mechanism of the electric heating system is determined by both electrothermal energy conversion and heat transfer. The specific dynamic equation for the temperature of the quartz lamp is as follows:

[0043] ;

[0044] in, Let be the first derivative of the quartz lamp temperature with respect to time. This refers to the temperature rise caused by the electrical power input of the quartz lamp. For electrothermal energy conversion efficiency, For equivalent heat capacity, For the first The control input for a quartz lamp, The nonlinear terms arising from heat exchange between quartz lamps, between the quartz lamp and the test specimen, and between the quartz lamp and the environment. Let be the temperature state vector of the quartz lamp. Let be the temperature state vector of the test specimen. For the first The total disturbance term of the temperature channel of a quartz lamp.

[0045] The The nonlinear terms arising from heat exchange between quartz lamps, between quartz lamps and the test specimen, and between quartz lamps and the environment, and include at least the following components:

[0046] The radiative heat transfer term between quartz lamps is expressed as:

[0047] ;

[0048] in, This represents the total number of quartz lamps. The surface emissivity of the quartz lamp, The Stefan Boltzmann constant is given. The effective heat exchange area of ​​the quartz lamp. This refers to the viewing angle coefficient between quartz lamps. For the first The temperature of a quartz lamp, For the first The temperature of a quartz lamp.

[0049] The convective heat transfer term between quartz lamps is expressed as:

[0050] ;

[0051] in, The convective heat transfer coefficient between quartz lamps is given. For the first The temperature of a quartz lamp.

[0052] The heat transfer term between quartz lamps is expressed as:

[0053] ;

[0054] in, The equivalent thermal resistance between quartz lamps.

[0055] The radiative heat transfer term between the quartz lamp and the test specimen is expressed as:

[0056] ;

[0057] in, The viewing angle coefficient between the quartz lamp and the test specimen. For the first The temperature of each test piece.

[0058] The convective heat transfer term between the quartz lamp and the test specimen is expressed as:

[0059] ;

[0060] in, The total number of test pieces The convective heat transfer coefficient between the quartz lamp and the test specimen is given.

[0061] The heat transfer term between the quartz lamp and the test specimen is expressed as follows:

[0062] ;

[0063] in, The equivalent thermal resistance between the quartz lamp and the test piece.

[0064] The radiative heat transfer term between the quartz lamp and the environment is expressed as:

[0065] ;

[0066] in, The viewing angle coefficient between the quartz lamp and the environment. The ambient temperature.

[0067] The convective heat transfer term between the quartz lamp and the environment is expressed as:

[0068] ;

[0069] in, The convective heat transfer coefficient between the quartz lamp and the environment.

[0070] The heat transfer term between the quartz lamp and its base is represented as follows:

[0071] ;

[0072] in, The equivalent temperature of the quartz lamp base. The equivalent thermal resistance between the quartz lamp and the base.

[0073] Furthermore, the heat exchange power term mentioned above is divided by the equivalent heat capacity of the quartz lamp. The rate of temperature change is obtained such that the Units and Consistent.

[0074] It should also be noted that the specific dynamic equation for the temperature of the test specimen is as follows:

[0075] ;

[0076] in, The first derivative of the specimen temperature with respect to time. This is a nonlinear coupled heat transfer term determined by both the temperature state of the quartz lamp and the temperature state of the test specimen. For the first The total disturbance term of the temperature channel of each test piece.

[0077] The It is a nonlinear coupled heat transfer term determined by both the temperature state of the quartz lamp and the temperature state of the test specimen, and includes at least the following components:

[0078] The radiative heat transfer term between the test specimen and the quartz lamp is expressed as follows:

[0079] ;

[0080] in, The viewing angle coefficient between the quartz lamp and the test specimen.

[0081] The convective heat transfer term between the test specimen and the quartz lamp is expressed as:

[0082] ;

[0083] in, The convective heat transfer coefficient between the quartz lamp and the test specimen is given.

[0084] The heat transfer term between the test specimen and the quartz lamp is expressed as follows:

[0085] ;

[0086] in, The equivalent thermal resistance between the quartz lamp and the test piece.

[0087] The radiative heat transfer term between test specimens is expressed as:

[0088] ;

[0089] in, The surface emissivity of the test specimen, The effective heat transfer area of ​​the test specimen. For the first Temperature of each test piece.

[0090] The convective heat transfer term between test specimens is expressed as:

[0091] ;

[0092] in, The convective heat transfer coefficient between the test specimens is denoted as . The effective heat exchange area of ​​the test specimen.

[0093] The heat transfer term between test specimens is expressed as follows:

[0094] ;

[0095] in, The equivalent thermal resistance between test specimens.

[0096] The radiative heat transfer term between the test specimen and the environment is expressed as:

[0097] ;

[0098] in, The viewing angle coefficient between the test specimen and the environment.

[0099] The convective heat transfer term between the test specimen and the environment is expressed as:

[0100] ;

[0101] in, The convective heat transfer coefficient between the test specimen and the environment.

[0102] The heat transfer term between the test specimen and the base is expressed as follows:

[0103] ;

[0104] in, The equivalent temperature of the test specimen base. The equivalent thermal resistance between the test piece and the base.

[0105] Furthermore, the aforementioned heat transfer power term is divided by the equivalent heat capacity of the test specimen. The rate of temperature change is obtained such that the Units and Consistency is achieved, thereby completing the establishment of a dynamic model 100 of the electrothermal system, which includes the temperature status of the quartz lamp, the temperature status of the test specimen, and the control input of the quartz lamp.

[0106] It should also be noted that by constructing a dynamic model of the electrothermal system that includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp, a precise description of the dynamic temperature characteristics under complex coupling conditions can be achieved. This solves the modeling problems caused by multi-physical coupling, strong nonlinearity, and underactuated characteristics, and provides a theoretical basis for subsequent high-precision temperature control strategies.

[0107] S2: Based on the dynamic equation of the test piece temperature and the preset test piece temperature reference trajectory T, a sliding mode function 200 is constructed with the test piece temperature tracking error and time derivative as variables. A state dependence coefficient related to the temperature state of the quartz lamp is introduced to reflect the nonlinear coupling relationship between the two.

[0108] Specifically, based on the dynamic equation of the test piece temperature and the preset test piece temperature reference trajectory T, a sliding mode function 200 is constructed with the test piece temperature tracking error and its time derivative as variables. The sliding mode function 200 introduces a state dependence coefficient related to the temperature state of the quartz lamp, so that the structure of the sliding mode function 200 reflects the nonlinear coupling relationship between the quartz lamp temperature and the test piece temperature.

[0109] Furthermore, constructing the sliding mode function 200 with the temperature tracking error and time derivative of the test specimen as variables includes: setting a continuous and at least second-order differentiable temperature reference trajectory T for each test specimen; defining the temperature tracking error of each test specimen based on the actual temperature state of the test specimen and the temperature reference trajectory T; obtaining the first derivative of the error by taking the time derivative of the temperature tracking error according to the temperature dynamic equation of the test specimen; and constructing the corresponding sliding mode function 200 for each test specimen, expressed as:

[0110] ;

[0111] in, For the first Sliding mode variables of each test piece, For the first The first derivative of the temperature tracking error of each test piece The sliding mode coefficient function is related to the system state. For the first Temperature tracking error of individual test specimens.

[0112] For each test specimen, a continuous and at least second-order differentiable temperature reference trajectory T is established. Based on the actual temperature state of the test specimen and the temperature reference trajectory T, the first... The temperature tracking error for each test piece is:

[0113] ;

[0114] in, For the first Temperature tracking error of individual test pieces For the first Temperature reference trajectory of each test piece Indicates the first The test piece at time The actual temperature.

[0115] Furthermore, based on the dynamic equation of the test specimen temperature, the time derivative of the temperature tracking error is calculated to obtain the first derivative of the error:

[0116] ;

[0117] in, For the first The first derivative of the temperature tracking error of each test piece For the first First time derivative of each test piece The temperature dynamic equation of the test specimen is given, which includes a nonlinear coupled heat transfer term determined by both the temperature state of the quartz lamp and the temperature state of the test specimen.

[0118] Based on this, construct the first The sliding mode function 200 corresponding to each test piece is:

[0119] ;

[0120] in, For the first Sliding mode variables of each test piece, For the temperature of all quartz lamps , , , The temperature state vector of the quartz lamp. The temperature state vector of the test specimen is composed of the temperatures of all test specimens. This is a sliding mode coefficient function that is related to the system state.

[0121] It should be noted that the sliding mode coefficient function is constructed to be related to the temperature state of the quartz lamp, and its specific form is defined as follows:

[0122] ;

[0123] in, Positive design parameters The surface emissivity of the quartz lamp, The Stefan Boltzmann constant is given. The effective heat exchange area of ​​the quartz lamp. The viewing angle coefficient 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.

[0124] Each term in the sliding mode coefficient function originates from the dependence of the nonlinear coupled heat transfer term of the quartz lamp test piece on the quartz lamp temperature in the dynamic equation of the test piece temperature. In this way, the sliding mode function 200 simultaneously includes the test piece temperature tracking error and its time derivative, and its coefficients are adaptively adjusted as the quartz lamp temperature changes. Thus, the structure of the sliding mode function 200 directly reflects the nonlinear coupling characteristics between the quartz lamp temperature and the test piece temperature, providing a foundation for establishing a direct relationship between the sliding mode variables and the quartz lamp control input through subsequent sliding mode dynamic differentiation.

[0125] It should also be noted that by constructing an adaptive sliding mode coefficient function that is strongly correlated with the temperature state of the quartz lamp, the nonlinear time-varying thermal coupling characteristics between the quartz lamp and the test piece are directly embedded into the sliding mode function 200. This solves the problem of insufficient performance of traditional fixed parameter sliding mode control when there is a strong state-dependent coupling relationship, and lays a key foundation for achieving high-precision and robust test piece temperature tracking control.

[0126] S3: Differentiate the constructed sliding mode function 200 and combine it with the dynamic equations of quartz lamp temperature and test piece temperature to establish a direct mapping relationship 300 between sliding mode dynamics and control input.

[0127] Specifically, the sliding mode function 200 is differentiated over time. By utilizing the dependence of the test specimen temperature dynamic equation on the quartz lamp temperature, a sliding mode dynamic expression containing the rate of change of the quartz lamp temperature is obtained. Furthermore, by combining the quartz lamp temperature dynamic equation, the sliding mode dynamic expression is transformed into a linear algebraic relationship with respect to the quartz lamp control input, thereby establishing a direct mapping relationship between the sliding mode dynamic and the control input.

[0128] Furthermore, establishing a direct mapping relationship 300 between the sliding mode dynamics and the control input includes taking the time derivative of the sliding mode function 200, and using the partial derivative relationship of the test specimen temperature dynamic equation with respect to the quartz lamp temperature, expressing the second derivative of the test specimen temperature in a form that includes the rate of change of the quartz lamp temperature. Substituting this into the time derivative of the sliding mode function 200, the sliding mode dynamic expression containing the rate of change of the quartz lamp temperature is obtained. Combining this with the quartz lamp temperature dynamic equation, the sliding mode dynamic expression is transformed into a linear algebraic relationship with respect to the quartz lamp control input, expressed as:

[0129] ;

[0130] in, The first derivative of the sliding mode variable. For the state coupling gain The coupling matrix formed The input vector consists of the control inputs of each quartz lamp. It is a vector term consisting of the system state, reference trajectory, and disturbance term.

[0131] For the construction of the first The sliding mode function 200 corresponding to each test piece is used with respect to time. Taking the first derivative, the time derivative of the sliding mode function 200 is expressed as:

[0132] ;

[0133] in, The time derivative of the sliding mode function. This is the second time derivative of the temperature tracking error of the test specimen. This is the derivative of the state-dependent sliding mode coefficient with respect to time.

[0134] According to the definition of error:

[0135] ;

[0136] The second derivative of the error is obtained:

[0137] ;

[0138] in, The second derivative of the temperature of the test specimen. This is the second-order time derivative of the preset test specimen temperature reference trajectory T.

[0139] Further differentiating the dynamic equation of the temperature of the test specimen with respect to time, the second derivative of the temperature of the test specimen is expressed using the chain rule as follows:

[0140] ;

[0141] in, For the dynamic equation of temperature of the test specimen, the first The partial derivative of the temperature of a quartz lamp. For the first Partial derivative of temperature for each test piece, For the first The total disturbance term of the temperature channel of each test piece.

[0142] It should be noted that, based on the structure of the dynamic equation for the temperature of the test specimen, the analytical form of the partial derivative of the dynamic equation for the temperature of the test specimen with respect to the temperature of the quartz lamp is defined as follows:

[0143]

[0144] in, The equivalent heat capacity of the test specimen.

[0145] This allows the second derivative of the specimen temperature to include the rate of change of the quartz lamp temperature. .

[0146] The above Substituting the expression into the time derivative of the sliding mode function 100, we obtain the sliding mode dynamic expression that includes the rate of change of the quartz lamp temperature:

[0147] ;

[0148] in, For the first The time first derivative of the sliding mode function corresponding to each test piece This is the state coupling gain term between the quartz lamp test specimens. , The remaining term is composed of the temperature state of the test specimen, the rate of change of the temperature of the test specimen, the reference trajectory and its derivative, and the disturbance term.

[0149] It can be represented as:

[0150] ;

[0151] in, This represents the total number of test specimens.

[0152] Substituting the rate of temperature change of the quartz lamp into the sliding mode dynamic expression, we obtain:

[0153] ;

[0154] in, This represents the total number of quartz lamps. This is the state coupling gain term. For electrothermal energy conversion efficiency, For a known nonlinear term that does not contain control input, it can be specifically expressed as:

[0155] ;

[0156] in, For the first The total disturbance term in the temperature channel of a quartz lamp. For the first The first derivative of the temperature of a test specimen with respect to time.

[0157] By combining the sliding mode dynamic equations of all test pieces, a linear algebraic form of the sliding mode dynamics is constructed:

[0158] ;

[0159] in, The first derivative of the sliding mode variable. For the state coupling gain The coupling matrix formed The input vector consists of the control inputs of each quartz lamp. It is a vector term consisting of the system state, reference trajectory, and disturbance term.

[0160] This establishes a direct mapping relationship between sliding mode dynamics and quartz lamp control input 300, providing a linear algebraic basis for subsequent inverse calculation of quartz lamp control input based on sliding mode dynamics.

[0161] It should also be noted that by differentiating the sliding mode function and analyzing the thermally coupled dynamic equations, the nonlinear sliding mode dynamics are transformed into a linear algebraic relationship with respect to the quartz lamp control input. This establishes a clear mapping between the sliding mode dynamics and the control input in a complex system with strong nonlinearity and variable coupling, providing a model basis for solving the design problem of multi-channel cooperative control in underactuated systems.

[0162] S4: Based on the sliding mode function 200, a second-order sliding mode control law 400 is designed. An auxiliary state is introduced and a continuous second-order sliding mode arrival law C is constructed to make the sliding mode variables and derivatives converge in a finite time. The quartz lamp control input is obtained by inverse calculation based on the linear algebraic relationship, so that the test piece can track the temperature reference trajectory T under disturbance.

[0163] Specifically, based on the established sliding mode dynamic linear algebraic relationship, auxiliary state variables corresponding to each sliding mode variable are introduced. And combine them into an auxiliary state vector:

[0164] ;

[0165] in, For auxiliary state vectors, For the first Auxiliary state components of sliding mode variables for each test piece.

[0166] Furthermore, the second-order sliding mode control law 400 includes introducing auxiliary state variables corresponding to each sliding mode variable and forming an auxiliary state vector to construct a continuous second-order sliding mode arrival law C for each sliding mode variable corresponding to the test piece; the second-order sliding mode arrival law C adopts a continuous form containing fractional terms and sets a control gain function related to the system state.

[0167] For each test piece, the sliding mode variable Construct a continuous second-order sliding mode arrival law C, wherein the second-order sliding mode arrival law C adopts a continuous form containing fractional terms, and is specifically defined as follows:

[0168] ;

[0169] in, A positive control gain function It is a symbolic function.

[0170] The arrival law is continuous near the sliding surface, thereby avoiding high-frequency chattering of the control input.

[0171] The control gain function is constructed to be dependent on the system state, and its specific form is:

[0172] ;

[0173] ;

[0174] in, The design parameter is positive, so that the gain of the second-order sliding mode arrival law is adaptively adjusted according to the temperature state of the quartz lamp.

[0175] It should be noted that the second-order sliding mode control law 400 includes expressing the second-order sliding mode arrival law C in vector form:

[0176] ;

[0177] ;

[0178] in, The first derivative of the sliding mode variable. It is a diagonal matrix composed of the corresponding gains of each test piece. For each component The vector function formed For auxiliary state vectors, For the time derivative of the auxiliary state vector, It is the sliding mode vector composed of the sliding mode variables of each test piece.

[0179] In the above expression Specifically, it is expressed as follows:

[0180] ;

[0181] ;

[0182] ;

[0183] ;

[0184] in, For the first The absolute value of each sliding mode variable Power of 1 It is a diagonal matrix.

[0185] Furthermore, by comparing the desired sliding mode dynamics with the linear algebraic relationship of the sliding mode dynamics, and making them equal, we obtain the linear algebraic equation for the quartz lamp control input:

[0186] ;

[0187] in, The coupling matrix is ​​formed by the state coupling gains. The input vector consists of the quartz lamp control inputs.

[0188] Based on this, while satisfying the coupling matrix Under the condition of full row rank, its right inverse matrix is ​​constructed. By inversely calculating the quartz lamp control input, we obtain the expression for the quartz lamp control law:

[0189] ;

[0190] in, It is the right inverse of the coupling matrix.

[0191] Finally, the quartz lamp control input is applied to the underactuated quartz lamp test specimen electrothermal system, so that the sliding mode variables and their time derivatives corresponding to each test specimen converge to zero simultaneously within a finite time, thereby realizing the coordinated control between multiple quartz lamps and multiple test specimens, and ensuring the stability and robustness of the overall operation of the electrothermal system under the conditions of system nonlinear coupling and external disturbances.

[0192] It should also be noted that the stability of the quartz lamp control input has been verified:

[0193] Assumption explanation:

[0194] A1: For all , , , Bounded.

[0195] A2: , , Bounded.

[0196] A3: All , Bounded, therefore Bounded.

[0197] A4: Complete the rank, and thus It is invertible, and its smallest eigenvalue has a positive lower bound:

[0198] ;

[0199] in, It is the positive lower bound of the eigenvalues.

[0200] A5: There are lower and upper bounds for the constant:

[0201] ;

[0202] ;

[0203] ;

[0204] prove Reach the small neighborhood within a limited time.

[0205] Get the Lyapunov function:

[0206] ;

[0207] Among them, due to , Zhengding.

[0208] right Differentiate:

[0209] ;

[0210] Further Substituting into the above equation, we get:

[0211] ;

[0212] For any nonnegative , , If established; then let , ,have to:

[0213] ;

[0214] Further calculations have to:

[0215] ;

[0216] because , , Bounded, satisfying:

[0217] ;

[0218] ;

[0219] ;

[0220] Further calculations have to:

[0221] ;

[0222] For any Using Young's inequality, we get:

[0223] ;

[0224] make Further calculations have to:

[0225] ;

[0226] in, .

[0227] By Hölder's inequality, for ,but .exist Take the minimum value under different circumstances:

[0228] ;

[0229] Further calculations have to:

[0230] ;

[0231] Among them, due to ,make ,but ,so Taking the power of 2 / 3 further, we get:

[0232] .

[0233] Further calculations have to:

[0234] ;

[0235] in, , , , This allows the error to converge to a small neighborhood.

[0236] Proof on the sliding surface Upper error .

[0237] When on the sliding surface , The error dynamics on the sliding surface are given: ,because ,and ,therefore: .

[0238] Get the Lyapunov function:

[0239] ;

[0240] Further First derivative:

[0241] ;

[0242] in, ,therefore , .

[0243] It should also be noted that by designing an adaptive second-order sliding mode control law based on a continuous fractional-order arrival law, the quartz lamp control input is solved by using the established linear mapping relationship, eliminating the high-frequency chattering of traditional sliding mode control, and realizing the precise tracking of the test piece temperature under strong nonlinear coupling and disturbance in finite time, which significantly improves the dynamic quality and control robustness of the system.

[0244] Example 2, refer to Figures 2-3 As an embodiment of the present invention, a state-coupled sliding mode control method for an underdriven electrothermal system is provided. To verify the beneficial effects of the present invention, scientific demonstration is carried out through economic benefit calculations and simulation experiments.

[0245] First, a nonlinear electrothermal coupling dynamic model of two quartz lamps and two test specimens was established under the same system structure and parameters, with uniform initial temperature, reference trajectory, physical constraints, and external disturbance forms. Then, the proposed method and the comparative method were implemented on these models, with all model parameters, reference input, disturbance settings, and control input limiting conditions remaining identical except for the control law structure. Next, the closed-loop system was simulated using numerical integration, recording the temperature change of the test specimen over time under both methods. Finally, the control effects of the two methods were compared and analyzed using indicators such as settling time, overshoot, and steady-state tracking error as evaluation criteria, thus objectively verifying the improved dynamic response speed and robustness of the proposed method.

[0246] Set the simulation parameters as follows:

[0247] , Simulation time is Initial temperature , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

[0248] The common sliding mode arrival law is: , , , .

[0249] like Figure 2 As shown, the temperature tracking process of the two test pieces under the method of the present invention can be seen. and The initial temperature rises rapidly and gradually conforms to their respective reference trajectories. , Throughout the entire process, there was no significant overshoot or oscillation. The curve essentially coincided with the reference curve in the middle and later stages, and the tracking error tended to be very small in the steady-state phase. This demonstrates that the closed-loop structure constructed in this invention, which utilizes a state-dependent sliding surface, a second-order sliding mode arrival law C, and an inverse control input, can maintain a smooth and stable tracking transition process even in the presence of nonlinear coupled heat transfer and disturbances.

[0250] like Figure 3 As shown, the tracking results are compared using the conventional sliding mode method. It can be observed that... During the upward phase, there was a significant overshoot (peaking above 400K), followed by a pullback and a prolonged retesting process; simultaneously The tracking also exhibited more pronounced transition bias and convergence tailing, with the reference trajectory deviating more significantly from the actual temperature curve in the early to mid-stages. This phenomenon indicates that, under the same model, parameters, disturbances, and control input inverse mapping conditions, the ordinary sliding mode arrival law is more likely to introduce stronger transient shocks and transition oscillations, thus leading to a decrease in tracking quality.

[0251] In summary, the present invention outperforms the comparative method in both dynamic and steady-state performance. Using the test specimen temperature entering and remaining within the ±1K error band of the reference trajectory as the settling time criterion, the settling time of the present invention is approximately 120–150 s, while that of the comparative method is approximately 220–260 s, representing a reduction of approximately 35%–45%. Regarding overshoot, the test specimen temperature of the present invention exhibits virtually no overshoot or only a very small overshoot (less than 1%), while the comparative method shows significant overshoot in the initial stage of heating. In the steady-state phase, the steady-state tracking error of the present invention converges to within ±0.2–0.3 K, while the steady-state error of the comparative method is approximately ±0.6–1.0 K. These results demonstrate that the present invention has significant advantages in convergence speed, overshoot suppression, and steady-state accuracy.

[0252] The comparison results show that, under completely identical system parameters, reference trajectory, disturbance settings, and input limiting conditions, the method of this invention exhibits superior dynamic and steady-state performance compared to the comparative method. On the one hand, it significantly suppresses overshoot and oscillation, allowing the temperatures of both test specimens to more smoothly approach the reference trajectory; on the other hand, it shortens the time to enter the steady-state alignment range, resulting in smaller and more consistent steady-state tracking errors. This demonstrates that by introducing a sliding surface coefficient related to the lamp temperature state and employing a second-order sliding mode arrival mechanism, this invention can more fully utilize the nonlinear coupling relationship between the quartz lamp and the test specimen, improving the tracking accuracy and robustness of the underdriven electrothermal system under disturbance and uncertainty conditions.

[0253] Example 3, an embodiment of the present invention, provides a state-coupled sliding mode control system for an underactuated electrothermal system, including a dynamic modeling module, a sliding mode function construction module, a mapping relationship construction module, and a control law design and implementation module.

[0254] The dynamic modeling module is used to construct a dynamic model 100 of the electrothermal system, which includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp. The dynamic equation of the quartz lamp temperature includes an electric power input term and a nonlinear term generated by heat exchange between quartz lamps, between the quartz lamp and the test piece, and between the quartz lamp and the environment. The dynamic equation of the test piece temperature includes a nonlinear coupled heat transfer term determined by the temperature of the quartz lamp and the temperature of the test piece.

[0255] The sliding mode function construction module is used to construct a sliding mode function 200 based on the dynamic equation of the test specimen temperature and the preset test specimen temperature reference trajectory T, with the test specimen temperature tracking error and its time derivative as variables. The sliding mode function 200 introduces a state dependence coefficient related to the temperature state of the quartz lamp, so that the structure of the sliding mode function 200 reflects the nonlinear coupling relationship between the quartz lamp temperature and the test specimen temperature.

[0256] The mapping relationship construction module is used to perform time derivative of the sliding mode function 200. By utilizing the dependence of the test piece temperature dynamic equation on the quartz lamp temperature, the sliding mode dynamic expression containing the rate of change of the quartz lamp temperature is obtained. Furthermore, by combining the quartz lamp temperature dynamic equation, the sliding mode dynamic expression is transformed into a linear algebraic relationship with respect to the quartz lamp control input.

[0257] The control law design and implementation module is used to design a second-order sliding mode control law 400 based on the sliding mode dynamic equation. By introducing auxiliary states and constructing a continuous second-order sliding mode arrival law C, the sliding mode variables and derivatives converge. At the same time, the quartz lamp control input is obtained by inverse calculation based on the linear algebra relationship and applied to the electrothermal system.

[0258] This embodiment also provides a computer device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it implements the state-coupled sliding mode control method for the underactuated electrothermal system proposed in the above embodiment.

[0259] This embodiment also provides a computer-readable storage medium storing a computer program thereon. When the computer program is executed by a processor, it implements the state-coupled sliding mode control method for the underactuated electrothermal system proposed in the above embodiment.

[0260] If the aforementioned functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, essentially, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0261] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-including system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device.

[0262] More specific examples of computer-readable media (a non-exhaustive list) include: electrical connections (electronic devices) having one or more wires, portable computer disk drives (magnetic devices), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Furthermore, computer-readable media can even be paper or other suitable media on which the program can be printed, because the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in computer memory.

[0263] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0264] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A state-coupled sliding mode control method for an underactuated electrothermal system, characterized in that, include: A dynamic model (100) of the electrothermal system is constructed, which includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp. The dynamic equation of the quartz lamp temperature includes the power input term and the heat exchange nonlinear term, and the dynamic equation of the test piece temperature includes the nonlinear coupled heat transfer between the quartz lamp and the test piece temperature. Based on the dynamic equation of the test piece temperature and the preset test piece temperature reference trajectory (T), a sliding mode function (200) with the test piece temperature tracking error and time derivative as variables is constructed, and a state dependence coefficient related to the temperature state of the quartz lamp is introduced to reflect the nonlinear coupling relationship between the two. By differentiating the constructed sliding mode function (200) and combining the temperature dynamic equation of the quartz lamp and the temperature dynamic equation of the test piece, a direct mapping relationship between the sliding mode dynamic and the control input is established (300). Based on the sliding mode function (200), a second-order sliding mode control law (400) is designed. An auxiliary state is introduced and a continuous second-order sliding mode arrival law (C) is constructed so that the sliding mode variables and derivatives converge in a finite time. The quartz lamp control input is obtained by inverse calculation based on the linear algebraic relationship, so that the test piece tracks the temperature reference trajectory (T) under disturbance. The sliding mode function (200) constructed using the temperature tracking error and time derivative of the test specimen as variables includes, For each test piece, a continuous and at least second-differentiable temperature reference trajectory (T) is established. Based on the actual temperature state of the test piece and the temperature reference trajectory (T), the temperature tracking error of each test piece is defined. According to the temperature dynamic equation of the test piece, the time derivative of the temperature tracking error is obtained to obtain the first derivative of the error. Based on this, the corresponding sliding mode function (200) for each test piece is constructed, expressed as: ， in, For the first Sliding mode variables of each test piece, For the first The first derivative of the temperature tracking error of each test piece The sliding mode coefficient function is related to the system state. For the first Temperature tracking error of individual test specimens; The establishment of a direct mapping relationship between sliding mode dynamics and control input (300) includes, The sliding mode function (200) is differentiated over time, and the second derivative of the test specimen temperature with respect to the quartz lamp temperature is expressed using the partial derivative relationship of the test specimen temperature dynamic equation. This second derivative of the test specimen temperature is then expressed in a form that includes the rate of change of the quartz lamp temperature. Substituting this into the time derivative of the sliding mode function (200), the sliding mode dynamic expression containing the rate of change of the quartz lamp temperature is obtained. Combined with the quartz lamp temperature dynamic equation, the sliding mode dynamic expression is transformed into a linear algebraic relationship with respect to the quartz lamp control input, expressed as: ， in, The first derivative of the sliding mode variable. For the state coupling gain The coupling matrix formed The input vector consists of the control inputs of each quartz lamp. It is a vector term consisting of the system state, reference trajectory, and disturbance term.

2. The state-coupled sliding mode control method for an underdriven electrothermal system as described in claim 1, characterized in that: The dynamic model (100) of the electrothermal system includes, The underdriven quartz lamp test specimen heating system consists of A quartz lamp and The test piece consists of a temperature vector composed of all the quartz lamp temperatures and the temperature vector composed of all the test piece temperatures. The control input in the electrothermal system is applied only to the quartz lamp channel, and the control input of all the quartz lamps is composed of an input vector. The dynamic model (100) of the electrothermal system is constructed based on the temperature change mechanism jointly determined by electrothermal energy conversion and heat transfer, including multiple heat exchange mechanisms and corresponding physical coefficients.

3. The state-coupled sliding mode control method for an underactuated electrothermal system as described in claim 1 or 2, characterized in that: The dynamic equation for the temperature of the quartz lamp includes, ， in, Let be the first derivative of the quartz lamp temperature with respect to time. This refers to the temperature rise caused by the electrical power input of the quartz lamp. For electrothermal energy conversion efficiency, For equivalent heat capacity, For the first The control input for a quartz lamp, The nonlinear terms arising from heat exchange between quartz lamps, between the quartz lamp and the test specimen, and between the quartz lamp and the environment. Let be the temperature state vector of the quartz lamp. Let be the temperature state vector of the test specimen. For the first The total disturbance term of the temperature channel of a quartz lamp.

4. The state-coupled sliding mode control method for an underactuated electrothermal system as described in claim 3, characterized in that: The dynamic equation for the temperature of the test specimen includes, ， in, The first derivative of the specimen temperature with respect to time. This is a nonlinear coupled heat transfer term determined by both the temperature state of the quartz lamp and the temperature state of the test specimen. For the first The total disturbance term of the temperature channel of each test piece.

5. The state-coupled sliding mode control method for an underdriven electrothermal system as described in claim 1, 2, or 4, characterized in that: The second-order sliding mode control law (400) includes, An auxiliary state variable corresponding to each sliding mode variable is introduced and formed into an auxiliary state vector to construct a continuous second-order sliding mode arrival law (C) for the sliding mode variable corresponding to each test piece. The second-order sliding mode arrival law (C) adopts a continuous form containing fractional terms and sets a control gain function that is related to the system state.

6. The state-coupled sliding mode control method for an underactuated electrothermal system as described in claim 5, characterized in that: The second-order sliding mode control law (400) includes, The second-order sliding mode arrival law (C) is expressed in vector form: ， ， in, The first derivative of the sliding mode variable. It is a diagonal matrix composed of the corresponding gains of each test piece. It is a vector function composed of its components. For auxiliary state vectors, For the time derivative of the auxiliary state vector, It is the sliding mode vector composed of the sliding mode variables of each test piece.

7. A state-coupled sliding mode control system for an underactuated electrothermal system, employing the state-coupled sliding mode control method for an underactuated electrothermal system as described in any one of claims 1 to 6, characterized in that: It includes a dynamic modeling module, a sliding mode function construction module, a mapping relationship construction module, and a control law design and implementation module; The dynamic modeling module is used to construct a dynamic model (100) of an electrothermal system that includes the temperature state of the quartz lamp, the temperature state of the test piece, and the control input of the quartz lamp. The dynamic equation of the quartz lamp temperature includes an electric power input term and a nonlinear term generated by heat exchange between quartz lamps, between the quartz lamp and the test piece, and between the quartz lamp and the environment. The dynamic equation of the test piece temperature includes a nonlinear coupled heat transfer term determined by the temperature of the quartz lamp and the temperature of the test piece. The sliding mode function construction module is used to construct a sliding mode function (200) with the temperature tracking error of the test piece and its time derivative as variables based on the dynamic equation of the test piece temperature and the preset test piece temperature reference trajectory (T). The sliding mode function (200) introduces a state dependence coefficient related to the temperature state of the quartz lamp, so that the structure of the sliding mode function (200) reflects the nonlinear coupling relationship between the temperature of the quartz lamp and the temperature of the test piece. The mapping relationship construction module is used to perform time derivative of the sliding mode function (200), and obtain the sliding mode dynamic expression containing the rate of change of quartz lamp temperature by using the dependence of the test piece temperature dynamic equation on the quartz lamp temperature. Furthermore, by combining the quartz lamp temperature dynamic equation, the sliding mode dynamic expression is transformed into a linear algebraic relationship with respect to the quartz lamp control input. The control law design and implementation module is used to design a second-order sliding mode control law (400) based on the sliding mode dynamic equation. By introducing auxiliary states and constructing a continuous second-order sliding mode arrival law (C), the sliding mode variables and derivatives converge. At the same time, the quartz lamp control input is obtained by inverse calculation based on the linear algebra relationship and applied to the electrothermal system.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the state-coupled sliding mode control method for the underactuated electrothermal system as described in any one of claims 1 to 6.

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