A distributed diffusion control method for a wide and thick plate hot working furnace flame nozzle

Abstract

The application discloses a distributed diffusion control method for a wide-thick plate hot processing furnace flame nozzle, and comprises the following steps: constructing a distributed parameter system based on temperature space-time transformation in the wide-thick plate hot processing furnace; combining a wide-thick plate hot processing target temperature curve to obtain a distributed parameter error system model; constructing a wide-thick plate hot processing furnace flame nozzle distributed diffusion controller; the distributed diffusion controller acts on the distributed parameter error system model to obtain a distributed parameter system closed-loop control model; based on Lyapunov stability theory, a sufficient condition for realizing gradual stability control of the distributed parameter error system model is solved; based on wide-thick plate hot processing furnace system parameters, the sufficient condition is combined to numerically calculate and solve distributed diffusion controller parameters; numerical simulation is carried out to verify the effectiveness of the control scheme. The application adopts the distributed diffusion control scheme, realizes control on the whole distributed parameter system space through energy transmission between space nodes, and the control effect is better.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of heating furnace, in particular to a distributed diffusion control method for flame nozzle of wide and heavy plate hot working furnace. BACKGROUND

[0002] Wide and heavy plate is widely used in the fields of ship, bridge, pressure vessel, engineering machinery, etc., and its production process involves multiple links such as smelting, rolling, heat treatment, etc., and the process needs to be strictly controlled to ensure uniformity of performance and minimization of defects. In the heat treatment process of wide and heavy plate, the core thermal equipment, the hot working furnace, undertakes the key task of heating the slab to the target process temperature and maintaining the necessary uniformity.

[0003] The wide and heavy plate hot working process needs to control the organization evolution and performance through accurate temperature-time curve, and scientific control of the heating process is the key to ensure the quality of the final product. At the same time, the heating uniformity of wide and heavy plate billet has a great influence on the performance of steel, if the heating is not uniform, the wide and heavy plate formed by rolling will have many pitting, non-metallic inclusion, crack, surface quality problems such as alligatoring, which seriously affects the performance of steel. Therefore, the wide and heavy plate hot working furnace needs to achieve the following in actual engineering: ① the furnace temperature state continuously fits the wide and heavy plate hot working target temperature-time curve; ② the furnace temperature state is stably and uniformly distributed.

[0004] Considering the actual engineering, in the hot working process of wide and heavy plate hot working furnace, the range of action of flame nozzle is difficult to directly cover every area of the hot working furnace, that is, in actual engineering, there is a part of space in the wide and heavy plate hot working furnace, and the temperature regulation of this part of space is controlled by the energy diffusion process of other areas. However, in the existing related researches, the researchers' research focus is more on designing different control schemes to realize the stable furnace temperature of the wide and heavy plate hot working furnace, and to meet the fitting of the hot working target temperature-time curve; few researchers focus on the direct control range of the flame nozzle in the control implementation process, that is, the researchers set all the space of the wide and heavy plate hot working furnace as the directly controllable space in the process of model construction, and ignore the non-directly controllable area. The model constructed in this way will greatly affect the accuracy of the description of the temperature and space variation law of the wide and heavy plate hot working furnace, and will further lead to a significant reduction in the control precision and stability performance.

[0005] Meanwhile, in the process of controlling the temperature of the wide-thick plate hot working furnace, the process of the output energy of the flame nozzle is a heat diffusion process (i.e. the space-time transformation process of the distributed parameter system), and the time delay factor in the heat energy transmission process is considered, so in the design process of the control scheme, the actual control process of the flame nozzle is needed to be considered, and the time delay factor is considered, and then the control scheme is constructed to realize the designed control scheme to fit the actual engineering. However, in the existing research, the researchers usually directly control and adjust the system state detected by the temperature sensing system; such scheme does not consider the actual implementation process of the flame nozzle of the wide-thick plate hot working furnace, so that the designed control scheme is difficult to accurately realize in the implementation process, and further seriously affects the control precision and stability performance of the control system. SUMMARY

[0006] In order to solve the above technical problems, the present application provides a distributed diffusion control method for the flame nozzle of the wide-thick plate hot working furnace, which has simple algorithm and high control precision.

[0007] The technical scheme for solving the above technical problems is: a distributed diffusion control method for the flame nozzle of the wide-thick plate hot working furnace, comprising the following steps:

[0008] S1: based on the theory of distributed parameter system, the internal space of the wide-thick plate hot working furnace is divided into subspaces, and based on whether the control device exists in the subspace, the distributed parameter system based on the temperature space-time transformation in the wide-thick plate hot working furnace is constructed based on the temperature transmission process in the subspace of the wide-thick plate hot working furnace, and the distributed parameter system includes two parts of controllable node subspace distributed parameter system and diffusion node subspace distributed parameter system;

[0009] In the step S1, based on the installation position of the flame nozzle in the wide-thick plate hot working furnace, the actual engineering structure of the wide-thick plate hot working furnace is considered; the direct effect of the flame nozzle is considered, and the internal space of the wide-thick plate hot working furnace is divided into subspaces, wherein subspaces contain the flame nozzle; each subspace is defined as a space node, and the subspaces containing the flame nozzle can directly adjust the temperature of the space node through the flame nozzle, and the space node containing the flame nozzle is defined as the controllable node subspace; and there are The space node without flame nozzle is defined as a diffusion node subspace; the diffusion node subspace realizes temperature control through heat transfer process between space nodes; considering the time-varying energy transfer in each subspace inside the furnace and the space-time characteristics of energy interaction between each space node, the wide and thick plate hot working furnace temperature system with space-time characteristics is defined as a distributed parameter system, that is, the space temperature transformation in the wide and thick plate hot working furnace is modeled and described by using a distributed parameter system;

[0010] S2: combining the target temperature curve of the wide and thick plate hot working, a distributed parameter error system model of the temperature change state of each subspace of the wide and thick plate hot working furnace and the target temperature state is obtained; the distributed parameter error system model includes two parts of a controllable node subspace distributed parameter error system model and a diffusion node subspace distributed parameter error system model;

[0011] S3: a distributed diffusion controller of the flame nozzle of the wide and thick plate hot working furnace is constructed to realize stable control of the distributed parameter error system model, and the distributed diffusion controller includes a controller system state generation sub-controller and a state feedback output sub-controller;

[0012] S4: the distributed diffusion controller of the flame nozzle of the wide and thick plate hot working furnace is applied to the controllable node subspace distributed parameter error system model, and the diffusion node subspace distributed parameter error system model is combined to obtain a distributed parameter system closed-loop control model;

[0013] S5: based on Lyapunov stability theory, a sufficient condition for realizing gradual stable control of the distributed parameter error system model is obtained;

[0014] S6: based on the system parameters of the wide and thick plate hot working furnace, the sufficient condition is combined to numerically calculate and solve the parameters of the distributed diffusion controller;

[0015] S7: numerical simulation is performed to verify the effectiveness of the control scheme.

[0016] The above-mentioned distributed diffusion control method of the flame nozzle of the wide and thick plate hot working furnace, in the step S1, the space-time change state of the space temperature distribution in the first represents the space variable; is a time variable, considering the space-time transformation characteristics of the internal space temperature of the wide and thick plate hot working furnace and the specific structure of each subspace node, a distributed parameter system is constructed to describe the temperature change rule of each subspace in the hot working furnace as follows:

[0017] The controllable node subspace distributed parameter system model is:

[0018] (1)​​

[0019] The diffusion node subspace distribution parameter system model is:

[0020] (2)

[0021] wherein, represents the spatial temperature distribution spatio-temporal variation state in the th subspace, represents the spatio-temporal region in the wide and thick plate hot working furnace, ; represents the positive real number space; is a bounded region with smooth boundary , i.e. the inner space of the wide and thick plate hot working furnace, , represents the Euclidean norm, is a numerical parameter satisfying the condition, represents positive infinity; , represents a one-dimensional real number space, represents a real number matrix space of dimension, represents the number of spatial dimensions; and , represents a spatial dimension measurement function, represents that the dimension required is greater than zero; is the temperature diffusion coefficient of the th subspace, ; is the distribution parameter system control input state of the th subspace; let , represent the temperature diffusion laplace operator inside the subspace; represents the total number of dimensions in space, represents the spatial dimension ordinal number, represents the dimension variable in space; is the temperature transfer time lag amount; represents the influence factor of the th subspace temperature distribution state on the th subspace temperature distribution state, represents the influence factor of the th subspace temperature distribution time lag state on the th subspace temperature distribution state; , represents the energy released by the th subspace to the th subspace; , represents the first subspace absorbs energy from the first subspace.

[0022] The distributed diffusion control method of the flame nozzle of the wide plate hot working furnace, in the step S2, the target temperature curve of the wide plate hot working is set to satisfy:

[0023] (3)

[0024] wherein represents a spatial target temperature variation function of the distributed parameter system; represents a time-varying rate function of the spatial target temperature of the distributed parameter system;

[0025] To achieve the control target, the error space-time variation state between the first subspace temperature variation function and the target temperature is set as , satisfying:

[0026] (4)

[0027] Combined with formula (1), formula (2) and formula (4), the following distributed parameter error system model can be obtained:

[0028] The controllable node subspace distributed parameter error system model is:

[0029] (5)

[0030] The diffusion node subspace distributed parameter error system model is:

[0031] (6)

[0032] wherein, represents an influence factor of the first subspace temperature error state on the first subspace temperature error state, represents an influence factor of the first subspace temperature error time delay state on the first subspace temperature error state;

[0033] When , it satisfies: , ;

[0034] When , it satisfies: , .

[0035] The distributed diffusion control method of the flame nozzle of the wide-thick plate hot working furnace, in the step S2, sets two control targets, the first control target is to achieve the furnace temperature of the wide-thick plate hot working furnace to continuously match the wide-thick plate hot working target temperature in engineering; the second control target is to achieve the uniform distribution of the furnace temperature of the wide-thick plate hot working furnace;

[0036] In the theoretical solving process, when the controllable node subspace distribution parameter error system model and the diffusion node subspace distribution parameter error system model reach the asymptotic stability, that is, the furnace temperature of the wide-thick plate hot working furnace is uniformly distributed; at the same time, when the distribution parameter error system model reaches the asymptotic stability, the following formula is satisfied According to the definition of in formula (4), when , the furnace temperature of the wide-thick plate hot working furnace continuously matches the wide-thick plate hot working target temperature; and when the distribution parameter error system model reaches the asymptotic stability under the action of the controller, the two control targets set are achieved.

[0037] In order to simplify the description, the distribution parameter error system model is rewritten in the form of a matrix as follows:

[0038] (7)

[0039] Wherein, represents a distribution parameter error matrix variable with a dimension of , , , the superscript represents the transpose of the matrix; represents a distribution parameter error matrix variable with a dimension of , , ; represents a matrix function with a dimension of composed of the function , , ; represents a matrix function with a dimension of composed of the function , , ; , both represent diagonal matrices, , represent the identity of the diagonal matrix, ; represents a control input matrix variable composed of the control input state of nodes, , ; represents a zero matrix; Indicates by The dimensions of the composition are The matrix parameters, , The dimension represented by the elements within the parentheses is... matrix; Indicates by The dimensions of the composition are The matrix parameters, And satisfy: , ; This is the coupling matrix between the temperature error state of the controllable node space and the temperature error state of the controllable node space. This is the coupling matrix between the time delay state of the controllable node space temperature error and the controllable node space temperature error state; This is the coupling matrix between the temperature error state of the controllable node space and the temperature error state of the diffuse node space; This is the coupling matrix between the time-delay state of the controllable node space temperature error and the state of the diffuse node space temperature error; The coupling diffusion matrix of the spatial temperature error state of the diffusion node to the spatial temperature error state of the diffusion node; This is the coupling diffusion matrix of the spatial temperature error time delay state of the diffusion node to the spatial temperature error state of the diffusion node.

[0040] In the above-mentioned distributed diffusion control method for the flame nozzle of a hot working furnace for wide and thick plates, in step S3, the internal structure of the hot working furnace is used as the control method. Based on the thermal diffusion of the flame nozzle within a subspace, the following distributed diffusion controller is designed:

[0041] Controller system state generation sub-controller:

[0042] (8)

[0043] Status feedback output sub-controller:

[0044] (9)

[0045] in, Indicates the first The basic state variables of the subspace controller system; The diffusion coefficient represents the state of the control system. Indicates the first The state of the spatial control system at node i affects the first The coupling influence coefficient of the state of a node-space control system; Indicates the first The spatial temperature error state of the nth node affects the... The coupling influence coefficient of the state of a control system; represents the coupling influence coefficient of the error system state of the first controllable node on the state of the first control system; represents the feedback gain of the control system;

[0046] The constructed distributed diffusion controller does not directly control all subspaces inside the wide and thick plate hot working furnace. The controller only directly acts on the controllable subspace inside the wide and thick plate hot working furnace, and realizes the furnace temperature control target of the wide and thick plate hot working furnace through energy diffusion between the subspace inside the wide and thick plate hot working furnace.

[0047] The above-mentioned distributed diffusion control method of the flame nozzle of the wide and thick plate hot working furnace, in step S3, the distributed diffusion controller is changed to a matrix model form:

[0048] (10)

[0049] wherein, represents the state matrix of the control system, and satisfies ; ; is the diffusion coefficient matrix of the control system, which is composed of , and satisfies: ; is the feedback gain matrix of the control system, and satisfies: ; represents the coupling influence coefficient matrix between the states of the control system, and satisfies: ; represents the coupling influence coefficient matrix between the states of the controllable node error system, and satisfies: ; represents the coupling influence coefficient matrix between the time delay states of the controllable node error system, and satisfies: ; represents the coupling influence coefficient matrix between the states of the diffusion node error system, and satisfies: ; represents the coupling influence coefficient matrix between the time delay states of the diffusion node error system, and satisfies .

[0050] The above-mentioned distributed diffusion control method of the flame nozzle of the wide and thick plate hot working furnace, in step S4, the formula (7) and the formula (9) are combined to obtain a distributed parameter system closed-loop control model:

[0051] (11)

[0052] wherein, represents the state matrix of the closed-loop control system, and satisfies: ; represents a diffusion coefficient matrix of the closed-loop control system, satisfying: , ; represents a coupling and control parameter matrix of the closed-loop control system, satisfying: , ; represents a time delay coupling and control parameter matrix of the closed-loop control system, satisfying: , ; represents a target control temperature transformation matrix of the closed-loop control system, satisfying: ;

[0053] Based on the actual engineering of the wide and thick plate hot working furnace, considering that the control system model and the distributed parameter error system model both satisfy the unified boundary condition, the initial boundary condition of the distributed parameter system closed-loop control model is set as:

[0054] , (12)

[0055] or:

[0056] , (13)

[0057] , (14)

[0058] wherein is a unit outer normal vector of , is a smooth function.

[0059] The distributed diffusion control method of the wide and thick plate hot working furnace flame nozzle, in the step S5, based on the Lyapunov stability theory, a sufficient condition for realizing the gradual stability control of the distributed parameter error system model is solved;

[0060] Conclusion 1: Given the system parameters and the Lipschitz condition parameters , if there exist , , , , , , satisfying the following inequalities, the distributed parameter system closed-loop control model is gradually stable.

[0061] (15)

[0062] (16)

[0063] wherein , , are intermediate variable matrices; , , are Lyapunov functional parameter matrices, ; ; ; is a Lyapunov functional parameter matrix; intermediate variable matrix ; intermediate variable matrix ; intermediate variable matrix ; intermediate variable matrix ; intermediate variable matrix inverse matrix of ; ; ; ; ; ; ; ; ; ; denotes the identity matrix; both formulae (15) and (16) are symmetric matrices, denotes the symmetric term of a symmetric matrix; the superscript -1 in ( ) denotes the inverse matrix, denotes the inverse matrix of .

[0064] The above-mentioned distributed diffusion control method for the flame nozzle of the heavy plate hot working furnace, in order to derive the sufficient condition, the following assumptions and lemmas are given in step S5:

[0065] Assumption 1: for , the following Lipschitz condition is satisfied:

[0066] (17)

[0067] wherein , ; denotes the system state matrix;

[0068] Lemma 1: let be a smooth subdomain, , if the variable , then:

[0069] (18)

[0070] wherein denotes Hamiltonian operator; denotes an area element of the boundary region; denotes a set of twice continuously differentiable functions in a smooth subdomain;

[0071] Lemma 2: For a given symmetric matrix , , , , is a matrix term of , the following three conditions are equivalent:

[0072] Condition 1: ;

[0073] Condition 2: ;

[0074] Condition 3: ;

[0075] By using Lyapunov stability theory, a sufficient condition for the existence of a distributed parameter system gradually stable distributed diffusion controller is obtained, and according to the sufficient condition, the control parameters of the distributed diffusion controller are solved, so that the distributed parameter system reaches gradually stable under the action of the distributed diffusion controller, that is, the temperature of each subspace in the wide and thick plate hot processing furnace reaches the target control temperature.

[0076] The beneficial effects of the present application are:

[0077] 1、The present application adopts a distributed diffusion control scheme, considering that the direct control area of the flame nozzle of the wide and thick plate hot processing furnace is limited, and all space areas cannot be directly controlled by the flame nozzle in actual engineering, therefore, the present application constructs a diffusion control scheme, the constructed controller is not aimed at all space nodes of the system, but only controls part of the space nodes, and the control of the entire distributed parameter system space (the space of the wide and thick plate hot processing furnace) is realized through energy transmission between the space nodes. In the design process, the direct thermal control area of the flame nozzle in the wide and thick plate hot processing furnace and the diffusion area are partitioned and modeled, so as to realize more accurate model construction, and then realize more accurate and stable control of the furnace temperature in the process of furnace temperature control of the wide and thick plate hot processing furnace.

[0078] 2, The application considers that when the flame nozzle outputs the flame, the flame energy has a heat transfer process (a distributed parameter system category), that is, in actual engineering, the flame nozzle cannot directly feedback the output to reach the target control state according to the temperature state in the space of the wide and thick plate hot working furnace, so, the application constructs a distributed diffusion control scheme, the control model includes two parts, that is, a controller system state generation sub-controller and a state feedback output sub-controller, and the state of the distributed parameter system control model is constructed according to the space energy diffusion process to carry out feedback control. The innovation of the design lies in that the designed distributed diffusion controller does not act on all subspaces of the wide and thick plate hot working furnace, but only acts on the internal controllable node subspaces of the wide and thick plate hot working furnace; the basic feedback control state of the constructed distributed diffusion controller is not the traditional system state, but the control system state generator considering the space energy diffusion process of the flame nozzle, and the state of the control system state generation sub-controller is taken as the basic state; the control gain is designed based on the control state basic variable, and the feedback output sub-controller is constructed; then the feedback output sub-controller acts on the controllable node subspace to realize direct adjustment control of the distributed parameter error system state of the controllable node subspace. In the diffusion node subspace, the space temperature error state in the diffusion node subspace is adjusted through the energy transfer coupling between the controllable node subspace and the diffusion node subspace; the control scheme designed by the application can be closer to the actual engineering change of the flame energy output of the flame nozzle, so that the control precision of the control system is more accurate, and the control performance is more stable. BRIEF DESCRIPTION OF DRAWINGS

[0079] Figure 1 The whole flow chart of the application.

[0080] Figure 2 The temperature error state change trend chart of the first controllable node subspace in the numerical simulation.

[0081] Figure 3 The temperature error state change trend chart of the second controllable node subspace in the numerical simulation.

[0082] Figure 4 The temperature error state change trend chart of the third controllable node subspace in the numerical simulation.

[0083] Figure 5 The temperature error state change trend chart of the fourth controllable node subspace in the numerical simulation.

[0084] Figure 6 The temperature error state change trend chart of the first diffusion node subspace in the numerical simulation.

[0085] Figure 7 The temperature error state change trend chart of the second diffusion node subspace in the numerical simulation.

[0086] Figure 8 This is a trend diagram of the control state changes in the first controllable node subspace during numerical simulation.

[0087] Figure 9 This is a trend diagram of the control state changes in the second controllable node subspace during numerical simulation.

[0088] Figure 10 This is a trend diagram of the control state changes in the third controllable node subspace during numerical simulation.

[0089] Figure 11 This is a trend diagram of the control state changes in the fourth controllable node subspace during numerical simulation. Detailed Implementation

[0090] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0091] like Figure 1 As shown, a distributed diffusion control method for a flame nozzle in a hot working furnace for wide and thick plates includes the following steps:

[0092] S1: Based on the theory of distributed parameter systems, the internal space of the thick plate hot working furnace is divided into... Each subspace is distinguished by whether there is a control device inside the subspace. Based on the temperature transfer process in the subspace of the thick plate hot processing furnace, a distributed parameter system based on the spatiotemporal transformation of temperature in the thick plate hot processing furnace is constructed. The distributed parameter system includes two parts: a controllable node subspace distributed parameter system and a diffusion node subspace distributed parameter system.

[0093] In step S1, based on the installation position of the flame nozzle inside the furnace of the heavy plate hot processing furnace, and taking into account the actual engineering structure of the heavy plate hot processing furnace; considering the direct effect of the flame nozzle, the internal space of the heavy plate hot processing furnace is divided into a total of Each subspace, in which Each subspace contains a flame nozzle; each subspace is defined as a spatial node, and thus contains a flame nozzle. Each subspace can have its temperature directly adjusted via a flame nozzle; a space node containing a flame nozzle is defined as a controllable node subspace; then, there exists... Each subspace does not contain a flame nozzle; the space nodes without flame nozzles are defined as diffusion node subspaces; the temperature of the diffusion node subspace is controlled through the heat transfer process between the space nodes; considering the time-varying energy transfer inside each subspace of the furnace and the spatiotemporal characteristics of the energy interaction between each space node, the temperature system of the wide and thick plate hot processing furnace with spatiotemporal characteristics is defined as a distributed parameter system, that is, the temperature transformation of the furnace space of the wide and thick plate hot processing furnace is modeled and described by a distributed parameter system.

[0094] Definition of the first The spatiotemporal variation of spatial temperature distribution in each subspace is as follows: , Represents spatial variables; Taking time as the variable, and considering the spatiotemporal transformation characteristics of the temperature inside the hot-working furnace for wide and thick plates, as well as the specific structure of each subspace node, the following distributed parameter system is constructed to describe the temperature variation law of each subspace inside the hot-working furnace:

[0095] The controllable node subspace distributed parameter system model is as follows:

[0096] (1)

[0097] The system model for the distributed parameters of the diffusion node subspace is as follows:

[0098] (2)

[0099] in, Indicates the first The spatiotemporal variation of spatial temperature distribution in each subspace This indicates the spatiotemporal region within a hot-working furnace for thick plates. ; Represents the space of positive real numbers; For having smooth boundaries The bounded area, namely the furnace space inside the thick plate hot processing furnace. , Describes the Euclidean norm. Numerical parameters that meet the conditions, Represents positive infinity; , Represents a one-dimensional real space. express 3D real matrix space, Indicates the number of spatial dimensions; and , The function representing the measurement of spatial dimension. Indicates a request The dimension is greater than zero; For the first Temperature diffusivity of individual subspaces ; is the control input state of the distributed parameter system of the first subspace; let , denote the temperature diffusion laplace operator inside the subspace; denote the total dimension of the space, denote the dimension ordinal of the space, denote the dimension variable in the space; is the temperature transfer time lag; denote the influence factor of the temperature distribution state of the first subspace on the temperature distribution state of the first subspace, denote the influence factor of the temperature distribution time lag state of the first subspace on the temperature distribution state of the first subspace; , denote the energy released by the first subspace to the first subspace; , denote the energy absorbed by the first subspace to the first subspace.

[0100] S2: combined with the target temperature curve of the wide and thick plate hot working, a distributed parameter error system model of the temperature change state of each subspace of the wide and thick plate hot working furnace and the target temperature state is obtained; the distributed parameter error system model includes two parts of a controllable node subspace distributed parameter error system model and a diffusion node subspace distributed parameter error system model.

[0101] The target temperature curve of the wide and thick plate hot working is set to satisfy:

[0102] (3)

[0103] wherein denotes a distributed parameter system space target temperature change function; denotes a distributed parameter system space target temperature change rate function;

[0104] In the wide and thick plate hot working process, according to the different material and process requirements of the wide and thick plate, the set target temperature curve is not the same, but it is required that the temperature of each heating area of the wide and thick plate is consistent (i.e. the temperature of each place of the hot working furnace is consistent) in the hot working process, so the present application sets the distributed parameter system space target temperature change function as when setting the target temperature curve of the hot working furnace, i.e. the target temperature curve is a function independent of the space variable .

[0105] To achieve the control objective, set the error between the first subspace temperature variation function and the target temperature as , which satisfies:

[0106] (4)

[0107] Combining formula (1), formula (2) and formula (4), the following distributed parameter error system model can be obtained:

[0108] The controllable node subspace distributed parameter error system model is:

[0109] (5)

[0110] The diffusion node subspace distributed parameter error system model is:

[0111] (6)

[0112] Wherein, represents the influence factor of the first subspace temperature error state on the first subspace temperature error state, represents the influence factor of the first subspace temperature error time delay state on the first subspace temperature error state;

[0113] When , it satisfies: , ;

[0114] When , it satisfies: , .

[0115] Set two control objectives, the first control objective is to achieve the continuous agreement of the furnace temperature of the wide-thick plate hot working furnace with the wide-thick plate hot working target temperature in engineering; the second control objective is the uniform distribution of the furnace temperature of the wide-thick plate hot working furnace;

[0116] In the theoretical solving process, when the controllable node subspace distributed parameter error system model and the diffusion node subspace distributed parameter error system model reach asymptotic stability, that is, the furnace temperature of the wide-thick plate hot working furnace is uniformly distributed; at the same time, when the distributed parameter error system model reaches asymptotic stability, it satisfies , according to the definition of in formula (4), it can be known that when At this time, the furnace temperature of the wide and thick plate hot working furnace continuously matches the target temperature of the wide and thick plate hot working, and the distributed parameter error system model reaches gradual stability under the action of the controller, that is, the two control targets are achieved.

[0117] For simplifying the description, the distributed parameter error system model is rewritten in the matrix form:

[0118] (7)

[0119] wherein, represents a distributed parameter error matrix variable with the dimension of , , the superscript represents the transpose of the matrix; represents a distributed parameter error matrix variable with the dimension of , , ; represents a matrix function with the dimension of composed of the function , , ; represents a matrix function with the dimension of composed of the function , , ; , both represent diagonal matrices, , represents the identity of the diagonal matrix, ; represents a control input matrix variable composed of the control input states of nodes, , ; represents an all-zero matrix; represents a matrix parameter with the dimension of composed of , , represents a matrix with the dimension of composed of the elements in the brackets; represents a matrix parameter with the dimension of composed of , ; and satisfies: , ; is a coupling matrix of the controllable node space temperature error state to the controllable node space temperature error state; a coupling matrix of the controllable node space temperature error state to the diffusion node space temperature error state; a coupling matrix of the controllable node space temperature error state to the diffusion node space temperature error state; a coupling matrix of the controllable node space temperature error time delay state to the diffusion node space temperature error state; a coupling diffusion matrix of the diffusion node space temperature error state to the diffusion node space temperature error state; a coupling diffusion matrix of the diffusion node space temperature error time delay state to the diffusion node space temperature error state.

[0120] S3: Constructing a distributed diffusion controller of the flame nozzle of the heavy plate hot working furnace to realize stable control of the distributed parameter error system model, the distributed diffusion controller including a controller system state generating sub-controller and a state feedback output sub-controller.

[0121] The internal space of the hot working furnace is divided into N subspaces, and the flame nozzle in each subspace is controlled by a sub-controller. The distributed diffusion controller is designed based on the thermal diffusion of the flame nozzle in each of the N subspaces.

[0122] The controller system state generating sub-controller includes:

[0123] (8)

[0124] The state feedback output sub-controller includes:

[0125] (9)

[0126] wherein, Xk represents the kth subspace controller system basic state variable; dk represents a diffusion coefficient of the control system state; ak represents a coupling influence coefficient of the kth node space control system state on the jth node space control system state; bk represents a coupling influence coefficient of the kth node space temperature error state on the jth control system state; ck represents a coupling influence coefficient of the kth node space temperature error time delay state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state; bk represents a coupling influence coefficient of the kth node space temperature error state on the jth control system state; ck represents a coupling influence coefficient of the kth node space temperature error time delay state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state; dk represents a coupling influence coefficient of the kth control system state on the jth control system state;

[0127] The constructed distributed diffusion controller does not directly control all subspaces acting on the inside of the wide and thick plate hot working furnace, and the controller only directly acts on the internal controllable subspace, and the furnace temperature control target of the wide and thick plate hot working furnace is realized through energy diffusion between the internal subspace of the wide and thick plate hot working furnace.

[0128] The distributed diffusion controller is changed into a matrix model form:

[0129] (10)

[0130] Wherein, The control system state matrix is represented, and satisfies , ; The control system diffusion coefficient matrix is represented by , and satisfies: ; The control system feedback gain matrix is represented, and satisfies: ; The control system state inter-coupling influence coefficient matrix is represented, and satisfies: ; The controllable node error system state inter-coupling influence coefficient matrix is represented, and satisfies: ; The controllable node error system time delay state inter-coupling influence coefficient matrix is represented, and satisfies: ; The diffusion node error system state inter-coupling influence coefficient matrix is represented, and satisfies: ; The diffusion node error system time delay state inter-coupling influence coefficient matrix is represented, and satisfies .

[0131] The control scheme constructed in the application is a distributed diffusion control scheme, and the important innovation is that:

[0132] The application considers that the direct control area of the flame nozzle of the wide and thick plate hot working furnace is limited, and all space areas cannot be directly controlled by the flame nozzle in actual engineering. Therefore, the diffusion control scheme is constructed, and the innovation of the scheme is that the constructed controller does not control all space nodes of the system, but only controls part of the space nodes, and the control of the entire distributed parameter system space (the space of the wide and thick plate hot working furnace) is realized through energy transmission between the space nodes.

[0133] This invention considers that when a flame nozzle outputs flame, its flame energy undergoes a heat transfer process (within the scope of a distributed parameter system). In practical engineering, the controller (flame nozzle) cannot directly achieve the target control state by providing feedback output based on the temperature state within the space of a thick plate heat treatment furnace. Therefore, this invention constructs a distributed diffusion controller, as shown in equations (8) and (9). Its control model comprises two parts: equation (8) is a distributed parameter system model, which generates the basic control state through a sub-controller based on the controller system state; equation (9) provides the control state feedback output. The controller constructed in this invention does not use the traditional feedback based on the system state, but rather uses the distributed parameter system control model state constructed based on the spatial energy diffusion process for feedback control, which more closely reflects the changes in flame energy output from the flame nozzle in practical engineering.

[0134] S4: The constructed distributed diffusion controller of the flame nozzle of the thick plate hot processing furnace is applied to the controllable node subspace distributed parameter error system model, and combined with the diffusion node subspace distributed parameter error system model, the closed-loop control model of the distributed parameter system is obtained.

[0135] Combining equations (7) and (9), the closed-loop control model for the distributed parameter system is obtained as follows:

[0136] (11)

[0137] in, The state matrix of the closed-loop control system satisfies: ; The diffusion coefficient matrix of the closed-loop control system satisfies: , ; The closed-loop control system coupling and control parameter matrix represent the following: , ; The time-delay coupled control parameter matrix of the closed-loop control system satisfies: , ; The target control temperature transformation matrix of the closed-loop control system satisfies: ;

[0138] Based on the actual spatial and temporal engineering of a thick plate hot working furnace, and considering that both the control system model and the distributed parameter error system model satisfy the same boundary conditions, the initial boundary conditions for the closed-loop control model of the distributed parameter system are set as follows:

[0139] , (12)

[0140] or:

[0141] , (13)

[0142] , (14)

[0143] where is the unit outer normal vector, is a smooth function.

[0144] S5: Based on Lyapunov stability theory, the sufficient condition for realizing the gradual stability control of the distributed parameter error system model is solved.

[0145] Based on Lyapunov stability theory, the sufficient condition for realizing the gradual stability control of the distributed parameter error system model is solved;

[0146] Conclusion 1: Given the system parameters and the Lipschitz condition parameters , if there are , , , , , , satisfy the following inequalities, then the closed-loop control model of the distributed parameter system is gradually stable.

[0147] (15)

[0148] (16)

[0149] wherein, , , are all intermediate variable matrices; , , are all Lyapunov functional parameter matrices, ; ; ; is the Lyapunov functional parameter matrix; the intermediate variable matrix ; the intermediate variable matrix ; the intermediate variable matrix ; the intermediate variable matrix ; represents the Lyapunov functional parameter matrix, which is a positive definite symmetric matrix, and its inverse matrix is ; ; ; ; ​; ; ; ; ; ; denotes the identity matrix; both formulae (15) and (16) are symmetric matrices, denotes the symmetric term of a symmetric matrix; the superscript -1 in the above formulae denotes the inverse matrix, denotes the inverse matrix of .

[0150] To derive the sufficient condition, the following assumptions and lemmas are given:

[0151] Assumption 1: for , the following Lipschitz condition is satisfied:

[0152] (17)

[0153] where , ; denotes the system state matrix;

[0154] Lemma 1: Let be a smooth subdomain, , if the variable , then:

[0155] (18)

[0156] where denotes the Hamiltonian operator; denotes the area element of the boundary region; denotes the set of second-order continuous differentiable functions in the smooth subdomain;

[0157] Lemma 2: for a given symmetric matrix , , , , to construct the matrix term of , the following three conditions are equivalent:

[0158] Condition 1: ;

[0159] Condition 2: ;

[0160] Condition 3: .

[0161] The proof process of Conclusion 1 is as follows:

[0162] Lyapunov function is constructed whose function form contains two parts as follows:

[0163] (19)

[0164] Lyapunov function, satisfies: where is the spatial integral infinitesimal element in is the Lyapunov function parameter matrix; Lyapunov function, satisfies: , is the time integral infinitesimal element, is the Lyapunov function parameter matrix, which is a positive definite symmetric matrix;

[0165] The derivative of , is ;

[0166]

[0167]

[0168] where, is the derivative of ;

[0169] According to assumption 1, there exists:

[0170] (20)

[0171] According to lemma 1, there exists:

[0172] (21)

[0173] where, is the spatial integral variable, is the spatial variable, is the spatial infinitesimal element, is the maximum eigenvalue of ;

[0174] Similarly, it can be obtained that:

[0175]

[0176] Therefore, the derivative of is :

[0177] (22) ​

[0178] where, denotes the variable matrix, satisfying ; is the intermediate matrix variable, satisfying: ;

[0179] Suppose that:

[0180] (23)

[0181] If the parameters designed by the distributed diffusion controller can make formula (23) hold, the system can reach the asymptotic stable state under the action of the controller, that is, the wide plate hot processing furnace can reach the target temperature curve under the action of the distributed diffusion controller. However, in formula (23), the control parameters in the distributed diffusion controller , , , , , are all implied in , , and cannot be directly solved by the solver to effectively use the control parameters. Therefore, formula (23) cannot solve the control parameters.

[0182] In order to effectively obtain the sufficient condition for solving the control parameters, the following will set the specific form of , , based on formula (23), and through theoretical reasoning, the controller parameters will be implied in the inequality, so as to solve the specific control parameters of the controller.

[0183] Set:

[0184]

[0185]

[0186] where, , are Lyapunov parameter matrices satisfying the conditions;

[0187] (24)

[0188] According to , we can get:

[0189] (25)

[0190] Define:

[0191] , (26)

[0192] Then there exists

[0193] (27)

[0194] Consider is a positive definite symmetric matrix, so the solution process of the sufficient condition needs to meet the following conditions:

[0195] (28)

[0196] To solve the control parameters, the control parameters implied in equation (23) are made explicit;

[0197] Multiply the left side of equation (23) by and multiply the right side by , we get:

[0198] (29)

[0199] According to equation (29), there exists:

[0200] (30)

[0201] According to (30), there exists:

[0202]

[0203] That is, under the condition that equation (15) is satisfied, the distributed parameter error system model of equation (7) reaches a state of asymptotic stability under the action of the distributed diffusion controller of equation (10); the wide and thick plate hot processing furnace can reach the target temperature curve under the action of the distributed diffusion controller. Therefore, conclusion 1 is proved.

[0204] S6: Based on the parameters of the wide and thick plate hot processing furnace system, combined with the obtained sufficient condition, the parameters of the distributed diffusion controller are numerically calculated and solved.

[0205] Using Lyapunov stability theory, the sufficient condition for the existence of a distributed diffusion controller for a distributed parameter system is obtained, and based on the sufficient condition, the control parameters of the distributed diffusion controller are solved, so that the distributed parameter system reaches a state of asymptotic stability under the action of the distributed diffusion controller, i.e., the temperatures of each subspace in the wide and thick plate hot processing furnace reach the target control temperature.

[0206] S7: Numerical simulation is performed to verify the effectiveness of the control scheme.

[0207] Taking the distributed parameter error system model of equation (7) as the control object, set the specific parameters as follows: set the internal space of the wide and thick plate hot processing furnace to contain 6 subspaces, of which 4 are controllable node subspaces and 2 are diffusion node subspaces, i.e., set , ;

[0208]

[0209] wherein, , , , , , , , , , .

[0210] According to the conclusion 1 condition, the LMI toolbox is used to solve formula (15) and formula (16), and the parameters in the matrix model of the distributed diffusion controller are obtained as follows:

[0211] ; ; ; ; ; .

[0212] By observing the temperature error state change trend chart of the controllable node subspace shown in Figures 2-5 , it can be found that the temperature error state of the controllable node subspace of the distributed parameter system reaches an asymptotically stable state under the direct control of the controller.

[0213] By observing the temperature error state change trend chart of the diffusion node subspace shown in Figures 6-7 , it can be found that the temperature error state of the diffusion node subspace of the distributed parameter system reaches an asymptotically stable state under the diffusion coupling effect of the system.

[0214] By observing the control state change trend chart of the controllable node subspace shown in Figures 8-11 , it can be found that the state of the control system eventually reaches an asymptotically stable state.

Claims

1. A distributed diffusion control method for flame nozzles in a hot working furnace for wide and thick plates, characterized in that, Includes the following steps: S1: Based on the theory of distributed parameter systems, the internal space of the thick plate hot working furnace is divided into... Each subspace is distinguished by whether there is a control device inside the subspace. Based on the temperature transfer process in the subspace of the hot plate processing furnace, a distributed parameter system based on the spatiotemporal transformation of temperature in the hot plate processing furnace is constructed. The distributed parameter system includes two parts: a controllable node subspace distributed parameter system and a diffusion node subspace distributed parameter system. In step S1, based on the installation position of the flame nozzle inside the furnace of the heavy plate hot processing furnace, and taking into account the actual engineering structure of the heavy plate hot processing furnace; considering the direct effect of the flame nozzle, the internal space of the heavy plate hot processing furnace is divided into a total of Each subspace, in which Each subspace contains a flame nozzle; each subspace is defined as a spatial node, and thus contains a flame nozzle. Each subspace can have its temperature directly adjusted via a flame nozzle; a space node containing a flame nozzle is defined as a controllable node subspace; then, there exists... Each subspace contains no flame nozzles; the space nodes without flame nozzles are defined as diffusion node subspaces. Temperature control is achieved through the heat transfer process between spatial nodes in the diffusion node subspace. Considering the time-varying energy transfer inside each subspace of the furnace and the spatiotemporal characteristics of energy interaction between each spatial node, the temperature system of the wide and thick plate hot processing furnace with spatiotemporal characteristics is defined as a distributed parameter system. That is, the temperature transformation of the furnace space of the wide and thick plate hot processing furnace is modeled and described by the distributed parameter system. S2: Based on the target temperature curve of the hot working of thick plates, the distributed parameter error system model of the temperature change state of each subspace of the hot working furnace of thick plates and the target temperature state is obtained; the distributed parameter error system model includes two parts: the distributed parameter error system model of the controllable node subspace and the distributed parameter error system model of the diffusion node subspace. S3: Construct a distributed diffusion controller for the flame nozzle of a thick plate hot working furnace to achieve stable control of the distributed parameter error system model. The distributed diffusion controller includes a controller system state generation sub-controller and a state feedback output sub-controller. S4: Apply the constructed distributed diffusion controller of the flame nozzle of the hot working furnace for wide and thick plates to the controllable node subspace distributed parameter error system model, and combine the diffusion node subspace distributed parameter error system model to obtain the closed-loop control model of the distributed parameter system. S5: Based on Lyapunov stability theory, the sufficient conditions for achieving asymptotic stability control of the distributed parameter error system model are obtained; S6: Based on the system parameters of the thick plate hot processing furnace and combined with the obtained sufficient conditions, the parameters of the distributed diffusion controller are numerically calculated and solved. S7: Perform numerical simulations to verify the effectiveness of the control scheme.

2. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 1, characterized in that, In step S1, the first... The spatiotemporal variation of spatial temperature distribution in each subspace is as follows: , Represents spatial variables; Taking time as the variable, and considering the spatiotemporal transformation characteristics of the temperature inside the hot-working furnace for wide and thick plates, as well as the specific structure of each subspace node, the following distributed parameter system is constructed to describe the temperature variation law of each subspace inside the hot-working furnace: The controllable node subspace distributed parameter system model is as follows: （1）； The system model for the distributed parameters of the diffusion node subspace is as follows: （2）； in, Indicates the first The spatiotemporal variation of spatial temperature distribution in each subspace This represents the spatiotemporal region within a heavy plate hot working furnace. ; Represents the space of positive real numbers; For having smooth boundaries The bounded area, namely the furnace space inside the thick plate hot processing furnace. , Denotes the Euclidean norm. Numerical parameters that meet the conditions, Represents positive infinity; , Represents a one-dimensional real space. express 3D real matrix space, Indicates the number of spatial dimensions; and , The function representing the measurement of spatial dimension. Indicates a request The dimension is greater than zero; For the first Temperature diffusivity of individual subspaces ; For the first The distributed parameter system controls the input state of each subspace; let , The Laplace operator represents the temperature diffusion within a subspace; Represents the total dimension of space. Represents the ordinal number of spatial dimensions. Represented as a dimension variable in space; This is the time delay for temperature transfer; Indicates the first The temperature distribution state of the subspace on the first Influencing factors of the temperature distribution state of each subspace Indicates the first The time lag state of temperature distribution in each subspace affects the first... Influencing factors of the temperature distribution state of each subspace; , Indicates the first Subspace for the first Each subspace releases energy; , Indicates the first Subspace for the first Each subspace absorbs energy.

3. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 2, characterized in that, In step S2, the target temperature curve for hot working of the thick plate is set to satisfy: （3）； in This represents the spatial target temperature variation function of a distributed parameter system; This represents the rate of change of the spatial target temperature over time in a distributed parameter system; To achieve the control objective, the first step is to set... The spatiotemporal variation state of the error between the temperature change function of each subspace and the target temperature is as follows: ,satisfy: （4）； Combining formulas (1), (2), and (4), the following distributed parameter error system model can be obtained: The system model for the error of the controllable node subspace distributed parameters is as follows: （5）； The system model for the error of the diffusion node subspace distribution parameters is as follows: （6）； in, Indicates the first The temperature error state of the first subspace is related to the first Influence factors on the temperature error state of each subspace Indicates the first The time lag state of the temperature error of each subspace on the first Influencing factors of individual subspace temperature error state; when When satisfied: , ; when When satisfied: , .

4. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 3, characterized in that, In step S2, two control objectives are set. The first control objective is to ensure that the furnace temperature of the hot processing furnace for thick plates continuously matches the target temperature for hot processing of thick plates. The second control objective is to ensure that the furnace temperature of the hot processing furnace for thick plates is uniformly distributed. In the theoretical solution process, when the controllable node subspace distributed parameter error system model and the diffusion node subspace distributed parameter error system model reach asymptotic stability, that is, the furnace temperature of the thick plate hot processing furnace is uniformly distributed; at the same time, when the distributed parameter error system model reaches asymptotic stability, it satisfies According to formula (4) According to the definition, when At that time, the furnace temperature of the hot working furnace for thick plates was kept in line with the target temperature for hot working of thick plates; when the distributed parameter error system model reached asymptotic stability under the action of the controller, the two set control objectives were achieved. To simplify the description, the distributed parameter error system model is rewritten in matrix form: （7）； in, The dimension is The distribution parameter error matrix variables, , superscript Represents the transpose of a matrix; The dimension is The distribution parameter error matrix variables, , ; Indicates by function The dimensions of the composition are Matrix functions, , ; Indicates by function The dimensions of the composition are Matrix functions, , ; , Both represent diagonal matrices. , Indicates the identifier of a diagonal matrix. ; Indicates by The control input matrix variable is composed of the control input states of each node. , ; Represents a matrix consisting entirely of zeros; Indicates by The dimensions of the composition are The matrix parameters, , The dimension represented by the elements within the parentheses is... matrix; Indicates by The dimensions of the composition are The matrix parameters, And satisfy: , ; This is the coupling matrix between the controllable node space temperature error state and the controllable node space temperature error state. This is the coupling matrix between the time delay state of the controllable node space temperature error and the controllable node space temperature error state; This is the coupling matrix between the temperature error state of the controllable node space and the temperature error state of the diffuse node space; This is the coupling matrix between the time-delay state of the controllable node space temperature error and the state of the diffuse node space temperature error; The coupling diffusion matrix of the spatial temperature error state of the diffusion node to the spatial temperature error state of the diffusion node; This is the coupling diffusion matrix of the spatial temperature error time delay state of the diffusion node to the spatial temperature error state of the diffusion node.

5. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 4, characterized in that, In step S3, the interior of the heat treatment furnace Based on the thermal diffusion of the flame nozzle within a subspace, the following distributed diffusion controller is designed: Controller system state generation sub-controller: （8）； Status feedback output sub-controller: （9）； in, Indicates the first The basic state variables of the subspace controller system; The diffusion coefficient represents the state of the control system. Indicates the first The state of the spatial control system at node i affects the first The coupling influence coefficient of the state of a node-space control system; Indicates the first The spatial temperature error state of the nth node affects the... The coupling influence coefficient of the state of a control system; Indicates the first The time lag state of the spatial temperature error of the nth node affects the first The coupling influence coefficient of the state of a control system; Indicates the feedback gain of the control system; The constructed distributed diffusion controller does not directly control all subspaces inside the thick plate hot processing furnace. The controller only directly acts on the controllable subspaces inside, and achieves the furnace temperature control target of the thick plate hot processing furnace through energy diffusion between the subspaces inside the furnace.

6. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 5, characterized in that, In step S3, the distributed diffusion controller is changed to a matrix model form: （10）； in, Represents the state matrix of the control system, satisfying , ; The diffusion coefficient matrix of the control system is given by... Composition, satisfying: ; The feedback gain matrix of the control system satisfies: ; The matrix representing the coupling influence coefficients between states of the control system satisfies: ; The matrix representing the coupling influence coefficients between states of a controllable nodal error system satisfies: ; The matrix representing the coupling influence coefficients between time-delay states of a controllable nodal error system satisfies: ; The matrix representing the coupling influence coefficients between the states of the diffusion node error system satisfies: ; The matrix representing the coupling influence coefficients between time-delay states of the diffusion node error system satisfies the following condition. .

7. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 6, characterized in that, In step S4, combining equations (7) and (9), the closed-loop control model of the distributed parameter system is obtained as follows: （11）； in, The state matrix of the closed-loop control system satisfies: ; The diffusion coefficient matrix of the closed-loop control system satisfies: , ; The closed-loop control system coupling and control parameter matrix represent the following: , ; The time-delay coupled control parameter matrix of the closed-loop control system satisfies: , ; The target control temperature transformation matrix of the closed-loop control system satisfies: ; Based on the actual spatial and temporal engineering of a thick plate hot working furnace, and considering that both the control system model and the distributed parameter error system model satisfy the same boundary conditions, the initial boundary conditions for the closed-loop control model of the distributed parameter system are set as follows: ， （12）； or: ， （13）； ， （14）； in for The unit outward normal vector, It is a smooth function.

8. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 7, characterized in that, In step S5, based on Lyapunov stability theory, sufficient conditions for achieving asymptotic stability control of the distributed parameter error system model are obtained. Conclusion 1: Given system parameters and Lipschitz condition parameters If it exists , , , , , , If the following inequalities are satisfied, the closed-loop control model of the distributed parameter system is asymptotically stable; (15)； (16)； in, , , All are intermediate variable matrices; , , All are Lyapunov functional parameter matrices. ; ; ; The Lyapunov functional parameter matrix; intermediate variable matrix Intermediate variable matrix Intermediate variable matrix Intermediate variable matrix Lyapunov functional parameter matrix inverse matrix ; ; ; ; ; ; ; ; ; ; Let represent the identity matrix; formulas (15) and (16) are both symmetric matrices. Represents the symmetric terms of a symmetric matrix; The superscript -1 in the matrix indicates the inverse matrix. express The inverse matrix.

9. The distributed diffusion control method for the flame nozzle of a thick plate hot working furnace according to claim 8, characterized in that, In step S5, to obtain sufficient conditions, the following assumptions and lemmas are given: Assumption 1: For It satisfies the following Lipschitz condition: （17）； in, , , Represents the system state matrix; Lemma 1: Let For a smooth subfield, If variable ,but: （18）； in Represents the Hamiltonian operator; A micro-element representing the area of ​​the boundary region; Let represent the set of twice continuously differentiable functions within a smooth subfield; Lemma 2: For a given symmetric matrix , , , , To constitute The following three conditions are equivalent for the matrix terms: Condition 1: ; Condition 2: ; Condition 3: ; Using Lyapunov stability theory, sufficient conditions for the existence of a distributed diffusion controller for asymptotically stable distributed parameter systems are derived. Based on these sufficient conditions, the control parameters of the distributed diffusion controller are solved, enabling the distributed parameter system to achieve asymptotic stability under the action of the distributed diffusion controller, i.e., the temperature of each subspace in the hot processing furnace of thick plates reaches the target control temperature.

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