A data-driven PPDE system event triggering H ∞ Optimization control method

Abstract

This application provides a data-driven PPDE system event trigger H ∞ The optimization control method includes: constructing an industrial sintering temperature field model based on time variables and spatial position variables and corresponding performance indicators considering external interference when the PPDE system considers external disturbances; deriving a low-order slow subsystem model of the temperature field corresponding to the industrial sintering temperature field model based on the time-space separation method; based on the low-order slow subsystem model of the temperature field, converting the optimal control problem of the PPDE system into the system optimal control strategy and the system worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism; constructing the HJI equation based on the system worst interference strategy and the system optimal control strategy; determining the PPDE system under data-driven conditions based on the HJI equation, the system worst interference strategy and the system optimal control strategy. H ∞ The event triggering conditions of the control significantly reduce the consumption of computing resources and communication bandwidth in the temperature field temperature control process.

Description

Technical Field

[0001] This application relates to the field of optimized control of industrial sintering temperature field, and in particular to a data-driven PPDE system event triggering Optimize control methods. Background Art

[0002] Optimizing and controlling the temperature field during industrial sintering is crucial for ensuring stable sintering, improving product quality, and enhancing product consistency. The temperature field distribution is closely related to time and space and is typically considered a complex system described by parabolic partial differential equations (PPDEs). PPDE systems are an important tool for describing complex industrial process systems and are widely used in complex industrial environments such as chemical engineering, energy, and metallurgy. However, actual industrial processes often face numerous unknown external disturbances, which severely test the stability of PPDE systems. Two approaches are commonly used when designing PPDE system controllers: the first is to simplify the system first and then design the controller; the second is to design the controller first and then simplify it. Both approaches rely on precise mathematical models, but actual industrial processes often face challenges in modeling and low model accuracy, and their limitations are becoming increasingly apparent.

[0003] Adaptive Dynamic Programming (ADP), as a data-driven control method, can effectively solve the above problems. However, ADP has high resource consumption and data transmission requirements. Especially when dealing with complex industrial processes, the computing resource consumption and resource utilization often cannot meet the actual needs. In addition, the PPDE system itself has the characteristics of infinite dimensions, nonlinearity, and strong spatiotemporal coupling. These characteristics put higher demands on computing power, making traditional time-triggered data-driven The computational complexity of the control method is further increased, which leads to data-driven periodic sampling The applicability of the control method to the actual industrial processes described by PPDE is limited.

[0004] Event-triggered control offers a new approach to addressing issues such as inefficient resource utilization and high computational complexity. By performing controller updates and system data transmission in a non-periodic manner, it significantly improves the efficiency of computing and data transmission processes while avoiding the instability of temperature distribution caused by frequent updates. Unlike traditional time-based periodic sampling, this method transmits current system information and controller updates only at specific triggering moments, significantly reducing communication frequency and computing resource consumption.

[0005] Currently, research on event-triggered control primarily focuses on systems of ordinary differential equations (ODEs), while research on PPDEs is still in its infancy. However, the complex temperature field in industrial sintering processes makes it difficult for traditional ODEs to accurately describe the system's dynamic characteristics. Addressing the low accuracy of temperature field models and the high computational resource consumption in actual industrial sintering processes presents a bottleneck that urgently needs to be overcome in the development of optimal control for PPDE systems. Summary of the Invention

[0006] In order to overcome the above technical defects, this application provides a data-driven PPDE system event trigger To achieve the above objectives, the present invention optimizes the control method by the following technical solutions:

[0007] This application provides a data-driven PPDE system event trigger The optimization control method is characterized by comprising:

[0008] Construct an industrial sintering temperature field model based on time variables and spatial position variables and corresponding performance indicators considering external disturbances for the PPDE system, and establish a corresponding relationship between the performance indicators and the optimal control problem of the PPDE system;

[0009] Based on the time-space separation method, a low-order slow subsystem model of the temperature field corresponding to the industrial sintering temperature field model is derived. Based on the low-order slow subsystem model of the temperature field, the optimal control problem of the PPDE system is converted into the system optimal control strategy and the system worst-case interference strategy of the industrial sintering temperature field model under the time trigger mechanism;

[0010] Constructing the HJI equation based on the worst interference strategy of the system and the optimal control strategy of the system;

[0011] Based on the HJI equation, the system's worst interference strategy and the system's optimal control strategy, determine the data-driven PPDE system The event triggering conditions of the control.

[0012] Optionally, the industrial sintering temperature field model is specifically:

[0013] ;

[0014] Where, is the state of the temperature field system, is the time variable, is the distribution of temperature field, is the periodic control input under time triggering, is the external disturbance of the temperature field, and is the boundary condition of the temperature field, is the initial state distribution of the temperature field, is a highly dissipative spatial differential operator, 、 are matrix functions describing the spatial distribution of control input and interference input, respectively. Represents the temperature measurement point in the internal space of the temperature field, which can be regarded as the target output. It is a matrix function describing the target output of the temperature field;

[0015] The performance indicators considering external interference include:

[0016] ;

[0017] Where, Represents the interference attenuation coefficient, which is a measure of The key parameters of the control system's anti-interference ability, matrix ;

[0018] The establishing of a corresponding relationship between the performance index and the optimal control problem of the PPDE system includes:

[0019] The optimal control problem of the PPDE system is described as: designing the optimal control strategy for the industrial sintering temperature field model under the studied PPDE system and the system's worst interference strategy , so that the performance indicators reach the most appropriate:

[0020] .

[0021] Optionally, the derivation of a temperature field low-order slow subsystem model corresponding to the industrial sintering temperature field model based on a time-space separation method includes:

[0022] By performing inner product operations on both sides of the industrial sintering temperature field model, the evolution process of the infinite-dimensional time mode of the temperature field is obtained as follows:

[0023] ;

[0024] Where, yes The derivative with respect to time t, represents the finite-dimensional slow mode part of the system, yes The derivative with respect to time t, represents the infinite-dimensional fast mode part of the system, represents the value of the finite-dimensional slow mode part of the system when t=0, represents the value of the infinite-dimensional fast mode part of the system when t=0, 、 is the function obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, and is the matrix obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, is a small positive constant, the matrix The bureau index is stable;

[0025] Based on the evolution process of the infinite-dimensional time mode of the temperature field, the low-order slow subsystem model of the temperature field of the PPDE system is obtained according to the fast-slow separation characteristics.

[0026] ;

[0027] Where, is the output of the low-order slow subsystem model of the temperature field;

[0028] Based on the low-order slow subsystem model of the temperature field, the optimal control problem of the PPDE system is converted into the system optimal control strategy and the system worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism, including:

[0029] The evolution process of the infinite-dimensional time mode of the temperature field and the performance index considering external interference are fused and calculated, and the performance index is converted to obtain the converted performance index:

[0030] ;

[0031] Where, is the performance index after conversion, is a preset constant and , It is a performance indicator sub-function that characterizes the correlation between the states of the fast and slow subsystems. Indicates correspondence Positive definite matrix of correlations;

[0032] According to the converted performance indicators, corresponding deductions are made to derive the performance indicators of the low-order slow subsystem model of the temperature field of the PPDE system:

[0033] ;

[0034] The temperature field low-order slow subsystem model and the temperature field low-order slow subsystem model performance index are integrated to construct the Hamiltonian function of the PPDE system under external interference:

[0035] ;

[0036] Where, is the Hamiltonian function of the PPDE system under external disturbance, is the transpose of the gradient of the value function;

[0037] Based on the Hamiltonian function of the PPDE system considering external interference, the system optimal control strategy and the system worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism are determined.

[0038] Optionally, determining the system optimal control strategy and the system worst-case interference strategy of the industrial sintering temperature field model under a time trigger mechanism based on the Hamiltonian function of the PPDE system considering external interference includes:

[0039] Based on the Hamiltonian function of the PPDE system under external disturbance, the periodic control input under time triggering is and external interference Obtain partial derivatives to determine the optimal control strategy and the worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism:

[0040] ;

[0041] ;

[0042] ;

[0043] Where, is the optimal control strategy of the system, is the worst interference strategy for the system, is a matrix The transposed matrix of is a matrix The transposed matrix of is the gradient of the optimal value function, is the optimal value function, It is the reciprocal of R.

[0044] Optionally, constructing the HJI equation based on the system worst interference strategy and the system optimal control strategy includes:

[0045] Obtain the first Hamiltonian equation model corresponding to the zero solution of the Hamiltonian function of the PPDE system under external interference, that is:

[0046] ;

[0047] The system worst interference strategy, the system optimal control strategy and the first Hamiltonian equation model are integrated to construct the HJI equation:

[0048] .

[0049] Optionally, the data-driven PPDE system is determined based on the HJI equation, the system worst interference strategy and the system optimal control strategy. Controlled event trigger conditions include:

[0050] Based on the optimal control strategy of the system, determine the PPDE system under the event trigger mechanism. Optimal control strategy:

[0051] ;

[0052] Where, For PPDE system under event trigger mechanism The optimal control strategy is , ;when , ;

[0053] Based on the PPDE under the event trigger mechanism The optimal control strategy, the system's worst interference strategy, and the industrial sintering temperature field model are integrated to obtain the event triggering under external disturbance considerations. Industrial sintering temperature field model of PPDE under control:

[0054] ;

[0055] Adopting adaptive dynamic programming technology, wherein the adaptive dynamic programming technology includes an evaluation network and an execution network;

[0056] The HJI equation is processed using the evaluation network to obtain the value function of the HJI equation:

[0057] ;

[0058] Where, is the activation function vector, is the weight vector, is the value function of the HJI equation, and the approximate error of NN is defined as ;

[0059] Based on the value function of the HJI equation, an estimate of the optimal value function is determined:

[0060] ;

[0061] Where, is the estimated value of the weight, is the estimated value of the optimal value function;

[0062] The estimated values ​​of the value function and the optimal value function of the HJI equation are respectively about Calculate the partial derivatives to obtain the first gradient of the value function of the HJI equation and the second gradient of the estimated value of the optimal value function:

[0063] ;

[0064] ;

[0065] Where, yes The first gradient of yes The second gradient, and They are and gradient;

[0066] For the execution network, based on the second gradient and the worst interference strategy of the system, by introducing the NN approximation technology, the time-triggered interference strategy of the data-driven PPDE system is obtained:

[0067] ;

[0068] Where, A time-triggered jamming strategy for data-driven PPDE systems;

[0069] For the execution network, based on the second gradient and the PPDE system under the event trigger mechanism The optimal control strategy is obtained by introducing NN approximation technology to obtain the data-driven PPDE system under the event trigger mechanism. Optimal control strategy:

[0070] ;

[0071] Where, For data-driven PPDE system under event trigger mechanism Optimal control strategy, Event triggering time gradient; The definition is: when hour, ;when Sometimes, there are ;

[0072] The first gradient and the data-driven PPDE system are combined under the event triggering mechanism The optimal control strategy and the time-triggered interference strategy of the data-driven PPDE system are substituted into the Hamiltonian equation to obtain a second Hamiltonian equation model corresponding to the first gradient:

[0073] ;

[0074] The second gradient and the data-driven PPDE system are controlled by the event trigger mechanism. H ∞ The optimal control strategy and the time-triggered interference strategy of the data-driven PPDE system are substituted into the Hamiltonian equation to obtain the third Hamiltonian equation model corresponding to the second gradient:

[0075] ;

[0076] The residual is defined as the difference between the third Hamiltonian equation model and the second Hamiltonian equation model:

[0077] ;

[0078] Where, is the residual;

[0079] The residual is processed using the gradient descent method and normalization technology to construct an update law for the estimated value of the evaluation network weights:

[0080] ;

[0081] Where, is the learning rate, , is an adaptive parameter; The value is 0 or 0.5. When , the value is 0.5, otherwise it is 0; the matrix , is a radially unbounded Lyapunov function;

[0082] Considering the event trigger under external disturbance Industrial sintering temperature field model of PPDE under control, update law of the evaluation network weight, time trigger interference strategy of PPDE under data drive, and event trigger mechanism under data drive The optimal control strategy is integrated and calculated to obtain the data-driven PPDE system. Controlled event trigger conditions:

[0083] ;

[0084] Where, is a positive real number to be designed, is a suitable positive real constant, where is the modal error, that is ,in, represents the system modes available to the controller, represents the true mode of the system, It represents the threshold condition for event trigger control. r is an adjustable matrix, is the square of the norm of the matrix r; when the above event triggering conditions are not met, an event trigger occurs, and the true mode of the system will be transmitted to the controller to update the control strategy.

[0085] This application has the following beneficial effects:

[0086] The proposed method dynamically designs event triggering conditions by integrating system status, control input, and external interference information. Based on these conditions, the controller's update timing and frequency are dynamically adjusted, significantly reducing the data transmission and computational burden of the PPDE system's optimization control process. By integrating neural network (NN) technology with the ADP architecture, precise optimization control of the PPDE system is achieved based on approximating the optimal value function. This significantly reduces the computational complexity of the temperature field control process and the consumption of computing resources and communication bandwidth.

[0087] In addition to the above-described purposes, features and advantages, the present application has other purposes, features and advantages. The present application will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0088] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0089] Figure 1 This is a data-driven PPDE system event trigger provided by the embodiment of the application Flowchart of the optimization control method;

[0090] Figure 2 is a schematic diagram of an interference signal provided in an embodiment of the present application;

[0091] Figure 3 Schematic diagram of the weight norm change trajectory of the evaluation NN provided by the experimental verification of the embodiment of the present application;

[0092] FIG4 is an event trigger provided by the experimental verification of the embodiment of the present application Schematic diagram of the control strategy learning process, Figure 4 (a) is triggered by the event Schematic diagram of the change process of the control input at the top of the temperature field under the control. Figure 4 (b) is the event trigger Schematic diagram of the change process of the control input at the bottom of the temperature field under the control;

[0093] FIG5 is a schematic diagram of the change process of the mode of the low-order slow subsystem model of the temperature field provided by the experimental verification of the embodiment of the present application. FIG5(a) is a schematic diagram of the evolution trend of the temperature field low-order mantle subsystem model describing the highest energy dominant behavior of the system. FIG5(b) is a schematic diagram of the evolution trend of the temperature field low-order mantle subsystem model describing the secondary dynamic characteristics of the system.

[0094] Figure 6 The event triggering provided by the embodiment of the present application is experimentally verified. Schematic diagram of the distribution of trigger time intervals under control;

[0095] Figure 7 The interference attenuation ratio designed by the experiment verification of the embodiment of the present application is designed Schematic diagram of the dynamic change process. DETAILED DESCRIPTION

[0096] The embodiments of the present application are described in detail below with reference to the accompanying drawings, but the present application can be implemented in many different ways as defined and covered by the claims.

[0097] Therefore, in order to solve the above problems, Figure 1 As shown, this application proposes a data-driven PPDE system event trigger Optimized control methods, including:

[0098] Step S101: constructing an industrial sintering temperature field model based on time variables and spatial position variables and corresponding performance indicators considering external disturbances for the PPDE system, and establishing a corresponding relationship between the performance indicators and the optimal control problem of the PPDE system;

[0099] First, the industrial sintering temperature field model is constructed, namely:

[0100] (1)

[0101] Where, is the state of the temperature field system, is the time variable, is the distribution of temperature field, is the periodic control input under time triggering, is the external disturbance of the temperature field, and is the boundary condition of the temperature field, is the initial state distribution of the temperature field, is a highly dissipative spatial differential operator, 、 are matrix functions describing the spatial distribution of control input and interference input, respectively. Represents the temperature measurement point in the internal space of the temperature field, which can be regarded as the target output. It is a matrix function describing the target output of the temperature field;

[0102] The characteristic of event-triggered control is that the available state of the controller is updated only at the intermittent moments when it is triggered, while the previously acquired system state information is maintained at other non-triggered moments. Define. hour, ;when hour, .when Indicates: Time t is the time when the event is triggered. At this time, the system is available is the system state at time t .when Indicates: time t is the last trigger time Until the next trigger The system has not reached the event triggering time. At this time, the system is available For the previous moment System status .

[0103] PPDE system under time-triggered control The control problem can be described as the system satisfies the designed feedback control strategy. Gain, that is:

[0104] (2)

[0105] Where, Represents the interference attenuation coefficient, which is a measure of The key parameters of the control system's anti-interference ability, matrix ;

[0106] The performance index considering external interference is defined as , which is defined as follows:

[0107] (3)

[0108] Establishing a corresponding relationship between the performance index and the optimal control problem of the PPDE system includes:

[0109] The optimal control problem of the PPDE system is described as: designing the optimal control strategy for the industrial sintering temperature field model under the studied PPDE system and the system's worst interference strategy , so that the performance indicators reach the most appropriate:

[0110] (4).

[0111] Step S102: Based on the time-space separation method, a low-order slow subsystem model of the temperature field corresponding to the industrial sintering temperature field model is derived. Based on the low-order slow subsystem model of the temperature field, the optimal control problem of the PPDE system is converted into a system optimal control strategy and a system worst-case interference strategy of the industrial sintering temperature field model under a time trigger mechanism.

[0112] PPDE system event triggering The control method is a control method that first reduces the order and then designs, which can trigger the event of PPDE system The optimal control problem is transformed into an event triggering of a low-order slow subsystem model of the temperature field Optimal control. Considering that the state of the PPDE system can be expressed by the spatial basis function and time mode Expressed as , so the PPDE system can be reduced to a certain order by the Karhunen–Loève decomposition (KLD) method.

[0113] For the industrial sintering temperature field model, that is, the inner product operation of both sides of the PPDE system of formula (1) with the spatial basis function is performed simultaneously, and the evolution process of the infinite-dimensional time mode of the temperature field is obtained by the KLD method and singular perturbation theory:

[0114] (5)

[0115] Where, yes The derivative with respect to time t, represents the finite-dimensional slow mode part of the system, yes The derivative with respect to time t, represents the infinite-dimensional fast mode part of the system, represents the value of the finite-dimensional slow mode part of the system when t=0, represents the value of the infinite-dimensional fast mode part of the system when t=0, 、 is the function obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, and is the matrix obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, is a small positive constant, the matrix The bureau index is stable; due to Very small, by introducing is the fast system constant, and let , then Established.

[0116] For the evolution process of the infinite-dimensional time mode of the obtained temperature field, according to the fast-slow separation characteristics, the low-order slow subsystem model of the temperature field under the PPDE system is obtained;

[0117] (6)

[0118] Where, is the output of the low-order slow subsystem model of the temperature field;

[0119] The evolution process of the infinite-dimensional time mode of the temperature field and the performance index considering external interference are integrated and calculated, and the performance index is converted to obtain the converted performance index:

[0120] (7)

[0121] Where, is the performance index after conversion, is a preset constant and , It is a performance indicator sub-function that characterizes the correlation between the states of the fast and slow subsystems; based on the above definition, , , , , .Depend on , it can be deduced that . , , Respectively indicate the corresponding , The positive definite matrix of the correlation.

[0122] Based on the temperature field low-order slow subsystem model of the PPDE system of formula (6), the obtained PPDE system of formula (2) The control problem description is fused and analyzed, which can be converted to satisfy Gain less than or equal to conditions, namely:

[0123] (8)

[0124] According to the converted performance indicators, and , further corresponding deduction can be carried out to obtain the performance index of the low-order slow subsystem model of the temperature field of the PPDE system ,Right now:

[0125] (9)

[0126] The temperature field low-order slow subsystem model and the temperature field low-order slow subsystem model performance index are integrated to construct the Hamiltonian function of the PPDE system under external interference:

[0127] (10)

[0128] Where, is the Hamiltonian function of the PPDE system under external disturbance, is the transpose of the gradient of the value function;

[0129] Combined with Bellman's optimality principle, The Hamilton-Jacobi-Isaacs (HJI) equation can be obtained, where is the optimal value function and satisfies .

[0130] Then, based on the Hamiltonian function of the PPDE system under external interference , for periodic control input under time trigger and external interference Find the partial derivative and get the corresponding extreme value, as shown in the following formula:

[0131] (11)

[0132] By Hamiltonian function Control The partial derivative is controlled The minimum value of the PPDE system under the time trigger mechanism is the optimal control strategy of the system for:

[0133] (12)

[0134] Where, is a matrix The transposed matrix of is the gradient of the optimal value function, is the optimal value function, It is the reciprocal of R.

[0135] Similarly, interference The partial derivative of the PPDE system under the time trigger mechanism can be obtained by for:

[0136] (13)

[0137] Where, is a matrix The transposed matrix of

[0138] Step S103: constructing an HJI equation based on the system worst interference strategy and the system optimal control strategy;

[0139] Obtain the first Hamiltonian equation model corresponding to the zero solution of the Hamiltonian function of the PPDE system under external interference, that is:

[0140] (14)

[0141] According to the optimal control strategy of the PPDE system under the time trigger mechanism and interference strategies By fusing the first Hamiltonian equation model, we can get the HJI equation:

[0142] (15)

[0143] Step S104: Based on the HJI equation, the system worst interference strategy and the system optimal control strategy, determine the data-driven PPDE system The event triggering conditions of the control.

[0144] According to the controller update mechanism triggered by events, the optimal control strategy of the PPDE system under the time trigger mechanism is , we can get the PPDE system under the event trigger mechanism Optimal control strategy ,Right now:

[0145] (16)

[0146] In the formula, when , ;when , .

[0147] Based on the PPDE under the event trigger mechanism The optimal control strategy, the system worst interference strategy, and the industrial sintering temperature field model are integrated to obtain the event triggering under external disturbance Industrial sintering temperature field model of PPDE under control:

[0148] (17)

[0149] Based on Equation (17) event triggering under external disturbance Industrial sintering temperature field model of PPDE under control, system interference strategy of PPDE system under time trigger mechanism (12) , Formula (15) PPDE system under event triggering mechanism Optimal control strategy The event triggering conditions of the PPDE system are constructed by integrating the multi-dimensional information of system status, control input and interference signal, namely:

[0150] (18)

[0151] and , is a suitable positive real constant. Then, when the event triggers During the control process, the closed-loop PPDE system is semi-globally uniformly ultimately bounded (SGUUB) and satisfies its Gain less than or equal to .

[0152] In this application, a data-driven adaptive dynamic programming method is used to design event triggering Controller. The implementation of this controller is based on the evaluation-execution network structure of adaptive dynamic programming, that is, the adaptive dynamic programming technology includes an evaluation network and an execution network. For the evaluation network, For the activation function vector, select the activation function vector and is the weight vector , the HJI equation is processed using the evaluation network to obtain the value function of the HJI equation :

[0153] (19)

[0154] Among them, the approximation error of NN (NN approximation technology usually refers to the technology of using neural network (NN) to approximate functions or data fitting. Its core is to approximate complex mathematical relationships or physical processes through the nonlinear mapping ability of neural networks) is defined as Since the ideal weights of NN It is difficult to obtain accurately, so the output of the evaluation network is actually is the estimated value of the optimal value function ,Right now:

[0155] (20)

[0156] in, is the estimated value of the weight.

[0157] The value function of the HJI equation of formula (19) is The estimated value of the optimal value function of formula (20) About Taking partial derivatives, we can get The first gradient of The second gradient is:

[0158] (twenty one)

[0159] (twenty two)

[0160] Where, yes The first gradient of yes The second gradient, and They are and gradient.

[0161] For the execution network, based on the second gradient and the worst interference strategy of the system, namely, Equation (22) and Equation (13), by introducing the NN approximation technology, the time-triggered interference strategy of the PPDE system under data-driven is obtained: :

[0162] (twenty three)

[0163] For the execution network, based on the second gradient and PPDE system under the event trigger mechanism The optimal control strategy, namely Equation (22) and Equation (16), is obtained by introducing NN approximation technology under the event triggering mechanism of the data-driven PPDE system. Optimal control strategy :

[0164] (twenty four)

[0165] Where, Event triggering time The gradient, The definition is: when hour, ;when Sometimes, there are .

[0166] Design a method for updating the NN weights in the evaluation network. Substitute the first gradient, the system's optimal control strategy, and the system's worst-case interference strategy into the Hamiltonian equation to obtain the second Hamiltonian equation model corresponding to the first gradient:

[0167] (25)

[0168] The second gradient, data-driven PPDE system under the event trigger mechanism The optimal control strategy and the time-triggered interference strategy of the data-driven PPDE system are substituted into the Hamiltonian equation to obtain the third Hamiltonian equation model corresponding to the second gradient:

[0169] (26)

[0170] The residual is defined as the difference between the third Hamiltonian equation model and the second Hamiltonian equation model:

[0171] (27)

[0172] Where, is the residual;

[0173] The residuals are processed using the gradient descent method and normalization technology to construct the update law for the estimated value of the evaluation network weights:

[0174] (28)

[0175] Where, is the learning rate, , is an adaptive parameter; The value is 0 or 0.5. When , the value is 0.5, otherwise it is 0; the matrix , is a radially unbounded Lyapunov function.

[0176] Event triggering under external disturbance Industrial sintering temperature field model of PPDE under control, update law of evaluation network weight, time trigger interference strategy of PPDE under data drive, and event trigger mechanism under data drive. The optimal control strategy is integrated and calculated to obtain the data-driven PPDE system. Controlled event trigger conditions:

[0177] (29)

[0178] Where, is a positive real number to be designed, is a suitable positive real constant. Where, is the modal error, that is ,in, represents the system modes available to the controller, represents the true mode of the system, It represents the threshold condition for event trigger control. r is an adjustable matrix, is the square of the norm of the matrix r; when the above event triggering conditions are not met, an event trigger occurs, and the true mode of the system will be transmitted to the controller to update the control strategy.

[0179] It can be seen that when the event is triggered Under control, the stability of the closed-loop PPDE system considering external disturbances is SGUUB under norm.

[0180] Experimental simulation analysis

[0181] In order to better verify the effectiveness and convergence of the method studied in this patent, the PPDE system event mentioned in this application is triggered The optimization control method is applied to the temperature field in the industrial sintering process. The industrial sintering temperature field model can be equivalently converted into the form of a PPDE system. The expressions of each function are shown in the following formula:

[0182] (30)

[0183] The physical and structural parameters of industrial sintering temperature are shown in Table 1. The initial value of the PPDE system is set to In the constructed PPDE system event triggering condition formula (25), the parameter and are set to 0.5 and 3.5 respectively. In the weight parameter update law (24) of the evaluation network, the learning rate and adaptive parameters Set to 1 and 0.04 respectively. is an identity matrix of appropriate dimensions. The interference signal is defined as ,in and is A random variable in the range Figure 2 As shown. At the same time, define the interference attenuation ratio for:

[0184] (31)

[0185] Table 1 Physical properties and structural parameters of industrial sintering temperature

[0186] ;

[0187] For the evaluation NN, construct the corresponding activation function vector Satisfaction relationship , weight vector satisfy Verify the PPDE event triggering proposed in this application in MATLAB Control methods, Figures 3 to 7 The corresponding simulation verification results are given.

[0188] The norm change trajectory of the weight vector of the evaluation NN is as follows Figure 3 As shown in Figure 4, according to the event triggering mechanism, the control strategy is updated only at the triggering moment and remains unchanged at other moments. and Represents the control inputs at the top and bottom of the temperature field, respectively. Under the control, the change process shows an irregular step-shaped change trend, indicating that the control strategy is indeed updated only when the trigger condition is not met, which is basically consistent with the mechanism of non-periodic update. Figure 5 describes the time mode change process of the low-order slow subsystem model of the temperature field. Its time mode is 2×1 order, that is: and ,As can be seen from the figure, the time mode of the ,low-order slow subsystem model of the temperature field eventually converges to zero. Figure 6 Describes the event triggering in the design The distribution of trigger time intervals under control. This shows that the difference from the traditional time-triggered control is that the event trigger studied in this paper is The controller does not need to perform frequent periodic sampling, which avoids the communication bandwidth and computing resource consumption caused by frequent periodic updates of the controller. Figure 7 Depicts the designed interference attenuation ratio The dynamic change process of the system can be clearly seen. It gradually and stably converges to around 0.0357, indicating that the PPDE system can fully meet the predetermined gain index under the designed event-triggered control strategy.

[0189] In summary, the proposed method dynamically designs event triggering conditions by integrating system status, control input, and external interference information. Based on these conditions, the controller's update timing and frequency are dynamically adjusted, significantly reducing the data transmission and computational burden of the PPDE system's optimization control process. By deeply integrating neural network (NN) technology with the ADP architecture, precise optimization control of the PPDE system is achieved based on approximating the optimal value function. This significantly reduces the computational complexity, computing resource consumption, and communication bandwidth consumption of the industrial sintering temperature field control process.

[0190] The above description is merely a preferred embodiment of the present application and is not intended to limit the present application. Various modifications and variations are possible for those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present application shall be included within the scope of protection of the present application.

Claims

1. A data-driven PPDE system event triggering H ∞ The optimization control method is characterized in that include: Construct an industrial sintering temperature field model based on time variables and spatial position variables and corresponding performance indicators considering external disturbances for the PPDE system, and establish a corresponding relationship between the performance indicators and the optimal control problem of the PPDE system; Based on the time-space separation method, a low-order slow subsystem model of the temperature field corresponding to the industrial sintering temperature field model is derived. Based on the low-order slow subsystem model of the temperature field, the optimal control problem of the PPDE system is converted into the system optimal control strategy and the system worst-case interference strategy of the industrial sintering temperature field model under the time trigger mechanism; Constructing the HJI equation based on the worst interference strategy of the system and the optimal control strategy of the system; Based on the HJI equation, the worst interference strategy of the system and the optimal control strategy of the system, the HJI equation of the PPDE system under data drive is determined. ∞ Controlled event triggering conditions; Wherein, the HJI equation is constructed based on the worst interference strategy of the system and the optimal control strategy of the system: Obtain the first Hamiltonian equation model corresponding to the zero solution of the Hamiltonian function of the PPDE system under external interference, that is: ； in, is the system mode, is the periodic control input under time triggering, is the external disturbance of the temperature field, is the gradient of the value function; The system worst interference strategy, the system optimal control strategy and the first Hamiltonian equation model are integrated to construct the HJI equation: ； Where, is the transpose of the gradient of the optimal value function, is the function obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, represents the finite-dimensional slow mode part of the system, and is the matrix obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, is the optimal control strategy, is the worst interference strategy for the system, Indicates correspondence The positive definite matrix of the correlation, the matrix , is a preset constant.

2. The method according to claim 1, characterized in that The industrial sintering temperature field model is specifically as follows: ； Where, is the state of the temperature field system, is the time variable, is the distribution of temperature field, is the periodic control input under time triggering, is the external disturbance of the temperature field, and is the boundary condition of the temperature field, is the initial state distribution of the temperature field, is a highly dissipative spatial differential operator, 、 are matrix functions describing the spatial distribution of control input and interference input, respectively. Represents the temperature measurement point in the internal space of the temperature field, which can be regarded as the target output. It is a matrix function describing the target output of the temperature field; The performance indicators considering external interference include: ； Where, To consider the performance indicators of external interference, Represents the interference attenuation coefficient, which is a measure of H ∞ Key parameters of the control system's anti-interference capability; The step of establishing a corresponding relationship between the performance index and the optimal control problem of the PPDE system includes: The optimal control problem of the PPDE system is described as: designing the optimal control strategy and the worst interference strategy for the industrial sintering temperature field model under the studied PPDE system so that the performance index reaches the most appropriate: 。 3. The method according to claim 1, characterized in that The temperature field low-order slow subsystem model corresponding to the industrial sintering temperature field model is derived based on the time-space separation method, including: By performing inner product operations on both sides of the industrial sintering temperature field model, the evolution process of the infinite-dimensional time mode of the temperature field is obtained as follows: ； Where, yes The derivative with respect to time t, yes The derivative with respect to time t, represents the infinite-dimensional fast mode part of the system, represents the value of the finite-dimensional slow mode part of the system when t=0, represents the value of the infinite-dimensional fast mode part of the system when t=0, is the function obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, 、 、 and is the matrix obtained by the inner product operation of the corresponding operator and matrix in the industrial sintering temperature field model described and the spatial basis function, is a small positive constant, the matrix The bureau index is stable; Based on the evolution process of the infinite-dimensional time mode of the temperature field, the low-order slow subsystem model of the temperature field of the PPDE system is obtained according to the fast-slow separation characteristics. ； Where, is the output of the low-order slow subsystem model of the temperature field; Based on the low-order slow subsystem model of the temperature field, the optimal control problem of the PPDE system is converted into the system optimal control strategy and the system worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism, including: The evolution process of the infinite-dimensional time mode of the temperature field and the performance index considering external interference are fused and calculated, and the performance index is converted to obtain the converted performance index: ； Where, is the performance index after conversion, A performance indicator sub-function that characterizes the correlation between the states of the fast and slow subsystems; According to the converted performance indicators, corresponding deductions are made to derive the performance indicators of the low-order slow subsystem model of the temperature field of the PPDE system: ； Where, is the performance index of the low-order slow subsystem model of the temperature field; The temperature field low-order slow subsystem model and the temperature field low-order slow subsystem model performance index are integrated to construct the Hamiltonian function of the PPDE system under external interference: ； Where, is the Hamiltonian function of the PPDE system under external disturbance, is the transpose of the gradient of the value function; Based on the Hamiltonian function of the PPDE system considering external interference, the system optimal control strategy and the system worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism are determined.

4. The method according to claim 3, characterized in that The method of determining the optimal control strategy and the worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism based on the Hamiltonian function of the PPDE system under the consideration of external interference includes: Based on the Hamiltonian function of the PPDE system under external disturbance, the periodic control input under time triggering is and external interference Obtain partial derivatives to determine the optimal control strategy and the worst interference strategy of the industrial sintering temperature field model under the time trigger mechanism: ； ； ； Where, is the optimal control strategy of the system, is the worst interference strategy for the system, is a matrix The transposed matrix of is a matrix The transposed matrix of is the gradient of the optimal value function, is the optimal value function, It is the reciprocal of R.

5. The method according to claim 4, characterized in that The HJI equation, the system worst interference strategy and the system optimal control strategy are used to determine the PPDE system H under data driving. ∞ Controlled event trigger conditions include: Based on the optimal control strategy of the system, determine the H of the PPDE system under the event trigger mechanism. ∞ Optimal control strategy: ； Where, H is the PPDE system under the event trigger mechanism ∞ The optimal control strategy is , ;when , ; Based on the H ∞ The optimal control strategy, the system worst interference strategy and the industrial sintering temperature field model are integrated to obtain the event triggering H under external disturbance. ∞ Industrial sintering temperature field model of PPDE under control: ； Adopting adaptive dynamic programming technology, wherein the adaptive dynamic programming technology includes an evaluation network and an execution network; The HJI equation is processed using the evaluation network to obtain the value function of the HJI equation: ； Where, is the activation function vector, is the weight vector, is the value function of the HJI equation, and the approximate error of NN is defined as ; Determining an estimate of an optimal value function based on the value function of the HJI equation; ； Where, is the estimated value of the weight, is the estimated value of the optimal value function; Taking partial derivatives of the value function of the HJI equation and the estimated value of the optimal value function with respect to s, respectively, to obtain a first gradient of the value function of the HJI equation and a second gradient of the estimated value of the optimal value function; ； ； Where, yes The first gradient of yes The second gradient, and They are and gradient; For the execution network, based on the second gradient and the worst interference strategy of the system, a time-triggered interference strategy of the data-driven PPDE system is obtained by introducing NN approximation technology; ； Where, A time-triggered jamming strategy for data-driven PPDE systems; For the execution network, based on the second gradient and the H of the PPDE system under the event trigger mechanism ∞ The optimal control strategy is obtained by introducing NN approximation technology to obtain the H of the PPDE system under the event trigger mechanism under data drive. ∞ Optimal control strategy; ； Where, H is the data-driven PPDE system under the event trigger mechanism ∞ Optimal control strategy, Event triggering time gradient; The definition is: when hour, ;when Sometimes, there are ; Substitute the first gradient, the system optimal control strategy, and the system worst interference strategy into the Hamiltonian equation to obtain a second Hamiltonian equation model corresponding to the first gradient: ； The second gradient and the data-driven PPDE system are H ∞ Substituting the optimal control strategy and the time-triggered interference strategy of the data-driven PPDE system into the Hamiltonian equation to obtain a third Hamiltonian equation model corresponding to the second gradient; ； The residual is defined as the difference between the third Hamiltonian equation model and the second Hamiltonian equation model: ； Where, is the residual; The residual is processed using the gradient descent method and normalization technology to construct an update law for the estimated value of the evaluation network weights: ； Where, is the learning rate, , is an adaptive parameter; The value is 0 or 0.

5. When , the value is 0.5, otherwise it is 0; the matrix , is a radially unbounded Lyapunov function; Considering the external disturbance event trigger H ∞ Industrial sintering temperature field model of PPDE under control, update law of the evaluation network weight, time trigger interference strategy of PPDE under data drive, H ∞ The optimal control strategy is integrated and calculated to obtain the data-driven PPDE system H ∞ Controlled event trigger conditions: ； Where, is a positive real number to be designed, is a suitable positive real constant, is the modal error, that is ,in, represents the system modes available to the controller, represents the true mode of the system, It represents the threshold condition of event trigger control; r is an adjustable matrix, is the square of the norm of the matrix r; when the above event triggering conditions are not met, an event trigger occurs, and the true mode of the system will be transmitted to the controller to update the control strategy.

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