Security Tracking Control Method and Device for Discrete-Time Two-Dimensional FM Model under Hybrid Attacks

Abstract

This invention relates to a secure tracking control method and apparatus for a discrete-time two-dimensional F-M model under hybrid attacks, belonging to the field of system network security technology. The invention introduces an error compensation communication mechanism with bidirectional evolution indices into the measurement channel of the discrete-time two-dimensional F-M model to schedule the communication of measurement data. Simultaneously, considering the impact of hybrid network attacks on system performance, a tracking controller is constructed under the combined influence of the hybrid network attack and the error compensation communication mechanism. Based on the error compensation communication mechanism and the tracking controller, a closed-loop augmented system model is constructed, and this control scheme is used to control the secure tracking of the discrete-time two-dimensional F-M model. This invention can effectively address the impact of hybrid network attacks on the discrete-time two-dimensional F-M model, ensuring that the output of the discrete-time two-dimensional F-M model can still stably track the reference signal under hybrid network attacks, while reducing the burden on the communication network and improving the adaptability to system uncertainties.

Description

Technical Field

[0001] This invention relates to the field of network security technology for two-dimensional systems, and more specifically to a method and apparatus for secure tracking and control of discrete-time two-dimensional FM models under hybrid attacks. Background Technology

[0002] With the rapid development of science and technology, the discrete-time two-dimensional Fornasini-Marchesini (FM) model is increasingly widely used in engineering practice. The discrete-time two-dimensional FM system is a core model describing two-dimensional signal and image processing, characterized by recursive state evolution in two independent directions. It is now widely applied in fields such as seismic data analysis, complex image processing, thermal process control, and circuit analysis. Compared to discrete-time one-dimensional systems, the bidirectional state evolution mechanism of the discrete-time two-dimensional FM model makes its dynamic behavior more complex. Traditional control theories and methods for discrete-time one-dimensional systems are difficult to directly transfer, which has largely driven the proposal and continuous optimization of classic two-dimensional systems such as the Roesser system and the FM system. Due to the deep integration of communication networks and control systems, while enhancing system openness and resource sharing capabilities, it also exposes them to severe information security threats. Currently, research on system security control mainly targets Denial of Service (DoS) attacks or False Data Injection (FDI) attacks. In networked control practice, to increase the success rate of network attacks, attackers often employ multi-source collaborative network attack patterns to launch attacks on the system, thereby maximizing the attack effect. Therefore, conducting research on security control based on the discrete-time two-dimensional FM model under hybrid network attack conditions is not only of great theoretical significance, but also provides necessary application support for the secure and stable operation of complex systems in real networked environments.

[0003] As is well known, in the field of networked control, the network overload problem caused by the traditional time-triggered mechanism due to "periodic forced transmission" is becoming increasingly prominent. Against this backdrop, event-triggered communication mechanisms have emerged. Their core idea is to determine whether to transmit data based on a preset data error magnitude, while ensuring system performance, thereby significantly reducing the occupancy rate of network communication resources. The bidirectional state evolution characteristics of the discrete-time two-dimensional FM model make its dynamic behavior strongly coupled, and data redundancy is more likely to cause network congestion. Existing error compensation communication is mostly based on the "unidirectional evolution" of discrete-time one-dimensional systems, making it difficult to adapt to the "bidirectional coupling" scenario of the discrete-time two-dimensional FM model. Therefore, it is necessary to consider dynamic auxiliary variables with bidirectional evolution characteristics and develop error compensation communication strategies suitable for the structural characteristics of the discrete-time two-dimensional FM model to ensure its stable operation.

[0004] Furthermore, output tracking control, a core problem in control theory and applications, aims to design tracking control strategies that enable the system output to accurately track a given reference signal. This problem has been widely applied in scenarios such as robot trajectory tracking, photovoltaic power regulation, and aircraft attitude control. However, most existing research focuses on discrete-time one-dimensional systems. Therefore, researching safe output tracking control methods and devices for discrete-time two-dimensional FM models is another objective of this invention.

[0005] Therefore, how to conduct in-depth research on error compensation communication security tracking control of discrete-time two-dimensional FM models under hybrid attacks, and how to help improve the security of seismic data analysis systems, complex image processing systems, thermal process control systems, or circuit analysis systems under hybrid network attacks, is a major research direction for those skilled in the art. Summary of the Invention

[0006] In view of this, the present invention provides a method and apparatus for secure tracking control of a discrete-time two-dimensional FM model under hybrid attacks; aiming to reduce communication pressure while considering the impact of hybrid network attacks on the system output, and effectively improve the control performance and stability of the system.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] On the one hand, this application provides a secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks, including the following steps: S1. Construct a discrete-time two-dimensional FM model of the monitored system and generate the reference signal to be tracked based on the reference model; S2. An error compensation communication mechanism based on dynamic auxiliary variables is introduced to update the state information of the discrete-time two-dimensional FM model and transmit it through a communication network. S3. Construct a tracking controller that is simultaneously affected by hybrid network attacks and the error compensation communication mechanism; S4. Substitute the information update rules of the error compensation communication mechanism and the tracking controller into the discrete-time two-dimensional FM model to construct a closed-loop augmented system model; S5. Solve for the gain matrix of the tracking controller based on the closed-loop augmented system model; S6. The tracking controller obtained by the solution is used to control the discrete-time two-dimensional FM model so that the output of the discrete-time two-dimensional FM model can stably track the reference signal under hybrid network attacks.

[0009] Furthermore, a discrete-time two-dimensional FM model of the monitored system is constructed in S1, specifically represented as the following model:

[0010] In the formula, and These represent indices of generalized spatial variables in the horizontal and vertical directions, respectively. Indicates the monitored system in the index The state information vector at that location; Indicates the monitored system in the index The output information vector at the location; Indicates process noise; To track the control input vector of the controller; , , , , , and These are known, well-defined constant matrices for the corresponding indices.

[0011] Furthermore, in S1, the monitored system includes an earthquake data analysis system, a complex image processing system, a thermal process control system, or a circuit analysis system.

[0012] Furthermore, S2 specifically includes: An error compensation communication mechanism based on dynamic auxiliary variables is introduced to update the state information of the discrete-time two-dimensional FM model, and the following state information update rules are formulated:

[0013]

[0014] In the formula, Indicates the first The index of the spatial variable of the monitored system when the event is triggered. Indicates the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; inf represents the lower bound function; This indicates the state information under the corresponding spatial variable index when the event is triggered. Status information with the current index position The difference; This indicates the reference signal under the corresponding spatial variable index when the reference model is triggered. Current position reference signal status information The difference; and It is a weighted positive definite matrix; Given a compensation threshold; For a given constant; Represents a dynamic auxiliary vector; The status information that meets the status information update rules is transmitted through the communication network.

[0015] Furthermore, the tracking controller constructed in S3, which is simultaneously affected by hybrid network attacks and the aforementioned error compensation communication mechanism, specifically includes the following tracking controllers:

[0016] In the formula, The tracking controller gain matrix represents the state information of a discrete-time two-dimensional FM model. The tracking controller gain matrix of the reference model; This provides the state information of the spatial variable index at the event trigger location in a discrete-time two-dimensional FM model. For the reference model, the state information of the spatial variable index at the event trigger location, and and All satisfy the error compensation communication mechanism, and the tracking controller obtains the latest transmitted status. and These are the false data signals injected by the attacker into the discrete-time two-dimensional FM model and the reference model at the event trigger location, respectively. Bernoulli random variable and These are used to describe the random transmission of denial-of-service attacks and spoofing injection attacks in a discrete-time two-dimensional FM model communication network, respectively, with the following probabilities:

[0017]

[0018] in Represents Bernoulli random variables Expected value Represents Bernoulli random variables Expected value This represents the probability of a Bernoulli random variable occurring. The mathematical expectation of a Bernoulli random variable; Bernoulli random variable and These are used to describe the random transmission of reference model denial-of-service attacks and spoofing injection attacks in the reference model communication network, respectively, with occurrence probabilities of 1 / 2 and 2 / 3 respectively.

[0019]

[0020] in, Represents Bernoulli random variables Expected value Represents Bernoulli random variables Expected value This represents the probability of a Bernoulli random variable occurring. This represents the mathematical expectation of a Bernoulli random variable.

[0021] Furthermore, S4, based on information update rules and tracking controllers, constructs a closed-loop augmented system model, specifically including: By introducing information update rules and a tracking controller into the discrete-time two-dimensional FM model, and fusing the gain of the tracking controller, the randomness of the attack, and the triggering mechanism of the information update rules, the following closed-loop augmented system model is obtained:

[0022] In the formula, This is a state augmentation vector composed of the state information of the discrete-time two-dimensional FM model and the state information of the reference model. To track the spoofed data augmentation vectors formed by the spoofed data signals injected by the attacker into the discrete-time two-dimensional FM model and the reference model in the controller, This is the augmented vector of state information difference, formed by the state information difference between the discrete-time two-dimensional FM model and the state information difference between the reference model. , These are the state matrices for the horizontal and vertical directions of the closed-loop augmented system, respectively. , The control input matrix is ​​the closed-loop augmented matrix; , This is the coefficient matrix of the nonlinear terms in the closed-loop augmented system model; , These are the coefficient matrices of the external disturbances in the closed-loop augmented system model; Indicates process noise; , These are the coefficient matrices of the reference input in the closed-loop augmented system; Indicates the spatial location of the reference input to the reference model; The output tracking error of the discrete-time two-dimensional FM model and the reference model; This is the output matrix of the closed-loop augmented system model.

[0023] Furthermore, in S5, the gain matrix of the tracking controller is solved based on the closed-loop augmented system model, specifically including: Using Lyapunov stability theory and linear matrix inequality techniques, we derive sufficient conditions for the closed-loop augmented system model to satisfy the final boundedness in the mean-square sense. The gain matrix of the tracking controller is obtained by solving the nonlinear minimization problem with linear matrix inequality constraints using the cone complement linearization algorithm.

[0024] Furthermore, the closed-loop augmented system model satisfies the sufficient condition for eventual boundedness in the mean-square sense, specifically expressed as:

[0025] In the formula, , , These are different matrix sub-blocks. It is a diagonal matrix.

[0026] Furthermore, S6 uses the solved tracking controller to control the discrete-time two-dimensional FM model, enabling the output of the discrete-time two-dimensional FM model to stably track the reference signal under hybrid network attacks, specifically including: The tracking controller gain matrix obtained from the solution of the discrete-time two-dimensional FM model state information. and the tracking controller gain matrix of the reference model. Integrate into the tracking controller; The latest state information is transmitted based on the error compensation communication mechanism using dynamic auxiliary variables. and reference signal status information Calculate the control input of the tracking controller , which acts on the discrete-time two-dimensional FM model; The output of the discrete-time two-dimensional FM model With the reference signal Tracking error between Bounded, and

[0027] In the formula, , It is a constant. It is a positive integer. As a scalar, It is a Lyapunov function. Indicates tracking error norm, Represents the tracking error norm The mathematical expectation.

[0028] On the other hand, the present invention discloses a secure tracking and control device for a discrete-time two-dimensional FM model under hybrid attacks, the device comprising: The data construction module is used to build a discrete-time two-dimensional FM model of the monitored system; The error compensation communication mechanism design module is used to introduce a discrete-time two-dimensional FM model for state information update and transmission by adding corresponding dynamic auxiliary vectors, based on the characteristics of information transmission in two directions in the discrete-time two-dimensional FM model. The closed-loop augmented system model building module utilizes an error compensation communication mechanism based on dynamic auxiliary variables to construct an augmented closed-loop system subjected to hybrid network attacks. The controller solver module is used to determine the control input based on the closed-loop augmented system model. The output tracking control module is used to implement the system output of the effective tracking reference signal for the discrete-time two-dimensional FM model by utilizing the closed-loop augmented system model.

[0029] In another aspect, this application provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, implement the discrete-time two-dimensional FM model security tracking control method under hybrid attacks as described above.

[0030] As can be seen from the above technical solution, the discrete-time two-dimensional FM model secure tracking control method under hybrid attacks disclosed in this invention first constructs a discrete-time two-dimensional FM model, and the reference signal being tracked is generated by the reference model; secondly, in order to save limited network resources, an error compensation communication mechanism with bidirectional evolution indicators is introduced. Simultaneously, considering that the measurement channel is subject to random denial-of-service attacks following a Bernoulli distribution and spoofing attacks, a tracking controller simultaneously affected by hybrid attacks and the error compensation communication mechanism is constructed, thereby establishing a closed-loop augmented system model; finally, based on the designed output tracking controller, the system is securely controlled, enabling the system output to stably track the reference signal even under hybrid network attacks. Compared with existing technologies, the discrete-time two-dimensional FM model designed in this invention is more in line with engineering practice under the security control of the discrete-time two-dimensional FM model because it is subject to uncertainties such as random denial-of-service attacks and fake data injection attacks that follow a Bernoulli distribution.

[0031] Furthermore, the secure tracking controller based on the error compensation communication mechanism designed in this invention can not only achieve the desired tracking control performance under the influence of hybrid network attacks, but also effectively improve the utilization rate of network communication resources, enhance the adaptability to the uncertainty of discrete-time two-dimensional FM models, and broaden the application scope of event-triggered communication mechanisms in multivariable systems. Attached Figure Description

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

[0033] Figure 1 This is a flowchart of the secure tracking and control method for discrete-time two-dimensional FM model under hybrid attacks provided by the present invention; Figure 2 This is the error compensation distribution location map provided by the present invention; Figure 3 This is a map showing the locations of hybrid network attacks provided by the present invention; Figure 4 This is a trajectory diagram of the output tracking error provided by the present invention; Figure 5 This is a comparative trajectory profile of the system output and the reference output when the evolution trajectory of the vertical component is limited to 15 according to the present invention. Figure 6 This is a comparative trajectory profile of the system output and the reference output when the evolution trajectory of the vertical component is limited to 16 according to the present invention; Figure 7 This invention provides a trend chart of the mean square error index of output tracking under different attack probabilities. Figure 8 This is a trend chart of the mean square error index of output tracking under different compensation thresholds provided by the present invention. Detailed Implementation

[0034] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0035] This invention proposes a security tracking control design strategy based on an error-compensated communication mechanism for a discrete-time two-dimensional FM model subjected to hybrid network attacks. The system studied is subject to the combined effects of random denial-of-service attacks following a Bernoulli distribution and spoofed data injection, making it more consistent with actual security control practices.

[0036] In this invention, i and j represent indices, i+1 represents the next index adjacent to index j, and j represents the next index adjacent to index j.

[0037] Example 1 This embodiment provides a secure tracking and control method for a discrete-time two-dimensional FM model under hybrid attacks, including the following steps: S1. Construct a discrete-time two-dimensional FM model of the monitored system and generate the reference signal to be tracked based on the reference model; S2. Within the framework of the discrete-time two-dimensional FM model, based on the transmission characteristics of the discrete-time two-dimensional FM model, an information update scheme for the discrete-time two-dimensional FM model based on error compensation communication is introduced. S3. Based on the introduced error compensation communication mechanism, construct a tracking controller that is simultaneously affected by hybrid attacks and the error compensation communication mechanism. S4. Based on the information update rules and tracking controller, construct a closed-loop augmentation system model; S5. Solve for the gain matrix of the tracking controller based on the closed-loop augmented system model; S6. The output tracking controller obtained by the solution is used to control the discrete-time two-dimensional FM model so that the output of the discrete-time two-dimensional FM model can stably track the reference signal under mixed network attacks.

[0038] This invention introduces an error-compensated communication mechanism with bidirectional evolution indices into the measurement channel of a discrete-time two-dimensional FM model to schedule the communication of measurement data. Simultaneously, considering the impact of hybrid network attacks on system performance, a tracking controller is constructed under the combined influence of hybrid network attacks and the error-compensated communication mechanism. Based on the introduced error-compensated communication mechanism and the tracking controller, a closed-loop augmented system model is constructed, and this control scheme is used to control the secure tracking of the discrete-time two-dimensional FM model. This invention can effectively address the impact of hybrid network attacks on the discrete-time two-dimensional FM model, ensuring that the output of the discrete-time two-dimensional FM model can still stably track the reference signal even under hybrid network attacks, while reducing the burden on the communication network and improving the adaptability to system uncertainties.

[0039] Specifically, in S1, the discrete-time two-dimensional FM model is as follows, and the system in Evolution within the range:

[0040] In the formula, These represent generalized spatial indices in the horizontal and vertical directions, respectively. Given a positive integer; the subscript indicates the spatial location where the system matrix and variables act, consistent with the location of the system state vector, where Indicates the current spatial location. Indicates adjacent spatial positions in the horizontal direction. Indicates the spatial positions of adjacent objects in the vertical direction; Let be the system state vector. To measure the output vector, To control the input vector; The noise represents a process noise sequence, which is a zero-mean Gaussian white noise sequence and satisfies the following conditions: ; , , , , , and These are known constant matrices of appropriate dimensions.

[0041] Reference signal Generated from the following reference model:

[0042] In the formula, the subscript Indicates the current spatial position of the reference model. Indicates the spatial positions of adjacent reference models in the horizontal direction. Indicates the spatial position of the reference model in the vertical direction; and These are the reference model state and the reference input, respectively. , and These are known constant matrices of appropriate dimensions. The boundary conditions of the reference model are zero, i.e., for all generalized spatial indices in the horizontal and vertical directions. They all Established.

[0043] Furthermore, the discrete-time two-dimensional FM model information update scheme for the error compensation communication includes: An information update rule for a discrete-time two-dimensional FM model based on error-compensated communication is as follows:

[0044]

[0045] In the formula, Indicates the first The location of the two-dimensional node where the system is located when the next event is triggered. Indicates the first The two-dimensional node position at the time of the next event-triggered transmission. and These represent the first and second parts of the reference model communication network, respectively. The position of the two-dimensional node when the next event is triggered and the first The position of the two-dimensional node when the event is triggered; and These are the status information of the trigger location. , Status information of the current location and The difference, and It is a weighted positive definite matrix. Given a compensation threshold, For a given constant, Represents a dynamic auxiliary vector; state information of the current position. and Status information of the trigger location , When the difference is large enough, the current point and Selected as the next trigger point , And transmit its state. Furthermore, define... , Then the dynamic auxiliary vector The update rules are as follows:

[0046] In the formula, , And the bounded initial conditions are satisfied. , And only when the status information , With trigger location status information and When the difference is large enough, the dynamic auxiliary vector This will reduce the likelihood of an information update plan being implemented.

[0047] Furthermore, the tracking controller, which combines the hybrid attack and error compensation communication mechanisms, includes: The following is a proposed method for constructing a tracking controller that is simultaneously affected by hybrid attacks and error compensation communication mechanisms:

[0048] In the formula, The gain matrix of the tracking controller for the system state. The tracking controller gain matrix of the reference model; This refers to the system's status information at the event trigger location. As a reference model, the state information at the event trigger location is transmitted only when the trigger condition of the error compensation communication mechanism is met, and the controller can only obtain the latest transmitted state. and These are the nonlinear signals, i.e., spoof data signals, injected by the attacker into the system and reference model at the event trigger location, respectively; for a given constant matrix... and The above attack signals and Each condition is satisfied , This invention utilizes mutually independent random variables that follow a Bernoulli distribution. and This paper describes the occurrence of denial-of-service attacks and spoofing attacks in discrete-time two-dimensional FM model communication networks, respectively, using independent random variables that follow a Bernoulli distribution. and Describe the occurrence of denial-of-service attacks and spoofing attacks in the reference model communication network, respectively. The probability of the above random variables occurring is: , , ,

[0049] in, Represents Bernoulli random variables and Expected value Represents Bernoulli random variables and Expected value Represents a Bernoulli random variable , , and The probability of occurrence Represents Bernoulli random variables , , and The mathematical expectation. In this invention, a first communication network can be used to represent the discrete-time two-dimensional FM model communication network, while a second communication network can be used to represent the reference system communication network. Therefore, there are a total of four possible transmission scenarios in this invention: when At that time, if If so, then the data in the first and second communication networks is transmitted normally; if , If the data transmission in the first communication network is normal, then the data in the second communication network is subjected to a fake data injection attack; if , If the data in the first communication network is subjected to a fake data injection attack, the data in the second communication network will be transmitted normally; if Then, both the first and second communication networks are vulnerable to data injection attacks. , At that time, if If the data in the first communication network is transmitted normally, the data in the second communication network will be subjected to a denial-of-service attack; if If this occurs, the data in the first communication network is vulnerable to a spoofed data injection attack, and the data in the second communication network is vulnerable to a denial-of-service attack. , At that time, if If the data in the first communication network is subjected to a denial-of-service attack, the data in the second communication network will be transmitted normally; if If this occurs, the data in the first communication network is vulnerable to a denial-of-service attack, and the data in the second communication network is vulnerable to a fake data injection attack. If this happens, the data in both the first and second communication networks will be affected by a denial-of-service attack, and the tracking controller will fail.

[0050] Furthermore, the closed-loop augmented system model includes: Specifically, by substituting the information update rules and tracking controller into the discrete-time two-dimensional FM model, and defining... , , The closed-loop augmented system described above can be obtained as follows:

[0051] In the formula, This is an augmented vector of the system state and the reference model state. Augmentation vector for false data signals injected by attackers into the controller. This is an augmented vector representing the difference between the trigger position and the current position state information. To output the tracking error; , These are the state matrices of the closed-loop augmented system in the horizontal and vertical directions, respectively, which include the effects of random mixed attacks; , These are the control input matrices of the closed-loop augmented matrix; ,

[0052] These are the coefficient matrices of the nonlinear terms in the closed-loop augmented system; , , , , , , These represent the nominal state matrices composed of the original matrix and the expected attack parameters, respectively, used to describe the impact of random attacks on the deviation of the system matrix. , , , , , These represent the nominal input matrix, used to describe the impact of random attacks on the deviation of the control input matrix; , , , , , , These represent the nominal nonlinear term matrices, respectively. , These are the coefficient matrices of the external disturbances in the closed-loop augmented system; , These are the coefficient matrices of the reference input in the closed-loop augmented system; This is the output matrix of the closed-loop augmented system.

[0053] Furthermore, to ensure that the aforementioned closed-loop augmented system is ultimately bounded in the exponential sense, this invention utilizes Lyapunov stability theory and linear matrix inequality techniques to analyze the ultimately bounded problem of the closed-loop system, and provides a cone-complement linearization algorithm to solve for the output tracking control gain matrix, thus ensuring the mean-square boundedness of the output tracking error; the specific process is as follows: For the aforementioned closed-loop augmented system, the Lyapunov function is defined as follows:

[0054] In the formula, , These are the selected Lyapunov functions, As a scalar, , , These are positive definite matrices.

[0055] Along any trajectory of the above closed-loop system, the difference of the Lyapunov function is obtained as follows:

[0056] Considering , can be obtained

[0057] In the formula, It is an augmented vector.

[0058] Then the sufficient condition for the closed-loop augmented system to be eventually bounded is:

[0059] In the formula, , , These are matrix sub-blocks; It is a diagonal matrix;

[0060] , , , , , ,

[0061] , , , , ,

[0062] , , , ,

[0063] , , ,

[0064] , ,

[0065] , , , , , , , , , , , , , , , , , , ; It is a diagonal matrix; , , They are positive definite matrices; , , , , , , , and All are scalars;

[0066] Let be a scalar, representing the upper bound of the system's eventual bound, defined by the trace of the external disturbance covariance and the reference input bound, respectively. And the influence of information update rule parameters. Among them, , , , , and The statistical properties of injecting random attack sequences into denial-of-service and spoofed data indicate that the closed-loop system is said to be exponentially bounded in the mean-square sense.

[0067] Furthermore, the mean-square boundedness of the output tracking error is ensured, and a cone-complement linearization algorithm is provided to solve for the output tracking control gain matrix, including: When the tracking error is bounded, the system output can track the reference signal and

[0068] In the formula, , It is a constant. It is a positive integer. As a scalar, It is a Lyapunov function. Indicates tracking error norm, Represents the tracking error norm The mathematical expectation of the output tracking error is given, and the asymptotic upper bound of the output tracking error is given. .

[0069] The solution to the output tracking control gain matrix is ​​obtained by using the cone complement linearization algorithm to solve the nonlinear minimization problem with linear matrix inequality constraints.

[0070] The specific steps of the cone complement linearization algorithm are as follows: 1) Set the maximum number of iterations Find a set that satisfies , Initial feasible solution Let the number of iterations be... ; 2) Solve the following nonlinear minimization problem with linear matrix inequality constraints: ,

[0071] Let its optimal solution be ; 3) Verify whether the optimal solution in step 2) satisfies 1). If it does, then the tracking control gain matrix is ​​obtained. and Exit the program; 4) If If there is no solution, exit; otherwise, let Proceed to step 2) to continue the program.

[0072] Next, the present invention further verifies the correctness and effectiveness of the proposed security control strategy through simulation examples. The specific implementation method is as follows: Consider the industrial heat exchange process described by the following partial differential equation:

[0073] in, It is related to spatial dimension and time dimension The relevant temperature function, where, and It is a positive scalar. It is a given force function. and These are the heat exchange coefficients in industrial heat exchange processes. (Definition) ,in , Then you can get , Therefore, the above equation can be approximated as a discrete-time two-dimensional FM model. , , ,

[0074] From a practical application perspective, the system may be affected by noise pollution from a non-ideal chemical reactor, thus impacting system performance. Subsequently, parameters are selected based on control practices. , , , The discrete-time two-dimensional FM model then has the following parameters: , , , , , ,

[0075] when At that time, the reference model has the following parameters: , ,

[0076] The initial state values ​​of the discrete-time two-dimensional FM model and the reference model are , , ,

[0077] In addition, the initial value of the dynamic auxiliary variable is set to The remaining parameters for event-triggered communication are set to , and The variance of the measured noise is... The reference input signal satisfies Hybrid network attacks occur in intervals. The probabilities of denial-of-service attacks and spoofed data injection attacks are respectively... and Furthermore, spurious data injection signals and constant matrices and for , , ,

[0078] Specific simulation graphics are attached. Figure 2-7 As shown Figure 2 The distribution locations of error compensation are given, where a vertical axis of "1" indicates an error-compensated location, and a vertical axis of "0" indicates that the current location is not compensated for communication. Figure 2 It can be seen that the error compensation communication strategy adopted in this paper effectively reduces the network communication burden. Figure 3 The occurrence in the interval is given The diagram shows a hybrid network attack, with a yellow asterisk indicating a denial-of-service attack and a blue triangle indicating a fake data injection attack. Figure 4 The trajectory of the output tracking error is given. To more intuitively illustrate the superior performance of the tracking control scheme designed in this paper, the evolution trajectory of the vertical component is now limited to between 15 and 16. The comparison trajectory profiles of the system output and the reference output are shown below. Figure 5 , Figure 6 As shown. Among them, Figure 5 When the evolution trajectory of the vertical component is restricted to 15, the evolution trajectory profiles of the system output and the reference output in the horizontal direction are compared. Figure 6 To limit the evolution trajectory of the vertical component to 16, a comparative trajectory profile of the system output and reference output in the horizontal direction is shown. Figure 5 , Figure 6 The green curve represents the evolution curve of the system output, and the orange curve represents the evolution curve of the reference output. Figure 5 , Figure 6 As can be seen, under the proposed error-compensated communication security tracking control scheme, even though the system is subjected to both DoS and FDI attacks, the output trajectory of the system can still effectively track the expected output trajectory during the industrial heat exchange process of the actual thermal process control system. This step verifies the effectiveness and practicality of the control scheme in this paper.

[0079] To quantitatively evaluate the impact of hybrid network attack strength and error compensation threshold on system tracking performance, the mean square error (MSE) index function of output tracking is defined as follows:

[0080] Figure 7 Table 1 and others respectively show the intervals... The probability of different denial-of-service attacks and spoofing attacks, and the changing trends of the MSE (Mean Search Engine Optimization) metric. From Figure 7As can be seen from Table 1, the output tracking MSE index will increase with the increase of the probability of hybrid network attacks. The higher the probability of network attacks, the greater the destructive impact. Figure 8 Table 2 and others respectively show the intervals. The changing trends of the MSE index are tracked using different compensation thresholds and outputs. Figure 8 As shown in Table 2, the output tracking MSE index increases with the increase of the compensation threshold. By comparing the occurrence probabilities of different hybrid network attacks, it can be found that the output tracking MSE index under the control strategy in this paper is always relatively small, which confirms the superiority of the tracking control scheme constructed in this embodiment.

[0081] Table 1. MSE index under different attack probabilities

[0082] Table 2 MSE index under different compensation thresholds

[0083] The security tracking and control method for discrete-time two-dimensional FM models under hybrid attacks disclosed in this invention can also be used in seismic data analysis systems. In this case, a discrete-time two-dimensional FM model of the seismic data analysis system is established, and the state vector of this discrete-time two-dimensional FM model is... Includes seismic wave propagation characteristic parameters and ground displacement vectors. The output vector of the seismic sensor includes the seismic waveform data collected by the seismic sensor. In the seismic data analysis system, the i-direction can correspond to the spatial domain and the j-direction can correspond to the time domain. Through the tracking control method of the present invention, the output u(i,j) of the tracking controller is obtained and input to the actuator of the seismic monitoring system. The actuator includes an earthquake early warning information release device and / or an earthquake monitoring equipment control device to maintain the stable operation and data integrity of the information transmission of the seismic monitoring system during denial-of-service attacks.

[0084] The method of this invention can be applied in other systems, such as complex image processing systems, thermal process control systems, or circuit analysis systems, by referring to the specific applications of the seismic data analysis system. These will not be described in detail here.

[0085] Example 2 On the other hand, the present invention discloses a secure tracking and control device for a discrete-time two-dimensional FM model under hybrid attacks, the device comprising: The data construction module is used to build the data model for the discrete-time two-dimensional FM model; The error compensation communication mechanism design module is used to introduce an information update scheme for the discrete-time two-dimensional FM model based on the error compensation communication mechanism by adding corresponding dynamic auxiliary vectors, based on the characteristics of information transmission in two directions of the discrete-time two-dimensional FM model. The closed-loop augmented system model building module is used to construct an augmented closed-loop system subjected to hybrid network attacks based on an error-compensated communication mechanism. The controller solver module is used to determine the control input based on the closed-loop augmented system model. The output tracking control module is used to implement the system output of the effective tracking reference signal for the discrete-time two-dimensional FM model by utilizing the closed-loop augmented system model.

[0086] Example 3 This application provides a computer-readable storage medium storing computer-executable instructions, which, when executed by a processor, implement the above-described secure tracking and control method for a discrete-time two-dimensional FM model under hybrid attacks.

[0087] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0088] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks, characterized in that, Includes the following steps: S1. Construct a discrete-time two-dimensional FM model of the monitored system and generate the reference signal to be tracked based on the reference model; S2. Introduce an error compensation communication mechanism based on dynamic auxiliary variables to update the state information of the discrete-time two-dimensional FM model and transmit it through a communication network; specifically, this includes formulating the following state information update rules: In the formula, Indicates the first The index of the spatial variable of the monitored system when the event is triggered. Indicates the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; inf represents the lower bound function; This indicates the state information under the corresponding spatial variable index when the event is triggered. Status information with the current index position The difference; This indicates the reference signal under the corresponding spatial variable index when the reference model is triggered. Current position reference signal status information The difference; and It is a weighted positive definite matrix; Given a compensation threshold; For a given constant; Represents a dynamic auxiliary vector; The status information that satisfies the status information update rules is transmitted through a communication network; S3. Construct a tracking controller that is simultaneously affected by hybrid network attacks and the error compensation communication mechanism; specifically, it includes the following tracking controller: In the formula, The tracking controller gain matrix represents the state information of a discrete-time two-dimensional FM model. The tracking controller gain matrix of the reference model; This provides the state information of the spatial variable index at the event trigger location in a discrete-time two-dimensional FM model. For the reference model, the state information of the spatial variable index at the event trigger location, and and All satisfy the error compensation communication mechanism; and These are the false data signals injected by the attacker into the discrete-time two-dimensional FM model and the reference model at the event trigger location, respectively. Bernoulli random variable and These are used to describe the random transmission of denial-of-service attacks and spoofing injection attacks in discrete-time two-dimensional FM model communication networks, respectively. Bernoulli random variable and These are used to describe the random transmission of reference model denial-of-service attacks and spoofed data injection attacks in the reference model communication network, respectively. S4. Substitute the information update rules of the error compensation communication mechanism and the tracking controller into the discrete-time two-dimensional FM model to construct a closed-loop augmented system model; S5. Solve for the gain matrix of the tracking controller based on the closed-loop augmented system model; S6. The tracking controller obtained by the solution is used to control the discrete-time two-dimensional FM model so that the output of the discrete-time two-dimensional FM model can stably track the reference signal under hybrid network attacks.

2. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 1, characterized in that, In S1, a discrete-time two-dimensional FM model of the monitored system is constructed, specifically represented as the following model: In the formula, and These represent indices of generalized spatial variables in the horizontal and vertical directions, respectively. Indicates the monitored system in the index The state information vector at that location; Indicates the monitored system in the index The output information vector at that location; Indicates process noise; To track the control input vector of the controller; , , , , , and These are known, well-dimensioned constant matrices corresponding to the indices.

3. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 1, characterized in that, In S1, the monitored system includes an earthquake data analysis system, a complex image processing system, a thermal process control system, or a circuit analysis system.

4. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 1, characterized in that, S4, based on information update rules and tracking controllers, constructs a closed-loop augmenting system model, specifically including: By introducing information update rules and a tracking controller into the discrete-time two-dimensional FM model, and fusing the gain of the tracking controller, the randomness of the attack, and the triggering mechanism of the information update rules, the following closed-loop augmented system model is obtained: In the formula, This is a state augmentation vector composed of the state information of the discrete-time two-dimensional FM model and the state information of the reference model. To track the spoofed data augmentation vectors formed by the spoofed data signals injected by the attacker into the discrete-time two-dimensional FM model and the reference model in the controller, This is the augmented vector of state information difference, formed by the state information difference between the discrete-time two-dimensional FM model and the state information difference between the reference model. , These are the state matrices for the horizontal and vertical directions of the closed-loop augmented system, respectively. , The control input matrix is ​​the closed-loop augmented matrix; , This is the coefficient matrix of the nonlinear terms in the closed-loop augmented system model; , These are the coefficient matrices of the external disturbances in the closed-loop augmented system model; Indicates process noise; , These are the coefficient matrices of the reference input in the closed-loop augmented system; Indicates the spatial location of the reference input to the reference model; The output tracking error of the discrete-time two-dimensional FM model and the reference model; This is the output matrix of the closed-loop augmented system model.

5. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 1, characterized in that, S5 involves solving for the gain matrix of the tracking controller based on the closed-loop augmented system model, specifically including: Using Lyapunov stability theory and linear matrix inequality techniques, we derive sufficient conditions for the closed-loop augmented system model to satisfy the final boundedness in the mean-square sense. The gain matrix of the tracking controller is obtained by solving the nonlinear minimization problem with linear matrix inequality constraints using the cone complement linearization algorithm.

6. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 5, characterized in that, The closed-loop augmented system model satisfies the sufficient condition for eventual boundedness in the mean-square sense, specifically expressed as: In the formula, , , These are different matrix sub-blocks. It is a diagonal matrix.

7. The secure tracking control method for a discrete-time two-dimensional FM model under hybrid attacks according to claim 1, characterized in that, S6 uses the solved tracking controller to control the discrete-time two-dimensional FM model, so that the output of the discrete-time two-dimensional FM model stably tracks the reference signal under hybrid network attacks, specifically including: The tracking controller gain matrix obtained from the solution of the discrete-time two-dimensional FM model state information. and the tracking controller gain matrix of the reference model. Integrate into the tracking controller; The latest state information is transmitted based on the error compensation communication mechanism using dynamic auxiliary variables. and reference signal status information Calculate the control input of the tracking controller , which acts on the discrete-time two-dimensional FM model; The output of the discrete-time two-dimensional FM model With the reference signal Tracking error between Bounded, and In the formula, , It is a constant. It is a positive integer. As a scalar, It is a Lyapunov function. Indicates tracking error norm, Represents the tracking error norm The mathematical expectation.

8. A secure tracking and control device for a discrete-time two-dimensional FM model under hybrid attacks, characterized in that, include: The data construction module is used to build a discrete-time two-dimensional FM model of the monitored system; The error-compensated communication mechanism design module, based on the characteristics of information transmission in two directions in a discrete-time two-dimensional FM model, introduces a state information update and transmission mechanism for the discrete-time two-dimensional FM model by adding corresponding dynamic auxiliary vectors. Specifically, it includes defining the following state information update rules: In the formula, Indicates the first The index of the spatial variable of the monitored system when the event is triggered. Indicates the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; In the reference model communication network, the first The index of the spatial variable when the event is triggered; inf represents the lower bound function; This indicates the state information under the corresponding spatial variable index when the event is triggered. Status information with the current index position The difference; This indicates the reference signal under the corresponding spatial variable index when the reference model is triggered. Current position reference signal status information The difference; and It is a weighted positive definite matrix; Given a compensation threshold; For a given constant; Represents a dynamic auxiliary vector; The status information that satisfies the status information update rules is transmitted through a communication network; The closed-loop augmented system model construction module utilizes an error compensation communication mechanism based on dynamic auxiliary variables to construct an augmented closed-loop system subjected to hybrid network attacks. Specifically, the information update rules of the error compensation communication mechanism and the tracking controller simultaneously affected by hybrid network attacks and the error compensation communication mechanism are substituted into the discrete-time two-dimensional FM model to construct the closed-loop augmented system model. The tracking controller specifically includes: In the formula, The tracking controller gain matrix represents the state information of a discrete-time two-dimensional FM model. The tracking controller gain matrix of the reference model; This provides the state information of the spatial variable index at the event trigger location in a discrete-time two-dimensional FM model. For the reference model, the state information of the spatial variable index at the event trigger location, and and All satisfy the error compensation communication mechanism; and These are the false data signals injected by the attacker into the discrete-time two-dimensional FM model and the reference model at the event trigger location, respectively. Bernoulli random variable and These are used to describe the random transmission of denial-of-service attacks and spoofing injection attacks in discrete-time two-dimensional FM model communication networks, respectively. Bernoulli random variable and These are used to describe the random transmission of reference model denial-of-service attacks and spoofed data injection attacks in the reference model communication network, respectively. The controller solver module is used to determine the control input based on the closed-loop augmented system model. The output tracking control module is used to utilize the closed-loop augmented system model to achieve effective tracking of the system output reference signal for the discrete-time two-dimensional FM model.

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