Safety synchronization control method and system of multiple coupling time-delay network by using impulsive perturbation

Abstract

The application provides a kind of multiple coupling time delay network's restraint pulse safety synchronization control method and system, including the technical scheme of establishing coupling time delay dynamic network model, constructing the mathematical description of random deception attack, designing restraint pulse safety controller and deducing Lyapunov stability discriminant, so that, by defining the error system of following network and target network, under the restraint pulse safety control framework, based on pulse system theory, analyze system stability, combine with bernoulli random variable to describe attack probability, finally design restraint pulse control node selection strategy, successfully realize safety synchronization control.The mechanism can effectively extract the time delay information with stabilizing effect in continuous system and integrate it into the design of attack-resistant restraint pulse safety controller, thereby significantly improving system synchronization safety, optimizing resource allocation efficiency, and weakening the impact of attacks using inherent time delay.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of complex network control, and in particular to a containment pulse safe synchronization control method and system for a multiple-coupling time-delay network. BACKGROUND

[0002] Complex coupled networks, as a core model to describe highly interconnected systems (e.g., biological, engineering, social systems), have been extensively studied in the fields of nonlinear dynamics, complex networks and control. Such networks are composed of a large number of nodes, each of which carries nonlinear dynamic behavior and interacts with other subsystems through a coupling topology (characterized by connection strength and structure). Their applications cover natural / engineering systems (e.g., smart grid control), artificial systems (e.g., information propagation modeling) and hybrid systems (e.g., robustness analysis of digital infrastructure). Network synchronization control is a core challenge in the study of such systems, and time-delay coupling is inevitable as an intrinsic property of the network. Due to non-uniform topology, differential communication time delays are caused, and external attacks or faults induce time delay disturbances. Ignoring time delays will lead to misjudgment of the real dynamic behavior of the network.

[0003] In complex dynamic networks composed of strongly coupled nodes, synchronization behavior not only reflects the self-organizing ability of the system, but also is the core basis for in-depth analysis of the internal operation mechanism of practical complex systems such as mechatronic system cooperation, stable operation of power networks, and discharge of biological neuron clusters. However, when a multiple-coupling network is affected by parameter perturbation, external disturbance or local fault, it is difficult to achieve reliable establishment, accurate maintenance and efficient recovery of the desired global / cluster synchronization state by relying solely on system self-organization. To improve the anti-interference performance, optimize the convergence speed and energy consumption index of the synchronization process, an active control strategy is needed to shape the network collective dynamic synchronization behavior.

[0004] Industrial control systems rely heavily on information communication technology, and their inherent network security vulnerability poses risks of operational anomalies, environmental damage and economic chain losses to critical infrastructure. Among them, the vulnerability of control data transmission channels is a core threat. Under the multiple constraints of confidentiality, availability and integrity, deception attacks (which damage system integrity by injecting hidden data) and denial of service attacks (DoS) constitute the main risk sources. Since deception attacks have the characteristics of hidden tampering, their harmfulness and detection difficulty are significantly higher than DoS attacks. To ensure the safe synchronization of complex coupled networks under deception attacks, a non-continuous pulse control mechanism compresses the attack exposure window, which is an effective solution to improve system flexibility and communication efficiency. Moreover, the inherent multiple time delay effects (communication delay, node processing delay, etc.) of industrial systems may induce instability phenomena such as chaotic oscillation.

[0005] Therefore, how to improve the security synchronization performance of complex networks under deception attacks, optimize the efficiency of control resources, and use time delay characteristics to weaken the impact of attacks are technical problems that urgently need to be solved by those skilled in the art. Summary of the Invention

[0006] This invention provides a method and system for secure synchronization control of restraint pulses in multi-coupled time-delay networks, which can improve the secure synchronization performance of complex networks under deception attacks, optimize control resource efficiency, and reduce the impact of attacks by utilizing time-delay characteristics.

[0007] On one hand, the present invention provides a method for safe synchronization control of a restraint pulse in a multi-coupled time-delay network, comprising: establishing a coupled time-delay dynamic network model, wherein the coupled time-delay dynamic network model includes a follower network model, a target network model, and an error system; the follower network model is a dynamic model of the state of the follower node relative to time and time delay; the target network model is a dynamic model of the state of the leader node relative to time and time delay; and the error system is the difference between the follower network model and the target network model.

[0008] Establish the Lyapunov stability criterion for synchronization error: Define the control input under the impulse control framework, model the error system, and derive the sufficient condition for the mean square stability of the error system as the Lyapunov stability criterion based on hybrid control theory and Lyapunov stability theory.

[0009] A deception attack model is introduced: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into communication channels.

[0010] Design of a restraint pulse security controller: Based on the Lyapunov stability criterion and the deception attack model, a restraint pulse security controller is designed so that the error system reaches mean square bounded stability under the influence of deception attacks, thereby achieving mean square secure synchronization between the follower network and the target network.

[0011] On the other hand, the present invention also provides a restraint pulse safety synchronization control system for a multi-coupled time-delay network, comprising:

[0012] The first model building module is used to build a coupled time-delay dynamic network model. The coupled time-delay dynamic network model includes a follower network model, a target network model, and an error system. The follower network model is a dynamic model of the state of the follower node relative to time and time delay. The target network model is a dynamic model of the state of the leader node relative to time and time delay. The error system is the difference between the follower network model and the target network model.

[0013] The discriminant establishment module is used to establish the Lyapunov stability discriminant for the synchronization error: under the pulse control framework, the control input is defined, the error system is modeled, and based on hybrid control theory and Lyapunov stability theory, the sufficient condition for the mean square stability of the error system is derived as the Lyapunov stability discriminant.

[0014] The second model building module is used to introduce a deception attack model: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into the communication channel.

[0015] The design module is used to design a restraint pulse security controller: based on the Lyapunov stability discriminant and the deception attack model, a restraint pulse security controller is designed so that the error system reaches mean square bounded stability under the influence of deception attacks, thereby achieving mean square secure synchronization between the follower network and the target network.

[0016] On the other hand, the present invention also provides an electronic device, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the restraint pulse safety synchronization control method of the multi-coupled time-delay network as described above.

[0017] On the other hand, the present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the restraint pulse safety synchronization control method for a multi-coupled time-delay network as described above.

[0018] On the other hand, the present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the restraint pulse safety synchronization control method for the multi-coupled time-delay network as described above.

[0019] The present invention provides a method and system for secure synchronization control of pinned pulses in multi-coupled time-delay networks. By constructing a multi-delay coupling model, a mathematical description of random spoofing attacks, designing a pinned pulse security controller, and deriving the Lyapunov stability criterion, the present invention can effectively extract time-delay information with stabilizing effects from continuous systems and integrate it into the design of the anti-attack controller. This significantly improves the system's synchronization security, optimizes resource allocation efficiency, and weakens the impact of attacks by utilizing inherent time delays. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0021] Figure 1 This is a flowchart illustrating the method for secure synchronization control of a multi-coupled time-delay network for confinement pulses provided in an embodiment of the present invention.

[0022] Figure 2a This is a schematic diagram of the simulation results of the restraint pulse safety synchronization control method of the multi-coupled time-delay network of the present invention;

[0023] Figure 2b yes Figure 2a (1) Enlarged view of the range from 80s to 250s;

[0024] Figure 2c yes Figure 2a (2) Enlarged view of the range from 80s to 250s in the middle;

[0025] Figure 3 This is a schematic diagram of the structure of the restraint pulse safety synchronization control system of the multi-coupled time delay network provided in the embodiment of the present invention;

[0026] Figure 4 This is a schematic diagram of the structure of the electronic device provided in an embodiment of the present invention. Detailed Implementation

[0027] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0028] In related technologies, secure synchronization control of complex coupled networks faces multiple challenges, especially in the presence of time delays and spoofing attacks. Traditional methods typically assume an idealized, time-delay-free environment and require control over all nodes, leading to high resource consumption and difficulty in responding to random attacks. Network nodes in industrial control systems often experience time delays due to communication latency and external interference, and existing technologies fail to effectively utilize the potential stabilizing properties of time delays while lacking robustness under attacks, making it difficult to achieve efficient and secure synchronization.

[0029] Furthermore, the core of restraint control lies in selectively applying control inputs to only a select few key nodes (or leader nodes) in the network. This leverages the network's inherent coupling and propagation characteristics to guide the entire network towards desired synchronization or dynamic behavior, significantly reducing the demand for control resources (such as energy, actuators, and communication bandwidth) and resolving the cost and implementation infeasibility of controlling all nodes in large-scale networks. Simultaneously, facing the extreme complexity introduced by node heterogeneity, topological non-uniformity, and multiple time-delay couplings, restraint control provides a structured and efficient framework. It focuses on regulating key influential nodes, avoiding the burden of handling complex interactions across the entire network. Crucially, in multi-time-delay environments, restraint control, by concentrating resources on key nodes and combining them with time-delay-aware control law design (such as predictive compensation and robust control), offers a more operational control scheme for effectively offsetting or adapting to multiple time-delay instability effects and achieving stable synchronization. Therefore, developing efficient restraint control strategies based on time-delay coupling models is a key theoretical and technological foundation for solving the synchronization control problem of large-scale complex networks and improving their controllability and robustness, and it has significant value for overcoming bottlenecks in related engineering applications.

[0030] Therefore, to address the aforementioned issues, research has revealed that time delay can produce a system stabilizing effect under specific conditions; however, current research has not effectively integrated this effect with security control strategies. Designing discontinuous control mechanisms to reduce the attack exposure window while simultaneously optimizing control efficiency using time delay characteristics has become a key breakthrough. By analyzing the time discreteness characteristics of pulse control, it was found that it can compress the attack duration. Combined with the node selection mechanism of the restraint strategy, a resource-efficient control framework is formed. Further exploration of the synergistic relationship between time delay parameters and pulse intervals, and the establishment of a time delay-dependent stability criterion, provides theoretical support for anti-attack synchronous control.

[0031] Therefore, this invention proposes a technical solution including establishing a coupled time-delay dynamic network model, deriving the Lyapunov stability criterion, constructing a deception attack model, and designing a restraining pulse security controller. This method defines the error system between the follower network and the target network, analyzes system stability within a pulse control framework, describes the attack probability using Bernoulli random variables, and finally designs a node selection strategy to achieve secure synchronization control.

[0032] Specifically, Figure 1 This is a flowchart illustrating the method for secure synchronization control of a multi-coupled time-delay network for confinement pulses provided in an embodiment of the present invention.

[0033] like Figure 1 As shown, the execution subject of the restraint pulse safety synchronization control method for multi-coupled time-delay networks provided in this embodiment of the invention can be an electronic device, and the method mainly includes the following steps:

[0034] 101. Establish a coupled time-delay dynamic network model, which includes a follower network model, a target network model, and an error system. The follower network model is a dynamic model of the state of the follower node relative to time and time delay.

[0035] Wherein, the target network model is a dynamic model of the leader node state relative to time and time delay; the error system is the difference between the follower network model and the target network model;

[0036] 102. Establish the Lyapunov stability criterion for synchronization error: Define the control input under the impulse control framework, model the error system, and derive the sufficient condition for the mean square stability of the error system based on hybrid control theory and Lyapunov stability theory as the Lyapunov stability criterion.

[0037] 103. Introduction of a deception attack model: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into the communication channel.

[0038] 104. Design of a restraint pulse security controller: Based on the Lyapunov stability criterion and the deception attack model, a restraint pulse security controller is designed so that the error system achieves mean-square bounded stability under the influence of deception attacks, thereby realizing mean-square secure synchronization between the follower network and the target network.

[0039] In a specific implementation, the coupled time-delay dynamic network model refers to a nonlinear dynamic model that includes time-varying and delayed states. Specifically, it can use differential equations to describe the changes in node states over time and historical states, providing a mathematical foundation for analyzing time-delay effects. The Lyapunov stability criterion is a stability criterion built upon an energy function, derived using exponential functionals combined with time-delay integral terms. This criterion quantifies the constraint relationship between time-delay parameters and control parameters. The spoofing attack model is a communication channel interference characterization method based on probability distributions. Specifically, it can use Bernoulli random variables to describe the probability of a successful attack, providing a stochastic process basis for controller anti-interference design. The restrained pulse safety controller is a discrete control strategy that selects the controlled object based on node error states. Specifically, it can use norm sorting to dynamically select key nodes for pulse excitation, achieving optimal allocation of control resources.

[0040] Specifically, firstly, a differential equation for the following network, including time-delay terms, is constructed, defining the target network state as the synchronization benchmark. An error system model is established by calculating the state difference between the two, containing time-delay coupling terms and nonlinear dynamic terms. Based on hybrid system theory, a pulse control input is designed to apply control to selected nodes at discrete times. A Lyapunov function containing time-delay integral terms is constructed, and the system energy change trend during continuous and pulsed phases is analyzed, deriving parameter constraints to ensure mean-square stability. A Bernoulli stochastic process is introduced to simulate a successful attack event, transforming the attack impact into system disturbances under expected computation. Based on the real-time error state ranking results, the nodes with the largest deviations are dynamically selected to form a control set. Pulse gain parameters are designed to satisfy stability conditions, ultimately achieving bounded stability of the error system under attack influence.

[0041] This invention incorporates time delay parameters into the stability analysis framework, quantifying the impact of delay on system energy through the time delay integral term, thereby achieving synergistic optimization of time delay characteristics and control parameters. Compared to the all-node pulse control scheme, the dynamic restraint strategy concentrates control action on key nodes, reducing control costs while ensuring synchronization performance. The joint design of the deception attack model and stability criterion enables the controller to adaptively adjust the pulse intensity to counteract attack disturbances.

[0042] Through the above technical solution, this invention can effectively extract time-delay information with stabilizing effects from continuous systems and integrate it into the design of an anti-attack controller, effectively solving the network synchronization control problem under the combined effects of multiple time delays and random attacks. The dynamic node selection mechanism significantly reduces control energy consumption, the time-delay-dependent stability criterion improves system convergence speed, and the probabilistic attack model enhances the robustness of the control strategy. This method provides a feasible solution for the secure operation of large-scale industrial networks, achieving efficient utilization of control resources while ensuring synchronization accuracy.

[0043] In a specific implementation process, the method for designing a restraint pulse security controller may include selecting n nodes from the follower network model to form a restraint node set, under the constraints of mean-square stability corresponding to the Lyapunov stability criterion and the constraint of cancelling the deception attack signal of the deception attack model; based on the restraint node set, the model of the restraint pulse security controller is obtained as follows: .in, Indicates the pulse control gain; Represents the set of restraining nodes, when When the i-th node is a controlled node, the i-th node exercises control; when When the i-th node is an unconstrained node, the i-th node does not exercise control. .

[0044] In a specific implementation, the restraint node set refers to a dynamically selected subset of nodes based on the norm of the error system. Specifically, this can be achieved by sorting N nodes according to the norm of their error states at the pulse moment and selecting a predetermined proportion of nodes. The pulse control gain refers to the adjustment coefficient applied by the controller at the pulse moment, which can be calculated using parameter constraints in the stability discriminant formula. It is used to adjust the control strength to counteract the attack's impact. The restraint pulse security controller model refers to a hybrid control structure combining pulse control and restraint strategies. Specifically, this can be implemented by applying pulse input only to selected nodes, guiding network-wide synchronization through periodic intervention at key nodes.

[0045] Specifically, at the pulse moment, the norm of the error state of all nodes in the following network is calculated and sorted in descending order. The top n nodes with the largest errors are selected according to a preset ratio to form a restraining node set. Nodes not selected receive no control input at this moment. For the selected nodes, a feedback control quantity adjusted by the pulse control gain is applied at the pulse moment. This gain must satisfy the constraints regarding time delay parameters and attack probability in the stability conditions. By periodically updating the restraining node set and adjusting the pulse gain, the controller can dynamically adapt to changes in network state and attack interference, minimizing control resource consumption while ensuring synchronization performance.

[0046] This method prioritizes controlling nodes with large deviations through an error sorting mechanism, and combines this with time-delay-dependent adjustment of pulse gain. This enables synchronization with fewer nodes even when an attack is present, while avoiding the risk of oscillation caused by over-control.

[0047] Through the above technical solution, this invention achieves dynamic optimization of control node selection and pulse intensity adjustment under deception attack environments, effectively reducing control energy consumption and improving synchronization accuracy. This method periodically updates the set of restraining nodes, ensuring that control resources are concentrated on the nodes with the greatest impact on system stability, significantly reducing the number of nodes requiring control while maintaining mean-square bounded stability. By combining pulse gain design with time delay parameters and attack probability, the control intensity can be adaptively adjusted to counteract attack interference, avoiding the performance degradation problem of traditional fixed-gain controllers when attack intensity changes.

[0048] In some embodiments, the dynamic node selection mechanism can be as follows: under the constraint of mean square stability corresponding to the Lyapunov stability discriminant and the constraint of canceling the deception attack signal of the deception attack model, a method for selecting n nodes from the follower network model to form a restraining node set includes sorting the N nodes at the k-th pulse time according to the norm of the error system, and selecting a set proportion of n nodes in descending order to form a restraining node set.

[0049] The error system's norm ranking refers to sorting nodes in descending order based on the Euclidean norm of their synchronization error with the target node. This can be achieved by calculating the squared L2 norm of each node's error vector, used to identify the node most severely deviating from the target state at the current moment. The predetermined proportion of n nodes refers to the number of nodes requiring pulse control, pre-determined based on the actual network size and control resource constraints. This can be achieved using a fixed proportion or an adaptively adjusted proportion; for example, when the total number of network nodes is large, the top 50% of nodes can be selected as the restraint set.

[0050] Specifically, at each pulse trigger moment, the synchronization error vector norm of all following nodes relative to the target node is first calculated, forming a sorted list of error amplitudes. The top n nodes with the largest error amplitudes are selected as the restraint control objects for the current pulse cycle, and pulse control signals are applied to them. This dynamic node selection mechanism can prioritize suppressing the nodes that have the greatest impact on overall synchronization performance, allowing limited control resources to be concentrated on key disturbance sources. For example, in a network containing 100 nodes, when the error norm of 50 nodes is detected to be significantly higher than that of other nodes, pulse control is applied only to these 50 nodes, while the remaining 50 nodes remain uncontrolled, thereby reducing control energy consumption while ensuring synchronization accuracy.

[0051] This method achieves dynamic node optimization through error norm sorting, ensuring that pulse control energy is precisely applied to the nodes that most affect synchronization stability, thereby improving error convergence speed under the same control cost. Simultaneously, the dynamic selection mechanism can adapt to network topology changes and attack interference, avoiding the performance degradation caused by the failure of fixed nodes.

[0052] Through the above technical solution, this invention can achieve adaptive matching between control resources and network synchronization requirements. In scenarios where deception attacks cause a sudden increase in errors at local nodes, it prioritizes suppressing the spread of deviations at critical nodes, effectively reducing the peak amplitude of the overall network synchronization error. This method effectively improves the energy utilization rate of pulse control and reduces the number of node controls while maintaining the same synchronization accuracy.

[0053] In some embodiments, the following description uses a follower network model with 100 nodes (i.e., N=100) as an example to illustrate the expression of the model involved. The present invention further proposes a specific expression for the follower network model as Equation (1):

[0054] (1)

[0055] in, This represents the rate of change of the state vector of the i-th node at time t. , Let represent the state vector of the i-th node at time t. Let represent the time-delayed state vector of the i-th node at time t, and , Represents the set of m-dimensional real-valued vectors; Indicates coupling strength. Indicates the time-delay coupling strength; Indicates time delay in a continuous system; Represents the internal coupling matrix. Represents the internal coupling matrix of the time delay. Represents an m-dimensional real matrix; Represents a nonlinear function; Indicates control input; Indicates coupling weight. This represents the time-delay coupling weight.

[0056] In a specific implementation, the state vector change rate refers to the rate at which the node state changes over time. This can be mathematically modeled using differential equations to quantify the node's dynamic characteristics. Coupling strength refers to the degree of influence of interactions between nodes, which can be calibrated using adjacency matrix elements to adjust the influence of network topology on the synchronization process. Time-delay coupling strength refers to the influence coefficient of coupling between nodes under time-delay conditions, which can also be calibrated using time-delay adjacency matrix elements to characterize the delay effect of historical states on current dynamics. Time delay refers to the delay time generated by data transmission or signal processing, which can be described using fixed values ​​or time-varying functions to reflect the actual transmission characteristics of the network communication channel. The internal coupling matrix describes the coupling relationship between the internal state components of a node, which can be implemented using diagonal or symmetric matrices to construct the interaction mechanism between multidimensional state variables. The nonlinear function describes the node's own dynamic characteristics, which can be modeled using Lipschitz continuous functions to characterize the nonlinear behavior of nodes in complex networks. The control input refers to the external intervention signal applied to the node, which can be implemented using pulse control signals to adjust the node state to converge towards the target trajectory.

[0057] Specifically, this technical solution precisely characterizes the dynamic behavior of a single node in a multi-coupled time-delay network by establishing a differential equation containing time-delay terms and non-linear terms. In the model, the coupling strength and the time-delay coupling strength respectively quantify the influence degrees of the current state and the historical state on the evolution of the node. The internal coupling matrix constructs the interaction rules between multi-dimensional state components, and the time-delay parameter reflects the inherent transmission delay characteristics of the network communication channel. The introduction of the non-linear function enables the model to describe the complex dynamic behaviors commonly existing in actual network nodes, and the control input term provides a mathematical interface for the subsequent design of the pulse controller. Through the corresponding relationship between the coupling weight matrix and the Laplace matrix, this model transforms the abstract network topology structure into computable mathematical parameters, establishing a strict mathematical foundation for stability analysis and controller design.

[0058] Compared with the prior art, traditional methods often ignore the time-delay coupling term or adopt linear approximation to deal with non-linear characteristics when establishing a network model, resulting in insufficient model accuracy. There are few comprehensive modeling methods in the existing literature that simultaneously consider multi-coupling, time-delay effects, and non-linear dynamics, and the matrix corresponding relationship between the coupling weight and the time-delay coupling weight is not clearly defined. The present invention constructs a dynamic equation with strict mathematical definitions by introducing double coupling strength parameters and a time-delay internal coupling matrix, which can more realistically reflect the time-delay characteristics and coupling mechanisms of actual complex networks.

[0059] Through the above technical solution, the present invention constructs a mathematical model that accurately describes the dynamic behavior of nodes in a multi-coupled time-delay network, clarifies the mapping relationship between network topology parameters and the dynamic equation, and provides a necessary mathematical foundation for subsequent stability analysis and pulse controller design. This model can accurately characterize the influence of time-delay effects and non-linear characteristics on the network synchronization process, enabling the design of control strategies to have clear parameter bases, and effectively improving the anti-interference ability and control accuracy of the synchronization control system.

[0060] In some embodiments, the present invention further proposes technical means for extracting the coupling weight based on the coupling weight matrix and the Laplace matrix, and for extracting the time-delay coupling weight based on the time-delay coupling weight matrix or the time-delay Laplace matrix. Specifically, it includes: the coupling weight matrix is defined in the form of an adjacency matrix, and the corresponding element is 1 when there is a connection between nodes, otherwise it is 0; the time-delay coupling weight matrix is defined using the same rule; the Laplace matrix is generated by subtracting the sum of the elements in the corresponding row from the diagonal element of the adjacency matrix, and the time-delay Laplace matrix is constructed using a similar method. <00>

[0061] In a specific implementation process, the coupling weight matrix is ,and the time-delay coupling weight matrix is When ,if there is a connection between the i-th node and the j-th node, then ,otherwise, ;when , ;

[0062] The Laplace matrix is: The time-delay Laplace matrix is: The Laplace matrix The elements are:

[0063] Laplace matrix of time delay The elements are:

[0064] .

[0065] In a specific implementation, the coupling weight matrix is ​​a matrix that describes the connection state between nodes. It can be implemented as an adjacency matrix, where each element takes a value of 0 or 1 to indicate whether a physical connection exists between nodes. This matrix provides a mathematical description of the network topology and is the basis for calculating coupling weights.

[0066] The Laplacian matrix is ​​a matrix that reflects the dynamic differences between nodes. Specifically, it can be achieved by subtracting the sum of the corresponding row elements from the diagonal elements of the adjacency matrix. This matrix quantifies the differences in the dynamic coupling strength between nodes, providing topologically dependent coupling weights for error system modeling.

[0067] The time-delay coupling weight matrix is ​​a matrix that describes the time-delay coupling topology, and it can be defined using the same rules as the adjacency matrix. This matrix is ​​used to capture the time-delay coupling effect and provides weight parameters for the time-delay error term.

[0068] The time-delay Laplace matrix is ​​a matrix that reflects the dynamic differences in time-delay coupling. Specifically, it can be achieved by subtracting the sum of the corresponding row elements from the diagonal elements of the time-delay adjacency matrix. This matrix transforms the time-delay coupling topology into a mathematically tractable operator form, supporting the stability analysis of the time-delay error term.

[0069] Specifically, when constructing a follower network model, it is necessary to define the mathematical expressions for coupling weights and time-delay coupling weights. For regular coupling weights, the corresponding elements of the coupling weight matrix are set to 1 when there is a physical connection between network nodes, and 0 otherwise. For time-delay coupling weights, the same rule is used to define the time-delay coupling weight matrix. The Laplace matrix is ​​constructed by setting the diagonal elements of the adjacency matrix to the negative of the sum of the off-diagonal elements of that row, while the off-diagonal elements retain the negative values ​​of the original adjacency matrix. The construction of the time-delay Laplace matrix follows the same logic, but is calculated based on the time-delay coupling weight matrix. This dual matrix structure enables the system to handle both instantaneous and time-delay coupling effects simultaneously, providing accurate topological parameter inputs for subsequent stability analysis.

[0070] Compared to existing technologies, traditional methods typically extract coupling weights using only the adjacency matrix, without establishing a joint extraction mechanism for the Laplace matrix and the time-delay coupling weight matrix. Existing technologies often simplify the processing of time-delay coupling weights to fixed proportional coefficients, failing to accurately reflect the dynamic characteristics of the time-delay topology. This invention, by constructing a parallel extraction framework for the coupling weight matrix and the time-delay coupling weight matrix, achieves a complete mathematical description of the network topology, significantly improving the accuracy of the synchronization error model.

[0071] Through the above technical solution, this invention effectively solves the technical problem of inaccurate extraction of coupling weight parameters in multi-coupled time-delay networks. By introducing a joint extraction mechanism of the Laplace matrix and the time-delay Laplace matrix, the dynamic coupling differences between nodes are accurately quantified, providing precise topological parameter support for the stability judgment of the error system. This method reduces the computational error caused by the traditional single adjacency matrix model, improves the accuracy of synchronization control in complex time-delay networks, and reduces the computational complexity of parameter extraction through matrix operation standardization.

[0072] In some embodiments, the present invention further proposes a specific mathematical expression for the target network model, which is a dynamic model of the leader node state relative to time and time delay, and its expression is Equation (2):

[0073] (2)

[0074] in, This represents the state vector of the target network. Let be the time-delayed state vector of the target network;

[0075] The error system is the difference between the follower network model and the target network model, and its expression is given by equation (3):

[0076] (3)

[0077] in, ;

[0078] .

[0079] In a specific implementation, the target network model refers to the mathematical equations describing the dynamic behavior of the leader node. Specifically, it is modeled using time-delay differential equations, introducing time-delay terms to characterize the impact of historical node state information on the current dynamics. This model can reflect the changes in dynamic characteristics caused by signal transmission delays or data processing lags in the actual system, providing an accurate reference benchmark for synchronization control.

[0080] An error system is a mathematical description of the state deviation between a follower node and a target node, specifically obtained by subtracting the state vectors of the follower network model and the target network model. The construction of this system transforms the synchronization problem into a stability analysis problem of error dynamics, making the control objective quantifiable as a proof of error convergence.

[0081] Specifically, the target network model provides the desired trajectory for the entire synchronization control process by defining the state evolution law of the leader node. In complex networks with time-delay coupling, the dynamics of each follower node are affected not only by its current state but also by its historical state. By establishing a target network model that includes time-delay terms, the true dynamics of the leader node can be accurately characterized, avoiding distortion of the reference trajectory caused by ignoring time-delay effects. The construction of the error system transforms the synchronization problem into a stability analysis of the error signal. By designing a controller, the error dynamics converge to zero or within a compact set, thereby achieving synchronization between the follower network and the target network. This error system comprehensively considers the effects of time-delay coupling terms and nonlinearity, and its mathematical form is a differential equation, which facilitates subsequent stability proof based on Lyapunov theory.

[0082] Compared to existing technologies, traditional synchronization control methods typically assume the target network is an idealized model without time delay, leading to a dynamic mismatch between the reference trajectory and the actual system. This invention introduces a time delay term to construct the target network model, making the reference trajectory more closely resemble the dynamic characteristics of the real physical system. Furthermore, existing error systems often neglect the influence of coupling time delays, considering only the current state deviation, while this invention explicitly includes a time delay coupling term in the error dynamics, enabling stability analysis to accurately reflect the impact of time delays on the synchronization process.

[0083] Through the above technical solution, this invention achieves accurate modeling of the target network dynamics and constructs an error system incorporating time-delay coupling effects, providing an accurate mathematical basis for subsequent design of anti-interference control strategies. This solution effectively solves the problem of reference trajectory distortion caused by neglecting time delays in traditional methods, improving the accuracy and anti-interference capability of synchronous control. Especially in industrial control scenarios with communication delays or data processing lags, it ensures that the stability analysis results of the error system are consistent with the actual system behavior.

[0084] In some embodiments, the present invention further proposes a combination of a restraint pulse control input and a Dirac function as the control input. The error system is modeled at the pulse moment, and sufficient conditions for mean-square stability of the error system are derived based on Lyapunov stability theory. These sufficient conditions involve mathematical constraints relating to the exponential Lyapunov function, a first constant, a second constant, and time-delay parameters.

[0085] In a specific implementation process, the control input is equation (4):

[0086] (4)

[0087] in, This indicates a control pulse input. Represents the Dirac function, Indicates the time of the k-th pulse;

[0088] Modeling the error system yields the functional expression (5):

[0089] (5)

[0090] Indicates the pulse timing controller;

[0091] The sufficient condition for the mean square stability of the error system is given by equation (6):

[0092] (6)

[0093] in, , indicating at the pulse moment The value of the right limit of the error state Subtract the value of the left limit of the error state , This represents the exponential Lyapunov function. Denotes the first constant, and , Denotes the second constant, and , All represent positive real numbers. Indicates taking and The largest of them, satisfy , Represents the time delay of a continuous system. Represents the set of non-negative integers. s is a time variable. This is the initial pulse time. This is the q-th pulse moment.

[0094] In a specific implementation, the restraint pulse control input refers to the control signal applied in pulse form. This can be achieved by applying control quantities in a concentrated manner at discrete time points using the Dirac function, for example, applying instantaneous control force to selected nodes via a controller at the pulse moment. The Dirac function characterizes the instantaneous nature of pulse control and can be implemented by suddenly applying control signals at discrete time points, such as instantaneously adjusting the system state at the pulse moment. Error system modeling refers to transforming the dynamic differences between the follower network and the target network into mathematical differential equations. This can be achieved by constructing dynamic equations using the differences in state variables, for example, substituting the differences between the node state and the target state into the coupled dynamic equations. The exponential Lyapunov function refers to the exponential energy function used for stability analysis. This can be achieved using a functional structure containing exponential decay terms, for example, constructing a composite function containing a quadratic error state and an exponential term. The mean-square stability sufficient condition refers to the mathematical constraint that guarantees the expected convergence of the error system state. This can be derived through Lyapunov function derivative analysis and stochastic differential equation theory, for example, establishing inequalities by combining pulse intervals and time delay parameters.

[0095] Specifically, within the pulse control framework, the control input is designed as an instantaneous action at discrete time points. Continuous-time control is transformed into a pulse sequence using the Dirac function, applying control to the error system at preset pulse moments. The dynamic model of the error system is constructed as a differential equation containing time-delay terms and pulse control terms, where time-delay state variables participate in the current dynamic process through historical state data. Based on the constructed exponential Lyapunov function, stability analyses are performed on the continuous dynamic stage and the discrete pulse stage of the error system, deriving inequality constraints containing time-delay parameters and pulse parameters. These constraints require the decay rate of the Lyapunov function to satisfy an exponential relationship, where the first constant controls the decay rate in the continuous stage, the second constant constrains the energy jump amplitude in the pulse stage, and the time-delay parameter influences the calculation of the stability boundary through the maximum time-delay term.

[0096] This invention introduces an exponential function structure, directly incorporating time delay parameters into the mathematical expression of the stability criterion, thus enabling the quantitative analysis of the impact of time delay on system stability. Compared to traditional fixed-gain pulse control methods, the time-delay-dependent stability condition established in this invention can dynamically adjust control parameters according to the actual system time delay.

[0097] Through the above technical solution, this invention can effectively handle the synchronization control problem in time-delay coupled networks, ensuring the mean square stability of the error system under pulse control. This solution provides a theoretical basis for matching the pulse control gain with the time delay parameters by establishing mathematical constraints, enabling the control system to maintain synchronization accuracy even with communication delays. Simultaneously, the use of exponential Lyapunov functions improves the accuracy of stability analysis, laying the foundation for optimizing control parameters in complex time-delay systems.

[0098] In some embodiments, the present invention further proposes using Bernoulli random variables on certain communication channels of the follower network model if a random spoofing attack exists. The model used to describe the probability of a deception attack occurring at any given moment is as follows: ,in, It is a constant. Represents a random variable at time t. The probability of taking the value 1 is equal to a constant. That is, the probability of a successful attack is constant. ; This represents the random variable at time t. The probability of taking the value 0 is equal to a constant. That is, the probability of an attack failing is constant. ,remember for The expectation can be represented by a diagonal matrix, i.e. .

[0099] In a specific implementation, a Bernoulli random variable refers to a discrete random variable that follows a two-point distribution, specifically implemented using binary values ​​of 0 or 1, used to characterize whether a spoofing attack event occurs. A spoofing attack model refers to quantitatively describing the randomness of an attack through probability parameters, specifically implemented using the mathematical expectation of the Bernoulli distribution, used to incorporate the statistical characteristics of the attack success probability into controller design.

[0100] Specifically, in scenarios where communication channels are subjected to spoofing attacks, attackers probabilistically tamper with or replace transmitted data. By introducing Bernoulli random variables, the attack event is modeled as a probabilistic event, where a value of 1 indicates a successful attack, and a value of 0 indicates that the attack did not occur. This model transforms the randomness of the attack into a mathematical expectation, enabling stability analysis of the error system within a probabilistic framework. Based on this model, the impact of the attack probability on synchronization performance can be explicitly considered during controller design. By adjusting control parameters, the expected deviation caused by the attack can be offset, ensuring the statistically bounded stability of the error system.

[0101] This invention establishes a probabilistic attack model using Bernoulli variables, which not only quantifies the statistical regularity of attack occurrences but also incorporates the influence of random attacks into Lyapunov stability analysis through expectation calculations, providing precise mathematical constraints for the design of anti-attack controllers.

[0102] Through the above technical solution, this invention solves the technical problem of synchronization control inaccuracy caused by the randomness of deception attacks, and achieves mean-square bounded stability of dynamic networks under random attacks. This model provides a probabilistic constraint basis for anti-attack control strategies, enabling the controller to adaptively adjust the control strength to match the attack probability, effectively suppressing synchronization error divergence caused by spoofed data injection.

[0103] In some embodiments, the present invention further proposes that under the influence of a deception attack, the error system achieves mean-square bounded stability, thereby realizing mean-square secure synchronization between the follower network and the target network. This includes: under the influence of a deception attack, the error system, under the action of a restraint pulse security controller, evolves its dynamic behavior into a specific model and achieves mean-square bounded stability; achieving mean-square bounded stability means that after the dynamic behavior evolution of the error system, the expected state of the evolved model eventually converges and stabilizes within a compact set.

[0104] The specific model of evolution is Equation (7):

[0105] (7)

[0106] The function of a compact set is given by equation (8) when the expected value of the state of the model evolved by the error system is less than or equal to a specific positive number.

[0107] (8)

[0108] in, It is a positive number. This indicates that the expected value of the state of the model evolved by the error system is less than or equal to... , This indicates that the i-th deception attack function occurs at the pulse moment. The function value; Represents the overall error vector. for The error vector of the i-th node at time t.

[0109] In a specific implementation, mean-square bounded stability means that the expected state of the error system evolves over time within a bounded range. This can be verified by constructing an exponential Lyapunov function and combining it with the theory of stochastic differential equations. Its role is to ensure that the system maintains controllable synchronization accuracy under random disturbances. A compact set refers to the set of mathematical closures to which the expected state of the error system eventually converges. This can be characterized by setting an upper bound parameter for the error. Its role is to quantify the limiting boundary of synchronization performance, providing a clear convergence target for controller design.

[0110] Specifically, when the system is subjected to a deception attack, the restraint pulse security controller selectively applies pulse control signals to critical nodes to suppress the spread of errors caused by the attack. The dynamic evolution model of the error system integrates time-delay coupling effects, pulse control effects, and attack randomness. By analyzing the mean square stability of this model, it can be proven that the expected system state will converge to a finite region defined by a compact set. The upper bound parameter of the compact set is jointly determined by the system time delay, pulse gain, and attack intensity. The synchronization accuracy can be optimized by adjusting the control parameters.

[0111] This invention achieves a quantitative correlation between control parameters and synchronization performance by constructing a time-delay-dependent compact set model, while utilizing time-delay characteristics to reduce the degree of damage to system stability caused by attacks.

[0112] Through the above technical solution, the present invention can ensure verifiable mean square secure synchronization of multi-coupled time-delay networks under random deception attack environment. By dynamically adjusting the proportion of restraining nodes and pulse control parameters, the control energy consumption is significantly reduced while maintaining synchronization accuracy, solving the problems of resource waste and insufficient anti-attack capability caused by full node control in traditional methods.

[0113] In a specific implementation process, the simulation results obtained by using the above method are as follows: Figures 2a-2c As shown, Figure 2a This is a schematic diagram illustrating the simulation results of the restraint pulse safety synchronization control method using a multi-coupled time-delay network according to the present invention. Figure 2b yes Figure 2a (1) Enlarged view of the range from 80s to 250s. Figure 2c yes Figure 2a (2) Enlarged view of the range from 80s to 250s, as shown Figures 2a-2c As shown, when the system is subjected to a deception attack, the control method of this invention enables the error system between each following time-delay network and the target dynamic network to achieve mean-square bounded stability. Specifically, under the premise of satisfying the stability condition, when... When, the error bound can be calculated. ;when When, the error bound can be calculated. As shown in Figure 2, increasing the time delay of a continuous system can significantly improve synchronization performance.

[0114] Based on the same general inventive concept, this invention also protects a restraint pulse safety synchronization control system for a multi-coupled time-delay network. The restraint pulse safety synchronization control system for a multi-coupled time-delay network provided by this invention will be described below. The restraint pulse safety synchronization control system for a multi-coupled time-delay network described below can be referred to in correspondence with the restraint pulse safety synchronization control method for a multi-coupled time-delay network described above.

[0115] Figure 3 This is a schematic diagram of the structure of the restraint pulse safety synchronization control system with a multi-coupled time-delay network provided in an embodiment of the present invention, as shown below. Figure 3 As shown, the restraint pulse safety synchronization control system of the multi-coupled time delay network in this embodiment includes a first model establishment module 31, a discriminant establishment module 32, a second model establishment module 33, and a design module 34.

[0116] The first model building module 31 is used to build a coupled time-delay dynamic network model, which includes a follower network model, a target network model, and an error system. The follower network model is a dynamic model of the state of the follower node relative to time and time delay; the target network model is a dynamic model of the state of the leader node relative to time and time delay; and the error system is the difference between the follower network model and the target network model.

[0117] The discriminant establishment module 32 is used to establish the Lyapunov stability discriminant of the synchronization error: under the pulse control framework, the control input is defined, the error system is modeled, and based on hybrid control theory and Lyapunov stability theory, the sufficient condition for the mean square stability of the error system is derived as the Lyapunov stability discriminant.

[0118] The second model building module 33 is used to introduce a deception attack model: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into the communication channel.

[0119] Design module 34 is used to design a restraint pulse security controller: based on the Lyapunov stability discriminant and the deception attack model, a restraint pulse security controller is designed so that the error system reaches mean square bounded stability under the influence of deception attacks, thereby achieving mean square secure synchronization between the follower network and the target network.

[0120] Figure 4This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present invention. The electronic device may include: a processor 410, a communication interface 420, a memory 430, and a communication bus 440. The processor 410, communication interface 420, and memory 430 communicate with each other via the communication bus 440. The processor 410 can call logical instructions in the memory 430 to execute a restraint pulse safety synchronization control method for a multi-coupled time-delay network.

[0121] Furthermore, the logical instructions in the aforementioned memory 430 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0122] On the other hand, the present invention also provides a computer program product, which includes a computer program that can be stored on a non-transitory computer-readable storage medium. When the computer program is executed by a processor, the computer is able to execute the restraint pulse safety synchronization control method for the multi-coupled time-delay network provided by the above methods.

[0123] In another aspect, the present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the restraint pulse safety synchronization control method for the multi-coupled time-delay network provided by the above methods.

[0124] It should be noted that all relevant information that may be involved in the various embodiments of the present invention is processed in strict accordance with the requirements of laws and regulations, following the principles of legality, legitimacy, and necessity, based on the reasonable purpose of the business scenario, and is information that users actively provide or generate during the use of the product / service, as well as information obtained with user authorization.

[0125] The information processed by this invention may vary depending on the specific product / service scenario and should be based on the specific scenario in which the user uses the product / service. This may involve user account information, device information, or other related information. This invention will treat the relevant information and its processing with the utmost diligence.

[0126] This invention places great emphasis on the security of relevant information and has adopted reasonable and feasible security protection measures that comply with industry standards to protect user information and prevent unauthorized access, public disclosure, use, modification, damage or loss of relevant information.

[0127] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.

[0128] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0129] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for safe synchronization control of restrained pulses using a multi-coupled time-delay network, characterized in that, include: A coupled time-delay dynamic network model is established, comprising a follower network model, a target network model, and an error system. The follower network model is a dynamic model of the follower node's state relative to time and time delay; the target network model is a dynamic model of the leader node's state relative to time and time delay; and the error system is the difference between the follower network model and the target network model. Establish the Lyapunov stability criterion for synchronization error: Define the control input under the impulse control framework, model the error system, and derive the sufficient condition for the mean square stability of the error system as the Lyapunov stability criterion based on hybrid control theory and Lyapunov stability theory. A deception attack model is introduced: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into communication channels. Design of a restraint pulse security controller: Based on the Lyapunov stability criterion and the deception attack model, a restraint pulse security controller is designed so that the error system reaches mean square bounded stability under the influence of deception attacks, thereby achieving mean square secure synchronization between the follower network and the target network.

2. The method for secure synchronization control of restrained pulses in a multi-coupled time-delay network according to claim 1, characterized in that, The follower network model is as follows: in, This represents the rate of change of the state vector of the i-th node at time t. , Let represent the state vector of the i-th node at time t. Let represent the time-delayed state vector of the i-th node at time t, and , Represents the set of m-dimensional real-valued vectors; Indicates coupling strength. Indicates the time-delay coupling strength; Indicates time delay in a continuous system; Represents the internal coupling matrix. Represents the internal coupling matrix of the time delay. Represents an m-dimensional real matrix; Represents a nonlinear function; Indicates control input; Indicates coupling weight. This represents the time-delay coupling weight.

3. The method for secure synchronization control of restrained pulses in a multi-coupled time-delay network according to claim 2, characterized in that, Also includes: The coupling weights are extracted based on the coupling weight matrix and the Laplace matrix; Furthermore, the time-delay coupling weights are extracted based on the time-delay coupling weight matrix and the time-delay Laplace matrix; Wherein, the coupling weight matrix is The time-delay coupling weight matrix is ,when If there is a connection between the i-th node and the j-th node, then ,otherwise, ; when , ; The Laplace matrix is: The time-delay Laplace matrix is: The Laplace matrix The elements are: , Laplace matrix of time delay The elements are: .

4. The method for secure synchronization control of restrained pulses in a multi-coupled time-delay network according to claim 2, characterized in that, The target network model: in, This represents the state vector of the target network. Let be the time-delayed state vector of the target network; The error system is as follows: in, ； 。 5. The method for secure synchronization control of a restraining pulse in a multi-coupled time-delay network according to claim 2, characterized in that, The control input is: in, This indicates a control pulse input. Represents the Dirac function, Indicates the time of the k-th pulse; Modeling the error system includes: Indicates the pulse timing controller; The sufficient condition for the mean-square stability of the error system is: in, , indicating at the pulse moment The value of the right limit of the error state Subtract the value of the left limit of the error state , This represents the exponential Lyapunov function. Denotes the first constant, and , Denotes the second constant, and , All represent positive real numbers. Indicates taking and The largest of them, satisfy , Represents the time delay of a continuous system. Represents the set of non-negative integers. s is a time variable. This is the initial pulse time. This is the q-th pulse moment.

6. The method for secure synchronization control of a restrained pulse in a multi-coupled time-delay network according to claim 5, characterized in that, Design a restraint pulse safety controller, including: Under the constraints of mean square stability corresponding to the Lyapunov stability criterion and the cancellation constraint on the deception attack signal of the deception attack model, n nodes are selected from the follower network model to form a restraint node set; wherein ; Based on the set of restraining nodes, the model of the restraining pulse safety controller is obtained as follows: in, Indicates the pulse control gain; Represents the set of restraining nodes, when When the i-th node is a controlled node, the i-th node exercises control; when When the i-th node is not controlled, the i-th node does not exercise control.

7. The method for secure synchronization control of a restrained pulse in a multi-coupled time-delay network according to claim 6, characterized in that, From the following network model, n nodes are selected to form a restraining node set, including: For N nodes at the k-th pulse time Sort according to the norm of the error system; Select n nodes in descending order of size to form a set of restraining nodes.

8. The method for secure synchronization control of a restraining pulse in a multi-coupled time-delay network according to claim 1, characterized in that, The probability of a successful spoofing attack is described using Bernoulli random variables, including: In certain communication channels of the aforementioned follower network model, if a random spoofing attack exists, a Bernoulli random variable is used. The model describing the probability of a deception attack occurring at any given time t is as follows: in, It is a constant. This represents the random variable at time t. The probability of taking the value 1 is equal to a constant. That is, the probability of a successful attack is constant. ; This represents the random variable at time t. The probability of taking the value 0 is equal to a constant. That is, the probability of an attack failing is constant. ,remember for The expectation, that is .

9. The method for secure synchronization control of a restrained pulse in a multi-coupled time-delay network according to claim 1, characterized in that, Under the influence of a deception attack, the error system reaches mean-square bounded stability, achieving mean-square secure synchronization between the follower network and the target network, including: Under the influence of a deception attack, the error system, under the action of the restraining pulse security controller, evolves its dynamic behavior into the following model and reaches mean-square bounded stability: Wherein, achieving mean-square bounded stability means that after the error system evolves through dynamic behavior, the expected state of the evolved model eventually converges and stabilizes within a compact set; The functional expression of the compact set is: in, It is a positive number. This indicates that the expected state of the model evolved by the error system is less than or equal to , This indicates that the i-th deception attack function occurs at the pulse moment. The function value; Represents the overall error vector. for The error vector of the i-th node at time t.

10. A restraint pulse safety synchronization control system with a multi-coupled time-delay network, characterized in that, include: The first model building module is used to build a coupled time-delay dynamic network model. The coupled time-delay dynamic network model includes a follower network model, a target network model, and an error system. The follower network model is a dynamic model of the state of the follower node relative to time and time delay. The target network model is a dynamic model of the state of the leader node relative to time and time delay. The error system is the difference between the follower network model and the target network model. The discriminant establishment module is used to establish the Lyapunov stability discriminant for the synchronization error: under the pulse control framework, the control input is defined, the error system is modeled, and based on hybrid control theory and Lyapunov stability theory, the sufficient condition for the mean square stability of the error system is derived as the Lyapunov stability discriminant. The second model building module is used to introduce a deception attack model: Bernoulli random variables are used to describe the success probability of a deception attack. The deception attack model aims to characterize the randomness of malicious attackers inserting false or tampered data into the communication channel. The design module is used to design a restraint pulse security controller: based on the Lyapunov stability discriminant and the deception attack model, a restraint pulse security controller is designed so that the error system reaches mean square bounded stability under the influence of deception attacks, thereby achieving mean square secure synchronization between the follower network and the target network.

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