A Distributed Pulse Control Method for Complex Networks Based on Dual-Channel Attacks

Abstract

This invention relates to the field of distributed pulse control technology, and in particular to a method, system, and computer-readable storage medium for distributed pinning pulse control of complex networks based on dual-channel attacks. The method includes: establishing a complex dynamic network and a leader node; selecting pinning nodes from the complex dynamic network; constructing a distributed delay pinning pulse controller; constructing an error network; and solving for the distributed control strength, pulse interval, and pulse delay of the distributed delay pinning pulse controller based on constraints that ensure the global exponential stability of the error network, thereby obtaining the target distributed delay pinning pulse controller. The distributed delay pinning pulse controller constructed by this invention enables the network to quickly achieve synchronization even under the combined effect of dual-channel fraud attacks, improving network security and reducing control costs at both the time and space levels.

Description

Technical Field

[0001] This invention relates to the field of distributed pulse control technology, and in particular to a distributed restraint pulse control method, system, and computer-readable storage medium for complex networks based on dual-channel attacks. Background Technology

[0002] Complex networks have received widespread attention and research in many scientific fields, such as social sciences, biology, physics, and engineering. Essentially, they are collectives composed of a large number of interconnected nodes with distinct dynamic characteristics. Among the many properties of complex networks, synchronization, as a typical collective behavior pattern, has high research value. In the specific implementation process, by adjusting network parameters or adding external control inputs, the nodes in the complex network can achieve orderly coordination in structure and function, thereby achieving a unity of rhythm and harmony within the network. In fact, synchronization enables networks to achieve their functions with higher efficiency and lower cost. To date, synchronization types such as cluster synchronization, projection synchronization, quasi-synchronization, and phase synchronization have all been explored and studied.

[0003] Taking distributed control of a power system as an example, smart substations in the power grid can be considered nodes in a complex network. A leader node is constructed using standard voltage / frequency reference signals issued by the power grid dispatch center as the standard state. Each smart substation collects real-time status data such as voltage / current amplitude, phase angle, frequency, power factor, and generator speed, and sends this data to the controller. The controller outputs control signals based on the error between the status of each node and the standard state of the leader node, sending these signals to the actuators of each smart substation, such as generator control valves, SVC reactive power compensation devices, and energy storage converters. Each actuator performs corresponding actions based on the control signals to adjust the status parameters, keeping various parameters of the power grid stable and ensuring that the status of each smart substation is synchronized with the standard state of the leader node. By synchronizing the status of each smart substation in the complex network with the standard state, phase differences in the power grid can be eliminated to avoid power oscillations, maintaining consistent voltage and frequency across the entire network to prevent grid disconnection accidents, and ensuring voltage stability to avoid cascading failures.

[0004] To achieve synchronization in complex networks, many control strategies, such as adaptive control, intermittent control, impulse control, and restraint control, have been introduced into synchronization control. Among these, impulse control and restraint control have attracted widespread attention from scholars due to their unique characteristics.

[0005] Compared to traditional continuous control methods, pulse control only makes discrete, instantaneous changes to the system state at specific moments, without requiring continuous system control. This characteristic results in lower resource consumption during the control process. For example, in wireless sensor networks, pulse control can extend the network's lifespan. Furthermore, in scenarios with high response speed requirements, pulse control can exert its control effect quickly, highlighting its efficiency and speed.

[0006] Compared to the efficiency of impulse control on the time scale, restraint control demonstrates a significant competitive advantage on the spatial scale. Many large controlled objects, such as power networks, have high generator node control costs, and controlling all nodes is extremely complex and uneconomical. Restraint control only applies control to a subset of key nodes in the network, indirectly influencing the synchronization state of the entire network through its coupling structure, thus reducing the economic and spatial costs associated with controlling all nodes.

[0007] The continuous and stable operation of modern information society relies heavily on the efficient and stable transmission and reception of information. However, due to vulnerabilities in network communication protocols, limited resources in some node devices, and significant attack points in distributed network topologies, information transmission within networks is highly susceptible to attacks. Various network attacks can cause negative impacts, including but not limited to damage to critical equipment, disruption of production processes, environmental pollution, and even personal injury or death.

[0008] Currently, network attacks can be broadly categorized into two types: DoS attacks and phishing attacks. DoS attacks send a large number of service requests to the target, exhausting hardware resources and preventing legitimate users from receiving services. Phishing attacks, on the other hand, often disrupt network communication by maliciously modifying data packets, compromising the authenticity and integrity of the data. Clearly, phishing attacks are more destructive than DoS attacks and require greater attention.

[0009] In complex network synchronization, fraud attacks on the sensor-to-controller loop are called substitution attacks, while fraud attacks on the controller-to-actuator loop are called injection attacks. Existing tethered pulse control only considers single-channel fraud attacks, and cannot effectively control complex networks to achieve synchronization when facing dual-channel attacks where both substitution and injection attacks coexist. Summary of the Invention

[0010] Therefore, the technical problem to be solved by the present invention is to overcome the problem that the existing technology only considers single-channel fraud attacks, and cannot effectively control the synchronization of complex networks when facing dual-channel attacks where substitution attacks and injection attacks coexist.

[0011] To address the aforementioned technical problems, this invention provides a distributed pinning pulse control method for complex networks based on dual-channel attacks, comprising:

[0012] Establish complex dynamic networks and leader nodes;

[0013] Select restraining nodes from a complex dynamic network; construct a distributed delayed restraining pulse controller, generate pulse control signals based on the state data of each node, and send them to the actuators of the restraining nodes; assume that the state data of each node is subjected to a substitution attack during the process of being sent to the distributed delayed restraining pulse controller, and that the pulse control signals are subjected to an injection attack during the process of being sent to the actuators of the restraining nodes.

[0014] Based on a distributed delay restraint pulse controller, an error network is constructed according to the synchronization error between nodes and the leader node in a complex dynamic network under the condition that both injection attacks and replacement attacks exist simultaneously.

[0015] Based on the constraint condition that the error network is globally exponentially stable, the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller are solved to obtain the target distributed delay restraint pulse controller.

[0016] The sensor at each node in the complex dynamic network collects the status data of each node in real time and sends it to the target distributed delay restraint pulse controller. The target distributed delay restraint pulse controller generates the target pulse control signal based on the status data of each node and the synchronization error between each node and the leader node. The actuator of the restraint node receives the target pulse control signal and adjusts the status parameters of the restraint node so as to dynamically adjust the status parameters of other nodes in the complex dynamic network according to the status parameters of the restraint node.

[0017] Preferably, a complex dynamic network and a leader node are established, including:

[0018] In a complex dynamic network, each node is represented as follows:

[0019]

[0020] Among them, z i (t)=(z i1 (t),z i2 (t),…,z im (t)) T Let z be the state vector of the i-th node at time t. im (t) represents the m-th state data of the i-th node at time t, where m is the number of states; For z i The first derivative of (t); A and B are both constant matrices; Let c be the nonlinear function of the i-th node at time t, used to represent the interaction relationship between nodes; c is the coupling strength; D = (d ij ) N×N Let d be a network topology matrix. If the j-th node is connected to the i-th node, then d ij >0, otherwise d ij =0; N is the total number of nodes in the complex dynamic network; Γ is the internal coupling matrix; z j (t-τ(t)) is the state vector of the j-th node at time t-τ(t); τ(t) is the coupling delay at time t; φ i (t) represents the injection attack at time t of the i-th node;

[0021] The leader node is represented as:

[0022]

[0023] Among them, z L (t)=(z L1 (t),z L2 (t),…,z Lm (t)) T Let be the state vector of the leader node at time t. Let be the nonlinear function of the leader node at time t. For z L The first derivative of (t).

[0024] Preferably, the distributed delay-controlled pulse controller is represented as follows:

[0025]

[0026] Among them, u i (t) represents the pulse control signal output by the distributed delay restraint pulse controller at time t; c p The distributed control strength is N; N is the total number of nodes in the complex dynamic network. For distributed control weights; t k Let k be the pulse moment, and k be the index of the pulse moment. It is the set of positive integers; For pulse delay, It is the set of positive real numbers; For the j-th node The state vector at time step j, j∈1,2,…,N; Let be the constraint gain of the i-th node. If the i-th node is connected to the constraint node, then... otherwise for The synchronization error between the i-th node and the leader node at time i.

[0027] Preferably, based on a distributed delay-restrained pulse controller, in the case of simultaneous injection and substitution attacks, an error network is constructed according to the synchronization errors between nodes and the leader node in the complex dynamic network, as shown in the formula:

[0028]

[0029] in, The synchronization error at non-pulse moments is represented by a matrix, z, which characterizes the synchronization errors between nodes and the leader node in a complex dynamic network under injection attacks. E (t), z E (t-τ(t)) and For time t, t-τ(t) and t-τ(t) respectively. The matrix formed by the synchronization errors between each node and the leader node in a complex dynamic network; τ(t) is the coupling delay at time t; I N Let F(z) be an N-dimensional identity matrix, where N is the total number of nodes in the complex dynamic network; A and B are both constant matrices; E (t) represents the nonlinear function of the error network at time t, and represents the matrix formed by the differences between the nonlinear functions of each node in the complex dynamic network at time t and the nonlinear function of the leader node; c is the coupling strength; D is the network topology matrix; Γ is the internal coupling matrix; θ x The intensity of the injected attack; γ ij I represents the probability of a successful injection attack. m It is an m-dimensional identity matrix; For the injection attack at time t, Let t be the transpose of the injected attack signal at time t for the i-th node, where i ∈ 1, 2, ..., N; k The pulse moment; Let ζ(t) represent the synchronization error at the pulse moment, which is a matrix representing the synchronization errors between each node and the leader node in a complex dynamic network under the presence of a substitution attack; ζ(t) = diag(ζ1(t), ζ2(t), ..., ζ N (t)), ζ i To replace the success probability of the attack; For t k Time-based replacement attack, For the i-th node t k The transpose of the attack signal at time step; G is the pulse gain matrix.

[0030] Preferably, the average delayed pulse gain is defined based on the pulse delay as follows:

[0031]

[0032] in, N is the average delayed pulse gain; N′(t,t0) is the pulse sequence {t k The number of pulses within the interval (t0, t], where t0 is the initial time. The nominal state deviation at the moment of the pulse; The compensation delay factor caused by actuator dynamics; For pulse delay, It is a real number.

[0033] Preferably, based on the constraint condition that makes the error network globally exponentially stable, the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller are solved, including:

[0034] The first constraint is:

[0035]

[0036] The second constraint is:

[0037]

[0038] The third constraint is:

[0039]

[0040] The fourth constraint is:

[0041] ι -1 θ x ασ max (Θ * )θ2≤0

[0042] The fifth constraint is:

[0043]

[0044] The sixth constraint is:

[0045]

[0046] The seventh constraint is:

[0047] δσ max (Θ ζ )θ2≤0

[0048] The eighth constraint is:

[0049]

[0050] The ninth constraint is:

[0051]

[0052] Where D is the network topology matrix; Γ is the internal coupling matrix; Let m be the Lyapunov weight matrix, and m be the number of states. θ1 and θ2 are positive real numbers, I m It is an m-dimensional identity matrix; I N Let A be an N-dimensional identity matrix, where N is the total number of nodes in the complex dynamic network; A and B are both constant matrices; c is the coupling strength. a1, a2, and ι are real numbers; θ x The intensity of the injected attack; σ is a diagonal matrix; α is the upper bound of the injection attack; max (Θ * ) is Θ * Spectral radius; Θ * =E[γ * (t)];c p For distributed control strength; ζ(t)=diag(ζ1(t),ζ2(t),…,ζ N (t)), ζ i ∈[0,1] is a constant; G is the pulse gain matrix; δ is the upper bound of the substitution attack; σ max (Θ ζ ) is Θ ζ spectral radius, Let be the expectation of the matrix; T represents the average delayed pulse gain. s The pulse interval; It is a positive real number;

[0053] Solve for the condition that satisfies the first, second, and third constraints. according to The obtained θ2 satisfies the fourth and seventh constraints; based on the fifth and sixth constraints and the obtained... Further solve c p Solve for the average delayed pulse gain based on the eighth and ninth constraints. and pulse interval T s The pulse delay is calculated based on the average delayed pulse gain.

[0054] Preferably, the criteria for selecting a restraining node from a complex dynamic network include: selecting the node with the highest degree centrality or the largest number of connected nodes in the complex dynamic network as the restraining node.

[0055] Preferably, the condition for the global exponential stability of the error network is:

[0056]

[0057] Among them, z i z(t) is the state vector of the i-th node at time t. L (t) is the state vector of the leader node at time t; ∈ is a positive constant; ∈ is a positive real number.

[0058] This invention also provides a distributed pinning pulse control system for complex networks based on dual-channel attacks, comprising:

[0059] The Complex Network Building Module is used to build complex dynamic networks and identify leader nodes;

[0060] The controller construction module is used to select restraining nodes from complex dynamic networks; it constructs a distributed delayed restraining pulse controller, generates pulse control signals based on the state data of each node, and sends them to the actuators of the restraining nodes; it is assumed that the state data of each node is subjected to a substitution attack during the process of being sent to the distributed delayed restraining pulse controller, and the pulse control signals are subjected to an injection attack during the process of being sent to the actuators of the restraining nodes.

[0061] The error network construction module is used to construct an error network based on the synchronization error between nodes and the leader node in a complex dynamic network, under the condition that both injection and substitution attacks exist simultaneously, based on a distributed delay restraint pulse controller.

[0062] The constraint solving module is used to solve the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller based on the constraint condition that makes the error network globally exponentially stable, so as to obtain the target distributed delay restraint pulse controller.

[0063] The control module is used to collect the status data of each node in real time using the sensors of each node in the complex dynamic network, and send it to the target distributed delay restraint pulse controller. The target distributed delay restraint pulse controller generates a target pulse control signal based on the status data of each node and the synchronization error between each node and the leader node. The actuator of the restraint node receives the target pulse control signal and adjusts the status parameters of the restraint node so as to dynamically adjust the status parameters of other nodes in the complex dynamic network according to the status parameters of the restraint node.

[0064] The present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the above-described distributed pinning pulse control method for complex networks based on dual-channel attacks.

[0065] Compared with the prior art, the above-described technical solution of the present invention has the following advantages:

[0066] This invention discloses a distributed pinning pulse control method for complex networks based on dual-channel attacks. It considers dual-channel fraud attacks on the sensor-to-controller loop and the controller-to-actuator loop in complex dynamic networks. The constructed distributed delay pinning pulse controller overcomes the limitations of single-channel fraud attacks, enabling the network to quickly achieve synchronization even under the combined effect of dual-channel fraud attacks. Furthermore, the distributed delay pinning pulse controller constructed in this invention implements discrete control on a few key nodes, i.e., pinning nodes, which, while ensuring network synchronization performance and improving network security, further reduces control costs at both the time and space levels.

[0067] Furthermore, this invention introduces the concept of average delay pulse gain based on pulse delay, comprehensively quantifies the cumulative effect of pulse delay and pulse gain over time, breaks through the limitations of traditional average pulse weight and average pulse delay concepts, can improve the dynamic response accuracy and anti-interference capability of the controller, realize the rapid and stable synchronization of each node and the leader node in complex dynamic networks, and further improve the security of complex dynamic networks. Attached Figure Description

[0068] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings, wherein:

[0069] Figure 1 This is a flowchart of a distributed restraint pulse control method for complex networks based on dual-channel attacks, according to the present invention.

[0070] Figure 2 This is a diagram illustrating a dual-channel spoofing attack;

[0071] Figure 3 This is a diagram of the Chua's network topology in an embodiment of the present invention;

[0072] Figure 4 This is an example diagram of the output of the distributed delay restraint pulse controller in an embodiment of the present invention;

[0073] Figure 5 This is an example diagram of the temporal distribution of injection attacks in an embodiment of the present invention;

[0074] Figure 6 This is an example diagram of the temporal distribution of the substitution attack in an embodiment of the present invention;

[0075] Figure 7 This is a diagram showing the evolution of synchronization error between each node and the leader node in an embodiment of the present invention.

[0076] Figure 8 This is a comparison curve of synchronization errors with different numbers of restraining nodes in an embodiment of the present invention. Detailed Implementation

[0077] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0078] Reference Figure 1 As shown, this application provides a distributed pinning pulse control method for complex networks based on dual-channel attacks, including:

[0079] S1: Establish complex dynamic networks and leader nodes;

[0080] S2: Select restraining nodes from complex dynamic networks; construct a distributed delayed restraining pulse controller, generate pulse control signals based on the state data of each node, and send them to the actuators of the restraining nodes; assume that the state data of each node is subjected to a substitution attack during the process of being sent to the distributed delayed restraining pulse controller, and that the pulse control signals are subjected to an injection attack during the process of being sent to the actuators of the restraining nodes.

[0081] S3: Based on a distributed delay restraint pulse controller, an error network is constructed according to the synchronization error between nodes and the leader node in a complex dynamic network under the condition that both injection attacks and replacement attacks exist simultaneously.

[0082] S4: Based on the constraint condition of global exponential stability of the error network, solve for the distributed control strength, pulse interval and pulse delay of the distributed delay restraint pulse controller to obtain the target distributed delay restraint pulse controller.

[0083] S5: Real-time acquisition of node status data by sensors in each node of the complex dynamic network is used and sent to the target distributed delay restraint pulse controller; the target distributed delay restraint pulse controller generates target pulse control signals based on the status data of each node and the synchronization error between each node and the leader node; the actuator of the restraint node receives the target pulse control signals and adjusts the status parameters of the restraint node so as to dynamically adjust the status parameters of other nodes in the complex dynamic network according to the status parameters of the restraint node.

[0084] Specifically, each node in a complex dynamic network is represented as:

[0085]

[0086] in, Let z be the state vector of the i-th node at time t. im (t) represents the m-th state data of the i-th node at time t, where m is the number of states; For z i The first derivative of (t); All are constant matrices; Let D be a nonlinear function of the i-th node at time t, used to represent the interaction relationship between nodes; the positive constant c is the coupling strength; D = (d ij ) N×N Let d be a network topology matrix. If the j-th node is connected to the i-th node, then d ij >0, otherwise d ij =0, and specifically, the diagonal elements of matrix D are defined as follows: N is the total number of nodes in the complex dynamic network; Γ is the internal coupling matrix; z j (t-τ(t)) is the state vector of the j-th node at time t-τ(t); the finite term τ(t) is the coupling delay at time t and satisfies 0<τ(t)<τ m ;φ i (t) represents the injection attack at time t of the i-th node.

[0087] Because the complex dynamic network in this application maintains a leader-follower structure, a target complex network node needs to be pre-defined as the leader node. In this application, the leader node is identified as follows:

[0088]

[0089] Among them, z L (t)=(z L1 (t),z L2 (t),…,z Lm (t)) T Let be the state vector of the leader node at time t. Let be the nonlinear function of the leader node at time t. For z L The first derivative of (t).

[0090] Let z Ei (t)=z i (t)-z L (t) represents the synchronization error between the i-th node and the leader node at time t. Considering the complex dynamic network (1) and the leader node (2), the error network is represented in the following form:

[0091]

[0092] in Let be the nonlinear function of the error network at time t.

[0093] In complex dynamic networks, a restraining node is selected to receive pulse control signals from a distributed delayed restraining pulse controller. The selection criteria for the restraining node include choosing the node with the highest degree centrality or the largest number of connected nodes in the complex dynamic network.

[0094] To achieve global exponential synchronization between the complex dynamic network (1) and the leader node (2), a distributed delay-restrained pulse controller is constructed, denoted as u. i (t):

[0095]

[0096] Among them, u i (t) represents the pulse control signal output by the distributed delay restraint pulse controller at time t; c p The distributed control strength is N; N is the total number of nodes in the complex dynamic network. The distributed control weights between the i-th node and the j-th node; t k Let k be the pulse moment, and k be the index of the pulse moment. Given a set of positive integers; assume a pulse sequence. Satisfy 0 <t k <t k+1 , and lim t→+∞ t k =+∞; For pulse delay, It is the set of positive real numbers; For the j-th node The state vector at time step j, j∈1,2,…,N; Let be the constraint gain of the i-th node. If the i-th node is connected to the constraint node, then... otherwise for The synchronization error between the i-th node and the leader node at time i.

[0097] Since the distributed delayed restraint pulse controller only operates on the restraint node, if the i-th node is the restraint node, then the pulse control signal received by the actuator of that node at time t is u. i (t); If the i-th node is not a restraining node, then the pulse control signal received by the controller of that node at time t is 0.

[0098] The distributed delay-controlled pulse controller generates pulse control signals based on the state data of each node and the synchronization error between each node and the leader node, and sends them to the actuators of the controlled nodes. The distributed delay-controlled pulse controller constructed in this application reduces control costs in both time and space by implementing discrete control on a few key nodes, namely the controlled nodes, while ensuring synchronization performance.

[0099] Under the control strategy of the distributed delay-controlled pulse controller, the error network can be represented as:

[0100]

[0101] in, This represents the synchronization error during non-pulse moments. The synchronization error at the pulse moment, z Ei z(t) represents the synchronization error at the initial time; z0(t) and z L0 (t) represents the initial conditions of the complex dynamic network (1) and the leader node (2), respectively, which belong to the interval [-v,0] to the m-dimensional vector. The set of all continuous functions of is denoted as .

[0102] Let the function z Ei (t) satisfies This indicates that it is a piecewise right-continuous function.

[0103] The dual-channel fraud attack considered in this application is as follows: Figure 2 As shown. Due to the fragility of communication protocols, the limited nature of equipment resources, and the heterogeneity of network devices, both the sensor-to-controller and controller-to-actuator channels are susceptible to fraud attacks. Substitution attacks and injection attacks are considered respectively.

[0104] Define injection attack φ in complex dynamic networks (1) i (t) is as follows:

[0105]

[0106] Where θ x For attack strength, α i (t) represents the injected attack signal at time t for the i-th node; γ ij Let (t) be a random variable representing the probability of a successful injection attack, which satisfies the following condition:

[0107]

[0108] Assume γ ij (t) are independent, and their probabilities are expressed as follows:

[0109]

[0110] Where γ ij It is a known constant and γ ii =0.

[0111] Let γ(t)=(γ ij (t)) N×N , Let the expectation of the matrix be denoted by:

[0112]

[0113] Under the influence of a substitution attack, distributed delay-based pinning pulse control is expressed as:

[0114]

[0115] in δ i (t), i∈1,2,…,N is t k A time-based substitution attack; G = L p +θ p This is the pulse gain matrix. For the restraint gain matrix; I m It is an m-dimensional identity matrix.

[0116] Assume a distributed random variable ζ i (t) is independent and meets the following conditions:

[0117]

[0118] Where, ζ i ∈[0,1] is a known constant used to represent the success probability of a substitution attack. Let ζ(t) = diag(ζ1(t),ζ2(t),…,ζ N (t)),

[0119] Under the influence of substitution attacks It can be represented as:

[0120]

[0121] in

[0122] Therefore, based on the distributed delay-restrained pulse controller, in the presence of both injection and substitution attacks, an error network is constructed according to the synchronization errors between nodes and the leader node in the complex dynamic network as follows:

[0123]

[0124] in, The synchronization error at non-pulse moments is represented by a matrix, z, which characterizes the synchronization errors between nodes and the leader node in a complex dynamic network under injection attacks. E (t), z E (t-τ(t)) and For time t, t-τ(t) and t-τ(t) respectively. The matrix formed by the synchronization errors between each node and the leader node in a complex dynamic network; τ(t) is the coupling delay at time t; I NLet F(z) be an N-dimensional identity matrix, where N is the total number of nodes in the complex dynamic network; A and B are both constant matrices; E (t) represents the nonlinear function of the error network at time t, and represents the matrix formed by the differences between the nonlinear functions of each node in the complex dynamic network at time t and the nonlinear function of the leader node; c is the coupling strength; D is the network topology matrix; Γ is the internal coupling matrix; θ x The intensity of the injected attack; γ ij I represents the probability of a successful injection attack. m It is an m-dimensional identity matrix; For the injection attack at time t, Let t be the transpose of the injected attack signal at time t for the i-th node, where i ∈ 1, 2, ..., N; k The pulse moment; Let ζ(t) represent the synchronization error at the pulse moment, which is a matrix representing the synchronization errors between each node and the leader node in a complex dynamic network under the presence of a substitution attack; ζ(t) = diag(ζ1(t), ζ2(t), ..., ζ N (t)), ζ i To replace the success probability of the attack; For t k Time-based replacement attack, For the i-th node t k The transpose of the attack signal at time step; G is the pulse gain matrix.

[0125] This application considers dual-channel fraud attacks in complex dynamic network synchronization, specifically the sensor-to-controller loop and the controller-to-actuator loop, with the two loops corresponding to substitution and injection attacks, respectively. The constructed error network model overcomes the limitations of single-channel fraud attacks, significantly improving network communication security, while also considering the differences in fraud attack probabilities among different nodes.

[0126] For example, in the distributed control of a power system, nodes can be power plants (thermal / hydropower), substations, renewable energy power plants (photovoltaic / wind power), and distribution stations. The role of each node is to constitute the physical topology of the power grid and undertake the functions of power production, transmission, and distribution. The state vector can include parameters such as voltage amplitude, phase angle, line current, and frequency, and its function is to characterize the real-time operating state of the nodes and determine system stability. Nonlinear functions are used to describe physical laws such as power interaction between nodes and electromagnetic transient processes. The leader node is the standard state composed of reference signals such as standard voltage / frequency issued by the power grid dispatch center. Sensors at each node collect state parameters such as voltage / current amplitude, phase angle, frequency, power factor, and generator speed, forming a state vector that is sent to the controller. The controller outputs a target pulse control signal based on the state vector and the synchronization error between each node and the leader node, and sends it to the actuators of the restraining nodes. The actuators of the restraining nodes, such as generator control valves, SVC reactive power compensation devices, and energy storage converters, perform corresponding actions according to the pulse control signal to adjust various state parameters to maintain synchronization and stability. Other nodes in the complex dynamic network can further dynamically adjust their state parameters based on the state parameters of the restraining nodes.

[0127] To investigate the constraints of the distributed delay-controlled pulse controller that enables error network synchronization, this application first provides the following three definitions:

[0128] Definition 1: In the presence of dual-channel fraud attack interference, a complex dynamic network (1) under the action of a distributed delay restraint pulse controller (4) can be said to be exponentially synchronized to the leader node (2), that is, the condition for the global exponential stability of the error network is:

[0129] exist ∈ satisfies the following conditions:

[0130]

[0131] Among them, z i z(t) is the state vector of the i-th node at time t. L (t) is the state vector of the leader node at time t; ∈ is a positive constant; ∈ is a positive real number.

[0132] Definition 2: Average delayed pulse gain is defined based on pulse delay:

[0133] For pulse delay sequences If a coefficient exists Make

[0134]

[0135] in, N is the average delayed pulse gain; N′(t,t0) is the pulse sequence {tk The number of pulses within the interval (t0, t], where t0 is the initial time. The nominal state deviation at the moment of the pulse; The compensation delay factor caused by actuator dynamics; For pulse delay, It is a real number.

[0136] This application proposes the concept of average delayed pulse gain, which comprehensively considers the cumulative effect of pulse delay and pulse gain over time, thus eliminating the limitations imposed on pulse delay and pulse gain by the concepts of average pulse weight and average pulse delay.

[0137] Definition 3: Assume (1) the function V(t,q) in each pulse interval from arrive Continuous, satisfying (t,q)→(t k+1 (2) For all t>0, V(t,0)=0 satisfies the Lipschitz property of V(t,q); (3) V(t,q) satisfies α1||q|| r ≤V(t,q)≤α2||q|| r Where α1, α2, r are positive scalars, and

[0138]

[0139] Then V(t,q) can be considered to belong to the function class. Where parameters

[0140] To investigate the constraints of a distributed delay-controlled pulse controller that enables error network synchronization, this application presents the following two lemmas:

[0141] Lemma 1: Let the function For functions It is for It is monotonically non-decreasing, and the function It is monotonically non-decreasing with respect to x, such that

[0142]

[0143] Considering t∈[-v,0], then This holds true for all t>0.

[0144] The proof of Lemma 1 is given below:

[0145] To prove that Lemma 1 holds for t>0, we first need to prove...

[0146]

[0147] Where t1 represents the first pulse moment.

[0148] consider If the expression is continuous on the interval [-v, t1), then if formula (8) is incorrect, then there must exist at least one time t in the interval [0, t1). * , making It holds true, and for t≤t * , Established.

[0149] Due to the function From the monotonicity of the expression, we can conclude that:

[0150]

[0151] It is obvious that this is related to The two equations are inconsistent. Therefore, equation (8) is reasonable.

[0152] Secondly, assuming This holds true for t∈[t] k-1 -v,t k ), k=1,2,…,m′, we can obtain t m′-1 -v≤t≤t m′ .

[0153] according to and The monotonicity can be obtained Therefore, when t m′-1 -v≤t≤t m′ hour, Established.

[0154] Using t∈[0,t1) and Using the same proof strategy, we can conclude that t∈[t m ,t m+1 ]hour Then, by using mathematical induction, the proof of Lemma 1 can be completed.

[0155] Lemma 2: Suppose any function V belongs to the class of functions defined in Definition 3 (7). If let Therefore, it can be inferred that:

[0156]

[0157] in, It represents the minimum upper bound of the pulse interval.

[0158] The proof of Lemma 2 is given below:

[0159] To prove the result of Lemma 2, we need to prove the following inequality:

[0160]

[0161] in

[0162] First, according to the first inequality in formula (7), we get

[0163]

[0164] Let k = 0, then This means that formula (9) holds for t∈[t0,t1).

[0165] Secondly, assume that for all k = δ, δ ≥ 1, (9) holds, that is... t∈[t δ ,t δ+1 ).

[0166] The next goal is to prove t∈[t δ+1 ,t δ+2 ).

[0167] When t = t δ+1 When the following formula holds true:

[0168]

[0169] Obviously, for t∈[t δ+1 ,t δ+2 This can be obtained from formula (10). At this stage, applying mathematical induction, for all t∈[t k ,t k+1 ), Established.

[0170] Combining Definition 2 and Formula (9), for all t>t0, we can obtain:

[0171]

[0172]

[0173] This completes the proof of Lemma 2.

[0174] To investigate the constraints of the distributed delay-controlled pulse controller that enables error network synchronization, this application makes the following two assumptions:

[0175] Assumption 1: Assume that the nonlinear function F(·) of the error network satisfies the following conditions:

[0176] ||F i (z1)-F i (z2)||≤l i ||z1-z2||, i = 1, 2, ..., N

[0177] Among them l i Positive scalar, vector And define L = diag{l1,l2,…,l N For subsequent use.

[0178] Assumption 2: Assume a fraud attack δ(t) k )and They are bounded, meaning they satisfy the following conditions:

[0179] ||δ(tk)|| 2 ≤δ

[0180]

[0181] Where δ and α are constants, representing the upper bounds for substitution attacks and injection attacks, respectively.

[0182] In order to stabilize the global exponential of the complex dynamic network (1) and the leader node (2), this application, based on the proposed distributed delay restraint pulse controller, gives the following theorem:

[0183] Assume that assumptions 1 and 2 are true. If there exist real numbers... ι,a1,a2 and Positive definite matrix If the following constraints are met, then the controlled error network (6) is globally exponentially stable, that is, under the action of the distributed delay restraint pulse controller (4), the complex dynamic network (1) and the leader node (2) can achieve global exponential synchronization under a dual-channel fraud attack.

[0184] The first constraint is:

[0185]

[0186] The second constraint is:

[0187]

[0188] The third constraint is:

[0189]

[0190] The fourth constraint is:

[0191] ι -1 θ x ασ max (Θ * )θ2≤0 (14)

[0192] The fifth constraint is:

[0193]

[0194] The sixth constraint is:

[0195]

[0196] The seventh constraint is:

[0197] δσ max (Θ ζ )θ2≤0 (17)

[0198] The eighth constraint is:

[0199]

[0200] The ninth constraint is:

[0201]

[0202] Where D is the network topology matrix; Γ is the internal coupling matrix; Let m be the Lyapunov weight matrix, and m be the number of states. θ1 and θ2 are positive real numbers, I m It is an m-dimensional identity matrix; I N Let A be an N-dimensional identity matrix, where N is the total number of nodes in the complex dynamic network; A and B are both constant matrices; c is the coupling strength. a1, a2, and ι are real numbers; θ x The intensity of the injected attack; σ is a diagonal matrix; α is the upper bound of the injection attack; max (Θ * ) is Θ * Spectral radius; Θ * =E[γ * (t)];c p For distributed control strength; ζ(t)=diag(ζ1(t),ζ2(t),…,ζ N (t)), ζ i ∈[0,1] is a constant; G is the pulse gain matrix; δ is the upper bound of the substitution attack; σ max (Θ ζ ) is Θ ζ spectral radius, Let be the expectation of the matrix; T represents the average delayed pulse gain. s The pulse interval; It is a positive real number.

[0203] When deploying a distributed delay-restrained pulse controller, solve for the condition that satisfies the first, second, and third constraints. according to The obtained θ2 satisfies the fourth and seventh constraints; based on the fifth and sixth constraints and the obtained... Further solve c p Solve for the average delayed pulse gain based on the eighth and ninth constraints. and pulse interval T s The pulse delay is calculated based on the average delayed pulse gain. The target distributed delay restraint pulse controller can then be obtained.

[0204] The proof of the above theorem is given below:

[0205] Choose the following Lyapunov functions:

[0206]

[0207] The time interval t∈[t] is calculated along the trajectory of the controlled error network (6). k ,t k+1 ), The Dini derivative of the inner V(t) can be obtained as follows:

[0208]

[0209] Consider assumption 1, assuming the existence of a positive definite matrix. and diagonal matrix This makes the following conclusions true:

[0210]

[0211]

[0212] By introducing the Kronecker product and condition (11) in Theorem 1, we obtain:

[0213]

[0214] set up Then there exists a real number ι that satisfies the following inequality:

[0215]

[0216] Combining (21), (22), (23) and (12)-(14) in Theorem 1, we can obtain:

[0217]

[0218] On the other hand, considering the substitution attack, for the pulse time t = t k , make We can obtain:

[0219]

[0220] Let υ = (c p ) 2 σ max (G T (I N -Θ ζ For the first term in (25), we can deduce:

[0221]

[0222] For the cross terms in equation (25), considering (15) and (16), we can obtain:

[0223]

[0224] Similarly, for the last term in equation (25), considering condition (17), it can be calculated as:

[0225]

[0226]

[0227] Substituting equations (26)-(29) into equation (25), we get:

[0228]

[0229] in

[0230] Based on equations (24) and (30), a comparison system that holds for any ε>0 and has a mathematical expectation is established:

[0231]

[0232] in It is the only solution for system (31).

[0233] According to Lemma 1, Established.

[0234] Using parametric variational equations, we can calculate... The solution is:

[0235]

[0236] Where W(t,s) represents the Cauchy matrix derived from the following impulse system:

[0237]

[0238] According to Lemma 2, the solution to the above pulse system can be obtained. for:

[0239]

[0240] where t∈[t0,+∞), Then W(t,s) can be estimated as:

[0241] W(t,s)≤e -ρ(t-s) ,t∈[t0-h,+∞) (33)where s∈[t0,t).

[0242] Integrating equations (32) and (33), we can derive:

[0243]

[0244] in

[0245] The parameter function is constructed as follows The next goal is to prove It has a unique zero solution in the positive real domain. This can be easily proven. and It is true. Considering... It is a continuous function. Combining its monotonically increasing property and the Intermediate Value Theorem, we can derive... There must be a unique solution.

[0246] Considering ∈>0, We can obtain:

[0247]

[0248] Next, assume there exists a specific time t. * >0, where equation (35) is invalid, that is, the following equation holds:

[0249]

[0250] At the same time, equation (35) is in [0,t * It still holds true within the domain of the definition of ).

[0251] Substituting equation (35) into equation (34), we get:

[0252]

[0253]

[0254] This is inconsistent with assumption (36). Therefore, it can be deduced that conclusion (35) is valid for all t.

[0255] When ε→0, we can obtain:

[0256]

[0257] According to Definition 1, it can be inferred that the complex dynamic network (1) successfully achieves global exponential synchronization with the leader node (2) through a distributed delay-controlled pulse controller (4) under the condition of suffering a two-channel fraud attack. This concludes the proof of the theorem.

[0258] This application introduces a comparison principle and parameter variation formula applicable to hybrid delay pulses, derives sufficient conditions for synchronization of complex dynamic networks, and fully demonstrates that under the combined effect of dual-channel fraud attacks, the distributed delay restraint pulse controller proposed in this application can also enable complex dynamic networks to achieve rapid synchronization.

[0259] In summary, the distributed pinning pulse control method for complex networks based on dual-channel attacks proposed in this invention considers dual-channel fraud attacks on the sensor-to-controller loop and the controller-to-actuator loop in complex dynamic networks. The constructed distributed delay pinning pulse controller overcomes the limitations of single-channel fraud attacks, enabling the network to quickly achieve synchronization even under the combined effect of dual-channel fraud attacks. Furthermore, the distributed delay pinning pulse controller constructed in this invention implements discrete control on a few key nodes, i.e., pinning nodes, which, while ensuring network synchronization performance and improving network security, can further reduce control costs at both the time and space levels.

[0260] Furthermore, this invention introduces the concept of average delay pulse gain based on pulse delay, comprehensively quantifies the cumulative effect of pulse delay and pulse gain over time, breaks through the limitations of traditional average pulse weight and average pulse delay concepts, can improve the dynamic response accuracy and anti-interference capability of the controller, realize the rapid and stable synchronization of each node and the leader node in complex dynamic networks, and further improve the security of complex dynamic networks.

[0261] To verify the effectiveness of the proposed distributed delay-controlled pulse control strategy under dual-channel fraud attacks, this embodiment uses a typical complex dynamic network model for numerical simulation.

[0262] Consider a complex dynamic network consisting of multiple Chua's circuits. The topology of the Chua's circuit network is as follows: Figure 3 As shown. Its dynamic characteristics are described below:

[0263]

[0264] Where h(z)=(2 / 7)z-(3 / 14)(|z+1|-|z-1|), k=9, l=14+(2 / 7), then the Lipschitz condition can be calculated as L=(2 / 7)I.

[0265] Let c = 2.5, Γ = I, θ x =1,ι=1.5, With the help of the LMI toolbox, the following can be calculated: The feasible solution is:

[0266]

[0267] Then the first constraint, the second constraint, and the third constraint are satisfied.

[0268] Let γ be the probability of a successful injection attack. ij =0.01,σ max (Θ * ) = 0.1. Considering the matrix A feasible solution is to let θ2 = 15.89 to satisfy... Then l -1 θ x ασ max (Θ * The fourth constraint is satisfied if θ² = 0.001 ≈ 0. Similarly, let ζ be the probability of a successful substitution attack. i =0.01,σ max (Θ ζ If ) = 0.01, and the upper bound of the replacement attack is δ = 0.01, then δσ max (Θ ζ θ² = 0.0016 ≈ 0 satisfies the seventh constraint. Let the distributed control strength c p =0.5, a1=a2=0.1, considering that only the first node is constrained, let θ p If the expression = diag(1,0,0,0,0,0,0,0), then the fifth and sixth constraints are satisfied. Select the pulse interval T. s =0.2, pulse delay Average delayed pulse gain Then it can be calculated therefore, and By satisfying the eighth and ninth constraints respectively, the target distributed delay restraint pulse controller can be obtained.

[0269] Figure 4 The control signal output by the distributed delay restraint pulse controller in this embodiment is shown. Figure 5 The time-domain distribution of injection attacks received by the first node from the other 7 nodes is shown over a 10-second simulation period. Figure 5 The discrete markers in the diagram represent injection attacks launched by each node against the first node at different times. The differences in injection attacks between nodes stem from the different connection weights between nodes in the network topology. Figure 6 The temporal distribution of the substitution attack is described in the form of binary signals. (Comparison) Figure 5 It can be seen that there is a significant overlap between substitution attacks and injection attacks, which will lead to more severe damage. Figure 7 The synchronization error of all nodes relative to the leader node is given, defined as follows: It can be seen that even when injection attacks and substitution attacks occur simultaneously, the network can still quickly reach a synchronization state under the action of the target distributed delay restraint pulse controller. Figure 8 The convergence speed of network synchronization error is demonstrated under different numbers of restraining nodes. Clearly, as the number of restraining nodes increases, the synchronization error converges at a faster rate. However, the control cost also increases accordingly. In scenarios with strict cost control requirements, the distributed delay restraining pulse controller proposed in this embodiment can still effectively complete the control task by controlling only a single node.

[0270] Based on the aforementioned distributed pinning pulse control method for complex networks based on dual-channel attacks, this application also provides a distributed pinning pulse control system for complex networks based on dual-channel attacks, comprising:

[0271] The Complex Network Building Module is used to build complex dynamic networks and identify leader nodes;

[0272] The controller construction module is used to select restraining nodes from complex dynamic networks; it constructs a distributed delayed restraining pulse controller, generates pulse control signals based on the state data of each node, and sends them to the actuators of the restraining nodes; it is assumed that the state data of each node is subjected to a substitution attack during the process of being sent to the distributed delayed restraining pulse controller, and the pulse control signals are subjected to an injection attack during the process of being sent to the actuators of the restraining nodes.

[0273] The error network construction module is used to construct an error network based on the synchronization error between nodes and the leader node in a complex dynamic network, under the condition that both injection and substitution attacks exist simultaneously, based on a distributed delay restraint pulse controller.

[0274] The constraint solving module is used to solve the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller based on the constraint condition that makes the error network globally exponentially stable, so as to obtain the target distributed delay restraint pulse controller.

[0275] The control module is used to collect the status data of each node in real time using the sensors of each node in the complex dynamic network, and send it to the target distributed delay restraint pulse controller. The target distributed delay restraint pulse controller generates a target pulse control signal based on the status data of each node and the synchronization error between each node and the leader node. The actuator of the restraint node receives the target pulse control signal and adjusts the status parameters of the restraint node so as to dynamically adjust the status parameters of other nodes in the complex dynamic network according to the status parameters of the restraint node.

[0276] This application also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of the above-described distributed pinning pulse control method for complex networks based on dual-channel attacks.

[0277] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A distributed pinning pulse control method for complex networks based on dual-channel attacks, characterized in that, include: Establish complex dynamic networks and leader nodes; Selecting restraining nodes from complex dynamic networks; Construct a distributed delayed restraint pulse controller, generate pulse control signals based on the status data of each node, and send them to the actuators of the restraint nodes; Assume that the status data of each node is subjected to a substitution attack during the process of being sent to the distributed delay restraint pulse controller, and the pulse control signal is subjected to an injection attack during the process of being sent to the actuator of the restraint node; Based on a distributed delay restraint pulse controller, an error network is constructed according to the synchronization error between nodes and the leader node in a complex dynamic network under the condition that both injection attacks and replacement attacks exist simultaneously. Based on the constraint condition that the error network is globally exponentially stable, the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller are solved to obtain the target distributed delay restraint pulse controller. The sensors of each node in the complex dynamic network are used to collect the status data of each node in real time and send it to the target distributed delay restraint pulse controller. The target distributed delay restraint pulse controller generates target pulse control signals based on the status data of each node and the synchronization error between each node and the leader node. The actuator of the restraining node receives the target pulse control signal and adjusts the state parameters of the restraining node so as to dynamically adjust the state parameters of other nodes in the complex dynamic network according to the state parameters of the restraining node.

2. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 1, characterized in that, Establishing complex dynamic networks and leader nodes includes: In a complex dynamic network, each node is represented as follows: Among them, z i (t)=(z i1 (t),z i2 (t),…,z im (t)) T Let z be the state vector of the i-th node at time t. im (t) represents the m-th state data of the i-th node at time t, where m is the number of states; For z i The first derivative of (t); A and B are both constant matrices; Let c be the nonlinear function of the i-th node at time t, used to represent the interaction relationship between nodes; c is the coupling strength; D = (d ij ) N×N Let d be a network topology matrix. If the j-th node is connected to the i-th node, then d ij >0, otherwise d ij =0; N is the total number of nodes in the complex dynamic network; Γ is the internal coupling matrix; z j (t-τ(t)) is the state vector of the j-th node at time t-τ(t); τ(t) is the coupling delay at time t; φ i (t) represents the injection attack at time t of the i-th node; The leader node is represented as: Among them, z L (t)=(z L1 (t),z L2 (t),…,z Lm (t)) T Let be the state vector of the leader node at time t. Let be the nonlinear function of the leader node at time t. For z L The first derivative of (t).

3. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 2, characterized in that, The distributed delay-controlled pulse controller is represented as follows: Among them, u i (t) represents the pulse control signal output by the distributed delay restraint pulse controller at time t; c p The distributed control strength is N; N is the total number of nodes in the complex dynamic network. For distributed control weights; t k Let k be the pulse moment, and k be the index of the pulse moment. It is the set of positive integers; For pulse delay, It is the set of positive real numbers; For the j-th node The state vector at time step j, j∈1,2,…,N; Let be the constraint gain of the i-th node. If the i-th node is connected to the constraint node, then... otherwise for The synchronization error between the i-th node and the leader node at time i.

4. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 3, characterized in that, Based on a distributed delay-controlled pulse controller, in the presence of both injection and substitution attacks, an error network is constructed according to the synchronization errors between nodes and the leader node in a complex dynamic network. The formula is as follows: in, The synchronization error at non-pulse moments is represented by a matrix, z, which characterizes the synchronization errors between nodes and the leader node in a complex dynamic network under injection attacks. E (t), z E (t-τ(t)) and For time t, t-τ(t) and t-τ(t) respectively. The matrix formed by the synchronization errors between each node and the leader node in a complex dynamic network; τ(t) is the coupling delay at time t; I N Let F(z) be an N-dimensional identity matrix, where N is the total number of nodes in the complex dynamic network; A and B are both constant matrices; E (t) represents the nonlinear function of the error network at time t, and represents the matrix formed by the differences between the nonlinear functions of each node in the complex dynamic network at time t and the nonlinear function of the leader node; c is the coupling strength; D is the network topology matrix; Γ is the internal coupling matrix; θ x The intensity of the injected attack; γ ij I represents the probability of a successful injection attack. m It is an m-dimensional identity matrix; For the injection attack at time t, Let t be the transpose of the injected attack signal at time t for the i-th node, where i ∈ 1, 2, ..., N; k The pulse moment; Let ζ(t) represent the synchronization error at the pulse moment, which is a matrix representing the synchronization errors between each node and the leader node in a complex dynamic network under the presence of a substitution attack; ζ(t) = diag(ζ1(t), ζ2(t), ..., ζ N (t)), ζ i To replace the success probability of the attack; For t k Time-based replacement attack, For the i-th node t k The transpose of the attack signal at time step; G is the pulse gain matrix.

5. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 4, characterized in that, The average delayed pulse gain is defined based on the pulse delay: in, N is the average delayed pulse gain; N′(t,t0) is the pulse sequence {t k The number of pulses within the interval (t0, t], where t0 is the initial time. The nominal state deviation at the moment of the pulse; The compensation delay factor caused by actuator dynamics; For pulse delay, It is a real number.

6. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 5, characterized in that, Based on the constraint of global exponential stability of the error network, the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller are solved, including: The first constraint is: The second constraint is: The third constraint is: The fourth constraint is: i -1 i x as max (I * )θ2≤0 The fifth constraint is: The sixth constraint is: The seventh constraint is: ss max (I ζ )θ2≤0 The eighth constraint is: The ninth constraint is: Where D is the network topology matrix; Γ is the internal coupling matrix; Let m be the Lyapunov weight matrix, and m be the number of states. θ1 and θ2 are positive real numbers, I m It is an m-dimensional identity matrix; I N Let A be an N-dimensional identity matrix, where n is the total number of nodes in the complex dynamic network; A and B are both constant matrices; c is the coupling strength. a1, a2, and ι are real numbers; θ x The intensity of the injected attack; σ is a diagonal matrix; α is the upper bound of the injection attack; max (Θ * ) is Θ * Spectral radius; Θ * =E[γ * (t)];c p For distributed control strength; ζ(t)=diag(ζ1(t),ζ2(t),…,ζ N (t)), ζ i ∈[0,1] is a constant; G is the pulse gain matrix; δ is the upper bound of the substitution attack; δ max (Θ ζ ) is Θ ζ spectral radius, Let be the expectation of the matrix; T represents the average delayed pulse gain. s The pulse interval; It is a positive real number; Solve for the condition that satisfies the first, second, and third constraints. according to The obtained θ2 satisfies the fourth and seventh constraints; based on the fifth and sixth constraints and the obtained... Further solve c p Solve for the average delayed pulse gain based on the eighth and ninth constraints. and pulse interval T s The pulse delay is calculated based on the average delayed pulse gain.

7. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 1, characterized in that, The criteria for selecting a restraining node in a complex dynamic network include: selecting the node with the highest degree centrality or the largest number of connected nodes in the complex dynamic network as the restraining node.

8. The distributed pinning pulse control method for complex networks based on dual-channel attacks according to claim 1, characterized in that, The condition for the global exponential stability of the error network is: Among them, z i z(t) is the state vector of the i-th node at time t. L (t) is the state vector of the leader node at time t; ∈ is a positive constant; ∈ is a positive real number.

9. A distributed restraint pulse control system for complex networks based on dual-channel attacks, characterized in that, include: The Complex Network Building Module is used to build complex dynamic networks and identify leader nodes; The controller construction module is used to select restraining nodes from complex dynamic networks; it constructs a distributed delayed restraining pulse controller, generates pulse control signals based on the state data of each node, and sends them to the actuators of the restraining nodes; Assume that the status data of each node is subjected to a substitution attack during the process of being sent to the distributed delay restraint pulse controller, and the pulse control signal is subjected to an injection attack during the process of being sent to the actuator of the restraint node; The error network construction module is used to construct an error network based on the synchronization error between nodes and the leader node in a complex dynamic network, under the condition that both injection and substitution attacks exist simultaneously, based on a distributed delay restraint pulse controller. The constraint solving module is used to solve the distributed control strength, pulse interval, and pulse delay of the distributed delay restraint pulse controller based on the constraint condition that makes the error network globally exponentially stable, so as to obtain the target distributed delay restraint pulse controller. The control module is used to collect the status data of each node in real time using the sensors of each node in the complex dynamic network and send it to the target distributed delay restraint pulse controller. The target distributed delay restraint pulse controller generates target pulse control signals based on the status data of each node and the synchronization error between each node and the leader node. The actuator of the restraining node receives the target pulse control signal and adjusts the state parameters of the restraining node so as to dynamically adjust the state parameters of other nodes in the complex dynamic network according to the state parameters of the restraining node.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the steps of the distributed pinning pulse control method for complex networks based on dual-channel attacks as described in any one of claims 1 to 8.

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