An Event-Triggered Impulse Control Method and System for Neural Networks under Fraud Attacks

Abstract

The present invention relates to an event-triggered impulsive control method and system for a neural network under fraud attacks, belonging to the technical field of neural network control. It includes: establishing a coupled neural network model with time-varying and time-delay, and determining the synchronization target of the coupled neural network model; obtaining the state information of the coupled neural network model, and establishing an error-coupled neural network system model according to the state information; designing a saturated impulsive controller according to the error-coupled neural network system model and the synchronization target; obtaining a first sufficient condition and a second sufficient condition based on the error-coupled neural network system model and the saturated impulsive controller; obtaining a first optimization mathematical model according to the first sufficient condition and the second sufficient condition; calculating the impulsive control gain matrix and the maximum estimation of the attraction domain by using the first optimization mathematical model. The present invention effectively solves the combined influence of actuator saturation, deception attacks, and impulsive effects on the overall performance of the system.

Description

Technical Field

[0001] The present invention relates to the technical field of neural network control, and in particular to an event-triggered pulse control method and system for a neural network under fraud attacks. Background Art

[0002] In the rapid development of modern technology, as a powerful computing model, neural networks have demonstrated their unique value and extensive application potential in multiple fields. The core advantages of neural networks lie in their adaptability, non-convexity, non-linearity, and excellent fault tolerance, which together form the broad application foundation in key fields such as frequency-domain image analysis, combinatorial optimization, and artificial intelligence. In particular, the synchronization characteristics of neural networks have attracted extensive attention in multiple disciplinary fields such as biomedical engineering, secure communication, and fluid dynamics. In these fields, the main challenges in achieving the synchronization of coupled neural networks include the dynamic behavior of nodes themselves and the complexity of the interconnections between nodes caused by network coupling. Currently, to achieve the synchronization of coupled neural networks, various control methods have been proposed, such as pinning control, sampling control, pulse control, etc. Among them, pulse control is an effective control strategy that promotes the synchronization of coupled neural networks through relatively small control gain pulses at specific discrete moments. This method has the advantages of strong robustness, simple structure, and low maintenance cost, which has inspired related research. However, the pulse disturbances caused by pulse control may have an adverse impact on the final synchronization, especially in practical engineering scenarios, such as bipedal robots and vehicle models traveling smoothly being affected by intermittent collisions. To solve this problem, an event-triggered control (ETC) mechanism can be adopted. The event-triggered control mechanism updates the control signal only when specific triggering conditions are met, while the time-triggered control mechanism updates the signal at fixed time intervals regardless of the contribution of the control signal to the dynamics. Therefore, event-triggered pulse control has received extensive attention because it reduces the amount of data transmission while ensuring system performance.

[0003] However, the system using event-triggered pulse control is driven by transmitting control signals through actuators. Due to physical, technical, and security limitations, saturation non-linearity is very common in physical systems. And the actuator saturation phenomenon will seriously weaken the system performance, specifically manifested as increased system oscillation, significantly increased overshoot, response lag, and even instability of the overall system. In addition, due to the open nature of the communication network, the controlled system also faces the threat of network attacks. Especially common network attacks such as spoofing attacks, which maliciously tamper with the original control instructions transmitted through the network channel, and such attacks are often difficult to be effectively detected. In recent years, although the research work in related fields has focused on actuator saturation or spoofing attacks to a certain extent, it cannot effectively solve the combined effects of actuator saturation, spoofing attacks, and pulse effects on the overall system performance. Summary of the Invention

[0004] To this end, the technical problem to be solved by the present invention is to overcome the influence of actuator saturation, deception attack and pulse effect on the overall performance of the system in the prior art without comprehensive consideration.

[0005] In a first aspect, to solve the above technical problem, the present invention provides an event-triggered pulse control method for a neural network under fraud attacks, including:

[0006] Establish a coupled neural network model with time-varying and time-delay, and determine the synchronization target of the coupled neural network model;

[0007] Obtain the state information of the coupled neural network model, and establish an error-coupled neural network system model according to the state information;

[0008] According to the error-coupled neural network system model and the synchronization target, design a saturation pulse controller, and the mathematical expression of the saturation pulse controller is:

[0009]

[0010] where, U i (t) is the saturation pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t k is the k-th of the pulse sequence, and t represents time;

[0011] Based on the error-coupled neural network system model and the saturation pulse controller, obtain the first sufficient condition and the second sufficient condition;

[0012] According to the first sufficient condition and the second sufficient condition, obtain the first optimization mathematical model; use the first optimization mathematical model to calculate the pulse control gain matrix and the maximum estimate of the attraction domain.

[0013] In an embodiment of the present invention, the mathematical expression of the coupled neural network model is:

[0014]

[0015] where, x i (t) represents the state variable information of the i-th neural network, and both represent the system matrix, N represents the number of agents, ε represents the coupling strength; Γ represents the inline coupling matrix; represents a non-linear vector-valued function, is the external input, τ(t) is the system time-delay, wij is the connection coefficient between node i and node j, and U i (t) is a saturation pulse controller, where both i and j represent different numbers, and t represents time.

[0016] In an embodiment of the present invention, the mathematical expression of the error-coupled neural network system model is:

[0017]

[0018] where z i (t) is the error vector; are different system matrices, f(z i (t)) is a non-linear vector-valued error function, f(z i (t - τ(t))) is the non-linear vector-valued error function at time t - τ(t), τ(t) is the system time delay, ε is the coupling strength, w ij is the connection coefficient between node i and node j, Γ is the inline coupling matrix, and N is the number of agents; both i and j represent different numbers, and t represents time.

[0019] In an embodiment of the present invention, the first sufficient condition is:

[0020]

[0021] LQ 2 L ≤ λ 2 P,

[0022]

[0023] where η, λ 2 and represent different positive constants, I N is the N-dimensional identity matrix, is the Kronecker product, H is the matrix to be solved, N is the number of agents, n is the total number of state variables, and P is a positive definite matrix; are different system matrices; Q 1 and Q 2 are different diagonal matrices, ρ is the probability value in the Bernoulli distribution, G is the known distribution matrix, T is the matrix transpose symbol, L is the Lipschitz matrix, is the convex hull matrix, ★ represents the symmetric omission symbol when the matrix is symmetric; Ξ 1 , Ξ 2 and Ξ 3 The calculation formulas are:

[0024]

[0025] Among them, γ, represent different positive constants, θ * represents the maximum pulse interval, is a known matrix, is the system matrix, W is the connection matrix of nodes, ε represents the coupling strength, and Γ represents the inline coupling matrix.

[0026] In an embodiment of the present invention, the second sufficient condition is:

[0027]

[0028] LQ 2 L ≤ λ 2 P,

[0029]

[0030]

[0031] Among them, η, λ 2 and represent different positive constants, I N is an N - dimensional identity matrix, is the Kronecker product, H is the matrix to be solved, N is the number of agents, n is the total number of state variables, and P is a positive definite matrix; are different system matrices; Q 1 and Q 2 are different diagonal matrices; ρ is the probability value in the Bernoulli distribution, G is a known distribution matrix, T is the matrix transpose symbol, L is the Lipschitz matrix, is the convex hull matrix, * represents the symmetric omission symbol when the matrix is symmetric; Ξ 4 、Ξ 2 and Ξ 3 The calculation formulas are:

[0032]

[0033] Among them, γ, represent different positive constants, θ * represents the maximum pulse interval, is a known matrix, is the system matrix, W is the connection matrix of nodes, ε represents the coupling strength, and Γ represents the inline coupling matrix.

[0034] In an embodiment of the present invention, the first optimization mathematical model is:

[0035]

[0036] Among them, is the variable to be solved, η, ξt and λ 2 represent different positive constants, P is the inverse matrix of a positive definite matrix, is a positive constant, I N is an N - dimensional identity matrix, N is the number of agents, and n is the total number of state variables, are different system matrices, is the Kronecker product, and both represent matrices, L is the Lipschitz matrix, G is the known distribution matrix, T is the matrix transpose symbol, ρ is the probability value in the Bernoulli distribution, l is a positive constant, and * represents the symmetric omission symbol when the matrix is symmetric; and The calculation formulas of are:

[0037]

[0038] where, γ, represent different positive constants, θ * represents the maximum pulse interval, is a known matrix, is the system matrix, W is the node connection matrix, ε represents the coupling strength, and Γ represents the inline coupling matrix.

[0039] In an embodiment of the present invention, after calculating the pulse control gain matrix and the maximum estimate of the attraction domain, it includes solving the maximum pulse interval using the first optimization mathematical model.

[0040] In an embodiment of the present invention, after calculating the pulse control gain matrix and the maximum estimate of the attraction domain, it further includes solving the minimum pulse interval using the second optimization mathematical model, and the second optimization mathematical model is:

[0041]

[0042]

[0043] where, is the variable to be solved, η, ξ i and λ 2 represent different positive constants, is the inverse matrix of a positive definite matrix, is a positive constant, I N is an N - dimensional identity matrix, N is the number of agents, and n is the total number of state variables, are different system matrices, is the Kronecker product, and All represent matrices, L is a Lipschitz matrix, G is a known distribution matrix, T is the matrix transpose symbol, ρ is the probability value in the Bernoulli distribution, l is a positive constant, and * represents the symmetric omission symbol when the matrix is symmetric; and The calculation formula of is:

[0044]

[0045] where, γ, represent different positive constants, θ * represents the maximum pulse interval, is a known matrix, is the system matrix, W is the connection matrix of nodes, ε represents the coupling strength, and Γ represents the inline coupling matrix.

[0046] In a second aspect, to solve the above technical problems, the present invention provides an event-triggered pulse control system for a neural network under fraud attacks, including:

[0047] A model establishment module, configured to establish a coupled neural network model with time-varying and time-delay, determine the synchronization target of the coupled neural network model; obtain the state information of the coupled neural network model, and establish an error-coupled neural network system model according to the state information;

[0048] A controller design module, configured to design a saturated pulse controller according to the error-coupled neural network system model and the synchronization target, and the mathematical expression of the saturated pulse controller is:

[0049]

[0050] where, U i (t) is the saturated pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t k is the k-th pulse sequence;

[0051] A condition acquisition module, configured to obtain a first sufficient condition and a second sufficient condition based on the error-coupled neural network system model and the saturated pulse controller;

[0052] An optimization calculation module, configured to obtain a first optimization mathematical model according to the first sufficient condition and the second sufficient condition; calculate the pulse control gain matrix and the maximum estimate of the attraction domain by using the first optimization mathematical model.

[0053] Thirdly, to solve the above technical problems, the present invention provides a controller, including the event-triggered pulse control system of a neural network under the above-mentioned fraud attack.

[0054] The above technical solution of the present invention has the following beneficial effects compared with the prior art:

[0055] (1) The present invention provides an event-triggered pulse control method and system for a neural network under fraud attack, adopting a hybrid event-triggered mechanism. The core feature of this mechanism lies in the introduction of a relaxation interval. This design effectively avoids the occurrence of Zeno behavior, that is, it prevents the problem of triggering an infinite number of control actions within a finite time. In addition, this mechanism exhibits excellent self-healing ability in the face of deception attacks, and can significantly improve the degradation of synchronization performance caused thereby. To better meet the requirements of engineering practical applications, the present invention constructs a comprehensive theoretical framework, which comprehensively considers various factors such as actuator saturation, deception attacks, and hybrid pulse effects, and effectively solves the combined influence of actuator saturation, deception attacks, and pulse effects on the overall performance of the system.

[0056] (2) The present invention conducts a detailed classification discussion on the relevant parameters of the pulse effect to distinguish the roles of pulse signals under different functions. Relying on this theoretical framework, the conditions for the mean-square synchronization of coupled neural networks are deeply studied, and the convergence rate of the system is analyzed. Through this process, the inherent elastic constraints between the pulse control gain and the probability and intensity of deception attacks are revealed, providing an important theoretical basis for the design of control systems.

[0057] (3) Based on the three optimization problems proposed, the present invention conducts in-depth mathematical derivations to obtain the maximum estimated value of the attraction domain, the feasible maximum pulse interval, and the allowable minimum pulse interval. These results not only enrich the theoretical basis of the hybrid event-triggered mechanism, but also provide practical references for the selection of relevant parameters, ensuring the effectiveness and reliability of the control system in practical applications. Through these innovative points, the present invention provides strong technical support for improving the robustness and synchronization performance of the control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] In order to make the content of the present invention easier to be clearly understood, the following further details the present invention according to the specific embodiments of the present invention in conjunction with the drawings, where

[0059] Figure 1 is a flowchart of an event-triggered pulse control method for a neural network under fraud attack in a preferred embodiment of the present invention;

[0060] Figure 2 is a topological graph of a coupled neural network composed of 11 nodes in a preferred embodiment of the present invention;

[0061] Figure 3 This is the state trajectory diagram of the coupled neural network and the error-coupled neural network without control input in the preferred embodiment of the present invention;

[0062] Figure 4 This is the state trajectory diagram of the coupled neural network and the error-coupled neural network with control input and deception attack in the preferred embodiment of the present invention;

[0063] Figure 5 This is the time series diagram of event triggering in the preferred embodiment of the present invention;

[0064] Figure 6 This is the state trajectory diagram of the coupled neural network and the error-coupled neural network with pulse disturbance in the preferred embodiment of the present invention. Specific implementation manner

[0065] The following further describes the present invention with reference to the drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited do not limit the present invention.

[0066] Embodiment 1

[0067] Referring to Figure 1 As shown, the embodiment of the present invention provides an event-triggered pulse control method for a neural network under fraud attack, including:

[0068] Establish a coupled neural network model with time-varying and time-delay, and determine the synchronization target of the coupled neural network model;

[0069] Obtain the state information of the coupled neural network model, and establish an error-coupled neural network system model according to the state information;

[0070] Design a saturated pulse controller according to the error-coupled neural network system model and the synchronization target;

[0071] Based on the error-coupled neural network system model and the saturated pulse controller, obtain the first sufficient condition and the second sufficient condition;

[0072] According to the first sufficient condition and the second sufficient condition, obtain the first optimization mathematical model; use the first optimization mathematical model to calculate the pulse control gain matrix and the maximum estimate of the attraction domain.

[0073] An embodiment of the present invention provides an event-triggered pulse control method for a neural network under fraud attacks. By establishing a time-varying delay-coupled neural network model and an error-coupled neural network model, a saturation pulse controller is designed to effectively regulate the system behavior. At the same time, to compensate for the impact of deception attacks, a hybrid event-triggered mechanism is proposed, and the occurrence of Zeno phenomenon is avoided by introducing a minimum event interval. In addition, according to the different functions of the pulse signal, the parameters related to the pulse effect are classified and discussed in the embodiment of the present invention, and the first sufficient condition and the second sufficient condition for the mean-square exponential synchronization of the time-varying delay-coupled neural network in two different cases are obtained. By establishing an optimization mathematical model based on these sufficient conditions, the embodiment of the present invention deeply analyzes the optimization problem to achieve resource conservation and system performance improvement. While fully considering the actual situation, the embodiment of the present invention simplifies the analysis process of the mean-square synchronization of the time-varying delay-coupled neural network. This method can not only effectively solve the combined effects of actuator saturation, deception attacks, and pulse effects on the overall system performance, but also ensure that the synchronization performance of the neural network is significantly improved in a fraud attack environment.

[0074] Specifically, a coupled neural network model with time-varying and time delay is established (the coupled neural network model can be simply referred to as the coupled neural network), and the mathematical expression of this model is:

[0075]

[0076] Where, i = 1, 2,..., N represents the state variable information of the i-th neural network, and T is the matrix transpose symbol; and both represent the system matrix, N represents the number of agents, represents the n×n-dimensional Euclidean space, represents the n-dimensional Euclidean space; ε represents the coupling strength; Γ represents the inline coupling matrix; represents the non-linear vector-valued function satisfying τ(t) is the system time delay and satisfies 0 ≤ τ(t) ≤ τ, where τ represents the maximum value of τ(t); the connection matrix of the nodes W = [w ij N×N , w ij is the connection coefficient between node i and node j, and is specifically defined as: if there is a connection between node i and node j, then w ij = w ji > 0, otherwise w ij = w ji > 0; in addition, the diagonal element represents the external input; U i ​(t) represents the controller to be designed, i.e., the saturation pulse controller to be designed; both i and j represent different numbers, and t represents time.

[0077] Specifically, the embodiment of the present invention designs an isolated neural network model with a target trajectory (the isolated neural network model with a target trajectory can be simply referred to as the target neural network), and its mathematical expression is:

[0078]

[0079] where, represents the state of the isolated neural network.

[0080] Specifically, the state information x of the neural network is obtained through the sensor device i (t) and an error system model is established. The mathematical expression of the defined error vector is:

[0081] z i (t) = x i (t) - y(t);

[0082] where, i = 1, 2,..., N.

[0083] Furthermore, according to Equation (1) and Equation (2), the error-coupled neural network system model can be calculated, and its mathematical expression is:

[0084]

[0085] where, z i (t) is the error vector; f(z i (t)) is a non-linear vector-valued error function, and its expression is f(z i (t - τ(t))) is the non-linear vector-valued error function at time t - τ(t), and its expression is i = 1, 2,..., N.

[0086] Specifically, according to the error-coupled neural network system model and the determined synchronization target, the embodiment of the present invention can design a saturation pulse controller applicable to the system, and its mathematical expression is:

[0087]

[0088] where, U i (t) is the saturation pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t kis the kth pulse sequence, t represents time, and the pulse sequence It is determined by the event trigger mechanism. Represents the set of positive integers.

[0089] Furthermore, the control signal u i The mathematical expression of (t) is:

[0090] u i (t) = sat(Kz i (t)); (5)

[0091] Where sat(·) represents the nonlinear saturation function, is the pulse control gain to be confirmed. The meaning of the nonlinear saturation function sat(·) is given as an example:

[0092] satisfy

[0093]

[0094] Specifically, when the communication network between the controller and the actuator is subjected to a spoofing attack, the attacker may send forged data information to the actuator, which replaces the real control signal originally sent by the controller. This attack behavior not only destroys the integrity of the control instructions, but also may cause the actuator to receive erroneous instructions, thereby causing a decrease in system performance, and may even cause the system to be unstable or completely out of control. Therefore, the embodiment of the present invention fully considers the impact of spoofing attacks on the system, and the pulse component (formula (5)) of the controller (formula (4)) can be rewritten as:

[0095]

[0096] in, is the new control signal; the random variable ρ(t) follows the Bernoulli distribution P(ρ i (t) = 1) = ρ, P (ρ i (t) = 0) = 1-ρ, ρ is the probability value in the Bernoulli distribution, and ρ∈[0,1]. In addition, g(z i (t)) is the attack signal from the adversary. Consider the deception attack g(z i (t)) has limited energy, assuming that the deception attack satisfies:

[0097] ||g(z i (t))||≤||Gz i (t)||;

[0098] Among them, G is a known distribution matrix.

[0099] Specifically, based on the error-coupled neural network system model (which can be simply referred to as the error-coupled neural network) and the designed saturation pulse controller (Equation (4)), the first sufficient condition and the second sufficient condition are obtained. The first sufficient condition is the sufficient condition for the mean-square synchronization of the coupled neural network under the action of the synchronization-promoting pulse, while the second sufficient condition is the sufficient condition for the mean-square synchronization of the coupled neural network under the action of the desynchronization pulse.

[0100] Furthermore, the method for obtaining the first sufficient condition is as follows:

[0101] The error vector can be written as

[0102] denotes the Nn-dimensional Euclidean space. Let denote the set of continuous functions from [-τ, 0] to , and use χ(t) to represent the initial value of z(t) and Under the deception attack, the controlled error-coupled neural network can be expressed in the following compact form:

[0103]

[0104] where f(z(t)) is a nonlinear function, f(z(t)) = (f(z 1 (t)), f(z 2 (t)),..., f(z N (t))) T ; the pulse control signal Here, z(t) is right-continuous, that is and at the pulse time t = t k , when, exists.

[0105] Next, the conditions for the mean-square synchronization of the coupled neural network model (Equation (1)) under the action of the hybrid event-triggered saturation pulse controller subject to the deception attack will be derived based on the controlled error network (Equation (7)). Construct a Lyapunov function, and its mathematical expression is:

[0106]

[0107] where, is a positive definite matrix. Assume that the initial value z(t) = z(t, t 0 , χ) is the solution of the error-coupled neural network. Now it will be proved that for any there exists

[0108]

[0109] Define Denote the constant. Prove that for any and \(t\geq t\) 0 -\(\tau\), it holds. First, for the number it is easy to prove. Next, assume that for \(t\in[t\) 0 , \(t\) 1 ), there exists If the assumption does not hold, then there exists such that and for it holds. Also, from the continuity of and the definition of \(t\) ＊ it can be deduced that there exists such that and for it holds. Therefore, for \(\theta\in[t\) 0 -\(\tau\), \(t\) ＊ and \(t\in[t\) ＊ , \(t\) ＊ , the inequality can be derived:

[0110]

[0111] For \(t\in[t\) ＊ , \(t\) ＊ , find the Dini derivative of along the trajectory of the error-coupled neural network (the Dini derivative is denoted as \(D\) + ), and we can get

[0112]

[0113] Furthermore, from the known inequality in the matrix field: deduce

[0114]

[0115] Now substitute Equation (9) and Equation (10) into Equation (8) to obtain:

[0116]

[0117] Then, according to and the matrix inequality

[0118]

[0119] where, represents the symmetric ellipsis symbol when the matrix is symmetric. Then, according to LQ 2 \(L\leq\lambda\) 2 \(P\), we get

[0120]

[0121] where ζ(t) = (z T (t) z T (t - τ(t))) T , and

[0122]

[0123] Furthermore, based on the maximum pulse interval θ * , it can be obtained that

[0124]

[0125] which contradicts the definition of t ＊ . Therefore, for t ∈ [t 0 , t 1 ), the inequality holds. Subsequently, assume that there exists satisfying for the case of t ∈ [t 0 - τ, t k ).

[0126] Furthermore, prove that the following inequality holds, and its expression is:

[0127]

[0128] Based on the inequality

[0129]

[0130] it can be proved that then holds. For the instantaneous trigger t = t k , from the controlled error-coupled neural network (7), it can be obtained that:

[0131]

[0132] Furthermore, from the matrix inequality

[0133]

[0134] and equation (14), it can be deduced that

[0135]

[0136] where is the convex hull matrix. In this case, it can be deduced that

[0137]

[0138] If the assumption (13) is false, then there must exist such that and In addition, there exists such that and Based on the maximum pulse interval θ * , it can be obtained that

[0139]

[0140] This is contradictory to the definition of t * . Therefore, the assumption (13) is proven to be correct. Finally, it can be deduced that

[0141]

[0142] This further shows that

[0143]

[0144] The above discussion shows that for the case of η ∈ (0, 1), the mean-square synchronization between the coupled neural network (1) and the target neural network (2) is finally obtained.

[0145] Thus, it can be concluded that if there exist two diagonal matrices Q 1 , a positive definite matrix and two matrices H, six positive constants ρ, γ, θ * , λ 2 , and η ∈ (0, 1) such that the following matrix inequalities hold,

[0146]

[0147] LQ 2 L ≤ λ 2 P, (18)

[0148]

[0149] where L is the Lipschitz matrix, denotes the interval symbol. Then in the mean-square sense, the trajectory of the error-coupled neural network (7) exponentially converges to zero at the convergence rate of . That is, under the hybrid event-triggered saturation pulse controller and the fraud attack framework, the coupled neural network (Equation (1)) can utilize the ellipsoid contained in the attraction domain Achieve the mean-square exponential synchronization with the isolated neural network (Equation (2)). And Equations (16) to (19) are the obtained first sufficient conditions.

[0150] Specifically, for the second sufficient condition, the obtaining method is as follows:

[0151] Select the above similar Lyapunov function and prove the following inequality:

[0152]

[0153] hold for the case of.

[0154] Similar to the definition of in the method of obtaining the first sufficient condition, the verification of Equation (20) can be transformed into the proof for t≥t 0 -τ and for any hold.

[0155] Similar to the proof process of obtaining the first sufficient condition, from the matrix

[0156]

[0157]

[0158] we can get

[0159]

[0160] where ζ(t)=(z T (t)z T (t - τ(t))) T , According to Equation (21), we can get

[0161]

[0162] And this is contrary to the definition of t ＊ and t ＊ . Therefore, it can be deduced that for t∈[t 0 -τ, t 1 )), there is hold. Next, assume that there exists such that

[0163]

[0164] First, assume If the assumption does not hold, then hold. In this case, according to we can get It holds for \(t\in[t k-1 ,t k ). Then it can be deduced that for \(t\in[t k-1 ,t k ). According to the parameters and the minimum pulse interval \(\theta * , it can be obtained that

[0165]

[0166] This is contrary to , so the assumption holds.

[0167] Furthermore, prove the following inequality:

[0168]

[0169] It can be understood that for the instantaneous \(t k at which triggering occurs, then it can be similarly deduced that

[0170] Similar to the derivation process of obtaining the first sufficient condition, it can be proved that under the condition that the assumption formula (23) holds, holds. Thus, it can be deduced that t≥t 0 −τ, and further obtain

[0171]

[0172] The above discussion shows that for the case of \(\eta\in(1,+\infty)\), the mean-square synchronization between the coupled neural network model (formula (1)) and the isolated neural network model (formula (2)) is finally obtained.

[0173] According to the above analysis, the conclusion can be drawn that if there exist two diagonal matrices \(Q 1 , a positive definite matrix and two matrices \(H\), six positive constants \(\rho,\gamma,\theta * ,\lambda 2 , and \(\eta\in(0,1)\) such that the matrix inequalities (16), (18), (19) and the following hold

[0174]

[0175] where Then in the mean-square sense, the trajectory of the error-coupled neural network (7) is with The convergence speed exponentially converges to zero. That is, in the framework of the hybrid event-triggered saturation pulse controller and the deception attack, the coupled neural network model (Equation (1)) can utilize the ellipsoid contained in the attraction domain to achieve the mean-square exponential synchronization with the isolated neural network (Equation (2)). And Equations (16), (18), (19) and (25) are the obtained second sufficient conditions.

[0176] According to the first and second sufficient conditions obtained above, further consider the estimation of the attraction domain when η ∈ (0, 1). Since the boundary of the attraction domain is convex and closed, it is difficult to represent the attraction domain analytically. Therefore, an embodiment of the present invention introduces a polyhedron to approximately measure the ellipsoid In this case, by finding the maximum μ that satisfies , the conservative estimate of the attraction domain can be maximized, where Υ is the symbolic representation of the required quantity during calculation.

[0177] The problem of estimating the maximum μ can be carried out through the following optimization problem

[0178]

[0179] Since there are multiple bilinear matrix inequalities in the optimization problem (26), the optimization problem (26) is transformed into an optimization problem based on linear matrices through certain operations. Among them, note that constraint (a) is equivalent to

[0180]

[0181] And constraint condition (b) is equivalent to

[0182]

[0183] Then, by multiplying before and after with constraint (c), we get

[0184]

[0185] where, is a positive constant. Similarly, by multiplying left and right with the matrix diag{P -1 ,P -1 ,P -1}, constraint (e) becomes

[0186]

[0187] where, In addition, from (d), it can be seen that LP -1 Q 2 P-1 L ≤ λ 2 P -1 。

[0188] Given that is a variable, which is not beneficial to subsequent analysis. Without loss of generality, select Let where it can be understood that is the variable to be solved, η, ξ i and λ 2 represent different positive constants, is the inverse matrix of a positive definite matrix, and both represent matrices. Based on this, the optimization problem (26) can be reformulated as

[0189]

[0190] where Therefore, equation (27) is the first optimization mathematical model.

[0191] Secondly, for the maximum pulse interval with η ∈ (0, 1), the first optimization mathematical model can be used to solve it, and the expression is as follows:

[0192]

[0193] Finally, for the acquisition of the minimum pulse interval with η ∈ (1, +∞), its mathematical expression is:

[0194]

[0195] Multiply on the left and right with The constraint (C) can be written as

[0196]

[0197] where Therefore, the optimization problem (2) can be reformulated as

[0198]

[0199] where Therefore, equation (30) is the second optimization mathematical model, and solving the second optimization mathematical model can obtain the minimum pulse interval.

[0200] By solving the optimization problem (27) based on linear matrix inequalities, the impulse control gain \(K\) and the maximum estimate of the domain of attraction can be derived. Further, according to the corresponding optimization problems (28) and (30), the maximum allowable impulse interval for \(\eta\in(0,1)\) and the minimum allowable impulse interval for \(\eta\in(1,+\infty)\) are given.

[0201] To more clearly and intuitively demonstrate the embodiments of the present invention, the following will be described in detail and verified by simulation through specific examples.

[0202] The system matrix parameters of the coupled neural network model are given as:

[0203]

[0204] And \(\varepsilon = 0.2\), \(\Gamma = I\) 2 , \(\tau(t)=e\) t / (1 + e t ), and \(q = 1,2\). In addition, the deception attack signal is And it is assumed that the deception attack probability \(\rho = 0.1\). The topological structure of the coupled neural network is as Figure 2 shown. For the simulation situation in the case of the state trajectory without control input, reference can be made to Figure 3 . Further analysis will be carried out for different situations below.

[0205] Case 1: Select \(\eta = 0.5\in(0,1)\); set to represent the zero matrix; select the polyhedron \(\Upsilon=\text{co}\{\xi\) 1 , \(\xi\) 2 \}, \(\xi\) 1 =(0.1, 0.2) T , \(\xi\) 2 =(-0.1, 0.2) T . Based on the proposed optimization problems (27) and (28), by selecting \(\gamma = 1.1\), \(\lambda\) 2 = 0.5, \(\theta\) * = 0.05, the maximum \(\mu = 10.0275\) can be derived. Under this condition, the other feasible solutions are as follows

[0206]

[0207] The maximum allowable impulse interval is 0.05. Select the weight matrix \(\Psi = I\) 2 , and the threshold parameter \(\kappa = 0.5\). Under the framework of the deception attack mixed event-triggered saturation impulse controller, the coupled neural network can synchronize to the target neural network in the mean square sense, as Figure 4 shown. In addition, the timing of the event trigger is as Figure 5As shown, this indicates that the collaboration between the general event-triggered scheme and the forced event-triggered scheme compensates for the impact of spoofing attacks, where Figure 5 the ordinate "impulsive effect" of Figure 5 represents the impulsive effect.

[0208] Case 2: Select η = 1.8 ∈ (1, ∞), γ = 1.1, λ 2 = 0.5, θ * = 0.24. Considering the impulsive effect of synchronization, let I 2 represent the 2D identity matrix. Based on the optimization problem (30), the minimum allowable impulsive interval is obtained as 0.231, and several matrices are derived as:

[0209]

[0210] Let θ * = 0.24, θ * = 3, and the constant κ = 0.9. Then, from Figure 6 it can be seen that mean-square synchronization is achieved. For the explanations of Figure 3 , Figure 4 and Figure 6 : Figure 3 , Figure 4 and Figure 6 In the upper subgraphs, the abscissa represents time and the ordinate represents the state of the coupled neural networks; Figure 3 , Figure 4 and Figure 6 In the lower subgraphs, the abscissa represents time and the ordinate represents the consensus error, where represents the error of the j-th state variable of the i-th neural network; Figure 4 In, the abscissa represents time and the ordinate represents the impulsive strength. Specifically, Figure 3 is the case of the state trajectory without control input, where the upper subgraph shows the trajectories of the coupled neural networks and the target neural network, and the lower subgraph shows the trajectory of the error-coupled neural network; Figure 4 is the state trajectory with control input and spoofing attack, where the upper subgraph shows the trajectories of the coupled neural networks and the target neural network, and the lower subgraph shows the trajectory of the error-coupled neural network; Figure 6 is the state trajectories of the coupled neural network and the error-coupled neural network with impulsive perturbation, where the upper subgraph shows the trajectories of the coupled neural networks and the target neural network, and the lower subgraph shows the trajectory of the error-coupled neural network. It should be noted that Figure 3 , Figure 4 and Figure 6 Each upper subgraph of represents the first one counted from top to bottom in that figure; Figure 3 , Figure 4 and Figure 6Each of the following sub - figures represents the second one from top to bottom in this figure.

[0211] The embodiment of the present invention proposes an innovative hybrid event - triggered mechanism, which is specially designed with a relaxation interval to effectively repair the problem of degraded synchronization performance that may be caused in the case of spoofing attacks. When facing the challenge of spoofing attacks, the embodiment of the present invention adopts a comprehensive method, jointly using advanced mathematical tools such as mathematical induction of saturated non - linearity, proof by contradiction, and polyhedron representation, and actively delves into the actuator saturation phenomenon in discrete control signals. In addition, the embodiment of the present invention takes into account both synchronous and desynchronous pulse effects in the design, and through precise mathematical analysis, successfully derives the sufficient conditions for achieving mean - square synchronization. More importantly, the embodiment of the present invention reveals, through numerical analysis, the inherent elastic constraint relationship between the pulse control gain and the probability and intensity of spoofing attacks, providing a new theoretical basis and practical guidance for improving the security and reliability of control systems.

[0212] Embodiment Two

[0213] Based on the same inventive concept, this embodiment provides an event - triggered pulse control system for a neural network under spoofing attacks. The principle of solving problems is similar to that of the event - triggered pulse control method for a neural network under spoofing attacks provided in Embodiment One, and the repeated parts will not be elaborated.

[0214] This embodiment provides an event - triggered pulse control system for a neural network under spoofing attacks, including:

[0215] A model - building module, used to build a coupled neural network model with time - varying and time - delay, determine the synchronization target of the coupled neural network model; obtain the state information of the coupled neural network model, and establish an error - coupled neural network system model according to the state information;

[0216] A controller - design module, used to design a saturated pulse controller according to the error - coupled neural network system model and the synchronization target. The mathematical expression of the saturated pulse controller is:

[0217]

[0218] where, U i (t) is the saturated pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t k is the k - th pulse sequence;

[0219] A condition acquisition module, configured to obtain a first sufficient condition and a second sufficient condition based on an error-coupled neural network system model and a saturation pulse controller;

[0220] An optimization calculation module, configured to obtain a first optimization mathematical model according to the first sufficient condition and the second sufficient condition; and calculate a pulse control gain matrix and a maximum estimation of an attraction domain by using the first optimization mathematical model.

[0221] Embodiment III

[0222] An embodiment of the present invention provides a controller, including the event-triggered pulse control system of a neural network under various fraud attacks provided in Embodiment II.

[0223] Those skilled in the art should understand that the embodiments of the present application may be provided as a method, a system, or a computer program product. Therefore, the present application may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0224] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, such that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one or more flows and / or blocks Figure 1 one or more blocks.

[0225] These computer program instructions can also be stored in a computer-readable memory capable of guiding a computer or other programmable data processing devices to work in a specific manner, such that the instructions stored in the computer-readable memory generate a manufactured article including instruction means, and the instruction means implements the functions specified in Figure 1 one or more flows and / or blocks Figure 1 one or more blocks.

[0226] These computer program instructions can also be loaded onto a computer or other programmable data processing apparatus, so that a series of operation steps are executed on the computer or other programmable apparatus to produce a computer-implemented process, thereby providing instructions for implementing the steps of the process Figure 1 one process or a plurality of processes and / or boxes Figure 1 steps for the functions specified in one box or a plurality of boxes.

[0227] Obviously, the above embodiments are only examples for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all implementation manners here. And the obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A method for controlling event-triggered pulses of a neural network under fraudulent attacks, characterized in that: include: Establishing a coupled neural network model with time variation and time delay, and determining a synchronization target of the coupled neural network model; Acquiring state information of the coupled neural network model, and establishing an error coupled neural network system model according to the state information; According to the error coupling neural network system model and the synchronization target, a saturation pulse controller is designed. The mathematical expression of the saturation pulse controller is: Among them, U i (t) is a saturation pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t k is the kth pulse sequence, t represents time; Based on the error coupled neural network system model and the saturation pulse controller, obtaining a first sufficient condition and a second sufficient condition; According to the first sufficient condition and the second sufficient condition, a first optimization mathematical model is obtained; and the maximum estimate of the pulse control gain matrix and the attraction domain is calculated using the first optimization mathematical model.

2. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The mathematical expression of the coupled neural network model is: Among them, x i (t) represents the state variable information of the i-th neural network, and All represent system matrices, N represents the number of agents, ε represents the coupling strength; Γ represents the inline coupling matrix; represents a nonlinear vector-valued function, is the external input, τ(t) is the system delay, w ij is the connection coefficient between node i and node j, U i (t) is a saturation pulse controller, i and j represent different numbers, and t represents time.

3. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The mathematical expression of the error coupling neural network system model is: Among them, z i (t) is the error vector; For different system matrices, f(z i (t)) is the nonlinear vector-valued error function, f(z i (t-τ(t))) is the error function of the nonlinear vector value at time t-τ(t), τ(t) is the system delay, ε is the coupling strength, and w ij is the connection coefficient between node i and node j, Γ is the internal coupling matrix, N is the number of agents; i and j both represent different numbers; t represents time.

4. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The first sufficient condition is: LQ2L≤λ2P, Among them, η, λ2 and Represents different positive constants, I N is the N-dimensional identity matrix, is the Kronecker product, H is the matrix to be determined, N is the number of agents, n is the total number of state variables, and P is a positive definite matrix; are different system matrices; Q1 and Q2 are different diagonal matrices, ρ is the probability value in Bernoulli distribution, G is the known distribution matrix, T is the matrix transpose symbol, L is the Lipschitz matrix, is the convex hull matrix, * indicates that the symbol is omitted symmetrically when the matrix is ​​symmetrical; the calculation formulas of Ξ1, Ξ2 and Ξ3 are: Among them, γ, Denotes different positive constants, θ * Indicates the maximum pulse interval, is a known matrix, is the system matrix, W is the node connection matrix, ε represents the coupling strength, and Γ represents the inline coupling matrix.

5. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The second sufficient condition is: LQ2L≤λ2P, Among them, η, λ2 and Represents different positive constants, I N is the N-dimensional identity matrix, is the Kronecker product, H is the matrix to be determined, N is the number of agents, n is the total number of state variables, and P is a positive definite matrix; are different system matrices; Q1 and Q2 are different diagonal matrices; ρ is the probability value in Bernoulli distribution, G is the known distribution matrix, T is the matrix transpose symbol, L is the Lipschitz matrix, is the convex hull matrix, ★ Indicates that the symbol is omitted symmetrically when the matrix is ​​symmetrical; the calculation formulas for Ξ4, Ξ2, and Ξ3 are: Among them, γ, Denotes different positive constants, θ * Indicates the maximum pulse interval, is a known matrix, is the system matrix, W is the node connection matrix, ε represents the coupling strength, and Γ represents the inline coupling matrix.

6. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The first optimization mathematical model is: in, is the variable to be determined, η, ξ1 and λ2 represent different positive constants, is the inverse matrix of a positive definite matrix, is a positive constant, I N is the N-dimensional identity matrix, N is the number of agents, n is the total number of state variables, For different system matrices, is the Kronecker product, and All represent matrices, L is the Lipschitz matrix, G is the known distribution matrix, T is the matrix transpose symbol, ρ is the probability value in the Bernoulli distribution, l is a positive constant, and * indicates symmetric omission of symbols when the matrix is ​​symmetric; and The calculation formula is: Among them, γ, Denotes different positive constants, θ * Indicates the maximum pulse interval, is a known matrix, is the system matrix, W is the node connection matrix, ε represents the coupling strength, and Γ represents the inline coupling matrix.

7. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: The calculation to obtain the maximum estimate of the pulse control gain matrix and the attraction domain includes solving the maximum pulse interval using the first optimization mathematical model.

8. The event-triggered pulse control method of a neural network under fraud attack according to claim 1, characterized in that: After the pulse control gain matrix and the maximum estimation of the attraction domain are obtained by calculation, the method further includes solving the minimum pulse interval by using a second optimization mathematical model, wherein the second optimization mathematical model is: in, is the variable to be determined, η, ξ1 and λ2 represent different positive constants, is the inverse matrix of a positive definite matrix, is a positive constant, I N is the N-dimensional identity matrix, N is the number of agents, n is the total number of state variables, For different system matrices, is the Kronecker product, and All represent matrices, L is the Lipschitz matrix, G is the known distribution matrix, T is the matrix transpose symbol, ρ is the probability value in the Bernoulli distribution, l is a normal number, ★ Indicates that the symbol is omitted symmetrically when the matrix is ​​symmetric; and The calculation formula is: Among them, γ, Denotes different positive constants, θ * Indicates the maximum pulse interval, is a known matrix, is the system matrix, W is the node connection matrix, ε represents the coupling strength, and Γ represents the inline coupling matrix.

9. An event-triggered pulse control system for a neural network under fraud attack, characterized in that: include: A model building module, used to build a coupled neural network model with time variation and time lag, and determine the synchronization target of the coupled neural network model; Acquiring state information of the coupled neural network model, and establishing an error coupled neural network system model according to the state information; The controller design module is used to design a saturation pulse controller according to the error coupling neural network system model and the synchronization target. The mathematical expression of the saturation pulse controller is: Among them, U i (t) is a saturation pulse controller, is a known matrix, z i (t) is the error vector, u i (t) is the control signal, δ(·) is the Dirac pulse function, t k is the kth pulse sequence; A condition acquisition module, used for obtaining a first sufficient condition and a second sufficient condition based on the error coupled neural network system model and the saturation pulse controller; The optimization calculation module is used to obtain a first optimization mathematical model according to the first sufficient condition and the second sufficient condition; and use the first optimization mathematical model to calculate the maximum estimate of the pulse control gain matrix and the attraction domain.

10. A controller, characterized in that: Including the event-triggered pulse control system of a neural network under a fraud attack as described in claim 9.

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