Event trigger pulse control method and system of neural network under fraud attack

Abstract

The invention relates to an event trigger pulse control method and system of a neural network under a fraud attack, and belongs to the technical field of neural network control. Comprising the following steps: establishing a coupling neural network model containing time variation and time delay, and determining a synchronization target of the coupling 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; designing a saturation pulse controller according to the error coupling neural network system model and the synchronization target; obtaining a first sufficient condition and a second sufficient condition based on the error coupling neural network system model and the saturation pulse controller; obtaining a first optimization mathematical model according to the first sufficient condition and the second sufficient condition; and calculating by using the first optimization mathematical model to obtain the maximum estimation of the pulse control gain matrix and the attraction domain. According to the method, the joint influence of actuator saturation, spoofing attack and pulse effect on the overall performance of the system is effectively solved.

Description

Technical Field

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

[0002] With the rapid development of modern science and technology, neural networks, as a powerful computing model, have demonstrated their unique value and wide application potential in many fields. The core advantages of neural networks are their adaptability, non-convexity, nonlinearity and excellent fault tolerance, which together form the basis for their wide application in key fields such as frequency domain image analysis, combinatorial optimization, and artificial intelligence. In particular, the synchronization characteristics of neural networks have attracted widespread attention in many disciplines such as biomedical engineering, secure communications, and fluid dynamics. In these fields, the main challenges faced in achieving synchronization of coupled neural networks include the dynamic behavior of the nodes themselves and the complexity of the interconnections between nodes caused by network coupling. At present, in order to achieve synchronization of coupled neural networks, a variety of control methods have been proposed, such as pinning control, sampling control, and pulse control. 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 stimulated related research. However, the pulse disturbance caused by pulse control may have an adverse effect on the final synchronization, especially in actual engineering scenarios, such as bipedal robots and smoothly moving vehicle models affected by intermittent collisions. To solve this problem, an event-triggered control (ETC) mechanism can be used. The event-triggered control mechanism updates the control signal only when a specific trigger condition is met, while the time-triggered control mechanism updates the signal at a fixed time interval, regardless of the contribution of the control signal to the dynamics. Therefore, event-triggered pulse control has attracted widespread 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 the actuator transmitting the control signal. Saturation nonlinearity is very common in physical systems due to physical, technical and safety limitations. The actuator saturation phenomenon will seriously weaken the system performance, which is manifested in the aggravation of system oscillation, significant increase in overshoot, response lag and even instability of the overall system. In addition, due to the open nature of the communication network, the controlled system is also facing the threat of network attacks. In particular, common network attacks such as deception attacks maliciously tamper with the original control instructions transmitted through the network channel, and such attacks are often difficult to detect effectively. In recent years, although research work in related fields has focused on actuator saturation or deception attacks to a certain extent, it has not been able to effectively solve the joint impact of actuator saturation, deception attacks and pulse effects on the overall performance of the system. Summary of the invention

[0004] To this end, the technical problem to be solved by the present invention is to overcome the problem in the prior art that the influence of actuator saturation, deception attack and pulse effect on the overall performance of the system is not comprehensively considered.

[0005] In a first aspect, in order to solve the above technical problems, the present invention provides an event-triggered pulse control method for a neural network under a fraud attack, comprising:

[0006] Establishing a coupled neural network model with time variation and time delay, and determining a synchronization target of the coupled neural network model;

[0007] Acquiring state information of the coupled neural network model, and establishing an error coupled neural network system model according to the state information;

[0008] 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:

[0009]

[0010] 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;

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

[0012] 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.

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

[0014]

[0015] 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, wij 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.

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

[0017]

[0018] 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 represent different numbers, and t represents time.

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

[0020]

[0021] LQ 2 L≤λ 2 P,

[0022]

[0023] Among them, η, λ 2 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; Q 1 and Q 2 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 when the matrix is ​​symmetric; 1 ,Ξ 2 and 3 The calculation formula is:

[0024]

[0025] Among them, γ, represents 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.

[0026] In one 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 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; Q 1 and Q 2 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 the symmetric omission symbol when the matrix is ​​symmetric; 4 ,Ξ 2 and 3 The calculation formula is:

[0032]

[0033] 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.

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

[0035]

[0036] in, are variables to be determined, η, ξt and λ 2 represents different positive constants, P is the inverse matrix of the 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:

[0037]

[0038] 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.

[0039] In one embodiment of the present invention, the step of calculating 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.

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

[0041]

[0042]

[0043] in, are variables to be determined, η, ξ i and λ 2 Represents 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:

[0044]

[0045] 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.

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

[0047] A model building module is used to build a coupled neural network model with time variation and time lag, determine the synchronization target of the coupled neural network model; obtain the state information of the coupled neural network model, and build an error coupled neural network system model according to the state information;

[0048] 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:

[0049]

[0050] 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;

[0051] 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;

[0052] 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.

[0053] In a third aspect, in order to solve the above-mentioned technical problem, the present invention provides a controller, including the above-mentioned event-triggered pulse control system of a neural network under a 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 a fraud attack, which adopts a hybrid event triggering mechanism. The core feature of the mechanism is 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 limited time. In addition, the mechanism exhibits excellent self-healing ability when facing a deception attack, and can significantly improve the synchronization performance degradation caused by this. In order to better meet the needs of practical engineering applications, the present invention constructs a comprehensive theoretical framework that comprehensively considers multiple factors such as actuator saturation, deception attacks, and hybrid pulse effects, and effectively solves the combined impact of actuator saturation, deception attacks, and pulse effects on the overall performance of the system.

[0056] (2) The present invention has carried out a detailed classification discussion on the relevant parameters of the pulse effect to distinguish the role of the pulse signal under different functions. Relying on this theoretical framework, the conditions for the coupled neural network to achieve mean square synchronization are deeply studied, and the convergence speed of the system is analyzed. Through this process, the inherent elastic constraints between the pulse control gain and the probability and intensity of the deception attack are revealed, which provides an important theoretical basis for the design of the control system.

[0057] (3) Based on the three proposed optimization problems, the present invention conducts in-depth mathematical derivation and obtains 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 trigger mechanism, but also provide a practical reference for the selection of relevant parameters, ensuring the effectiveness and reliability of the control system in practical applications. Through these innovations, 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 more clearly understood, the present invention is further described in detail below according to specific embodiments of the present invention in conjunction with the accompanying drawings, wherein

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

[0060] Figure 2 A topological diagram of a coupled neural network consisting of 11 nodes in a preferred embodiment of the present invention;

[0061] Figure 3 It is a state trajectory diagram of the coupled neural network and the error coupled neural network when there is no control input in the preferred embodiment of the present invention;

[0062] Figure 4 It is a state trajectory diagram of the coupled neural network and the error coupled neural network when there is a control input and a deception attack in the preferred embodiment of the present invention;

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

[0064] Figure 6 This is a state trajectory diagram of a coupled neural network and an error coupled neural network when there is a pulse disturbance in a preferred embodiment of the present invention. DETAILED DESCRIPTION

[0065] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

[0066] Embodiment 1

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

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

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

[0070] According to the error coupled neural network system model and synchronization target, a saturation pulse controller is designed;

[0071] Based on the error coupling neural network system model and the saturation pulse controller, the first sufficient condition and the second sufficient condition are obtained;

[0072] According to the first sufficient condition and the second sufficient condition, a first optimization mathematical model is obtained; and the maximum estimation of the pulse control gain matrix and the attraction domain is calculated by using the first optimization mathematical model.

[0073] The embodiment of the present invention provides an event-triggered pulse control method for a neural network under a fraud attack. By establishing a time-varying time-delay coupled neural network model and an error-coupled neural network model, a saturated pulse controller is designed to achieve effective regulation of system behavior. At the same time, in order to compensate for the impact of deception attacks, a hybrid event triggering mechanism is proposed, and the occurrence of the Zeno phenomenon is avoided by introducing a minimum event interval. In addition, according to the different functions of the pulse signal, the embodiment of the present invention classifies and discusses the parameters related to the pulse effect, and obtains the first sufficient condition and the second sufficient condition for the time-varying time-delay coupled neural network to achieve mean square exponential synchronization under two different conditions. By establishing an optimization mathematical model based on these sufficient conditions, the embodiment of the present invention conducts an in-depth analysis of the optimization problem to achieve resource saving 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 time-delay coupled neural network. This method can not only effectively solve the joint impact of actuator saturation, deception attacks and pulse effects on the overall performance of the system, but also ensure that the synchronization performance of the neural network is significantly improved under the fraud attack environment.

[0074] Specifically, a coupled neural network model with time variation and time lag (coupled neural network model, referred to as coupled neural network for short) is established. The mathematical expression of the model is:

[0075]

[0076] in, i=1,2,…,N represents the state variable information of the i-th neural network, and T is the matrix transpose symbol; and All represent system matrices, N represents the number of agents, represents n×n dimensional Euclidean space, represents n-dimensional Euclidean space; ε represents coupling strength; Γ represents the inline coupling matrix; The vector-valued function representing nonlinearity satisfies τ(t) is the system delay and satisfies 0≤τ(t)≤τ, where τ represents the maximum value of τ(t); the node connection matrix W = [w ij ] N×N , w ij is the connection coefficient between node i and node j, which 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 elements Indicates external input; U i(t) represents the controller to be designed, that is, the saturation pulse controller to be designed; i and j both 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 may be referred to as a target neural network for short), and its mathematical expression is:

[0078]

[0079] in, Represents the state of an isolated neural network.

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

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

[0082] in, i=1,2,…,N.

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

[0084]

[0085] Among them, z i (t) is the error vector; f(z i (t)) is the nonlinear vector-valued error function, and its expression is f(z i (t-τ(t))) is the error function of the nonlinear vector value at time t-τ(t), and its expression is i=1,2,…,N.

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

[0087]

[0088] 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 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 referred to as error coupled neural network for short) and the designed saturation pulse controller (Formula (4)), the first sufficient condition and the second sufficient condition are obtained, wherein the first sufficient condition is a sufficient condition for the mean square synchronization of the coupled neural network under the action of the facilitating synchronization pulse, and the second sufficient condition is a sufficient condition for the mean square synchronization of the coupled neural network under the action of the desynchronizing pulse.

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

[0101] The error vector can be written as

[0102] represents Nn-dimensional Euclidean space. Let represents the range from [-τ, 0] to A set of continuous functions, using χ(t) to represent the initial value of z(t) and Under spoofing attack, the controlled error coupled neural network can be expressed as 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 ; Pulse control signal Here, z(t) is right continuous, that is And at the pulse time t = t k , hour, exist.

[0105] Next, we will derive the conditions for achieving mean square synchronization of the coupled neural network model (Equation (1)) under the action of a mixed event-triggered saturated pulse controller subject to a fraud attack based on the controlled error network (Equation (7)). The Lyapunov function is constructed, and its mathematical expression is:

[0106]

[0107] in, 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 we will prove that for any exist

[0108]

[0109] definition represents a constant. Prove that for any and t ≥ t 0 -τ, First, for the number It is easy to prove. Next, suppose that for t∈[t 0 ,t 1 ),exist If the assumption does not hold, then there is Make And for Established. And, by Continuity and t ＊ The definition of Make And for holds. Therefore, for θ∈[t 0 -τ,t ＊ ] and t∈[t ＊ ,t ＊ ], we can derive the inequality:

[0110]

[0111] For t∈[t ＊ ,t ＊ ], along the trajectory of the error coupled neural network The Dini derivative of + ), we can get

[0112]

[0113] Furthermore, from the matrix field, we know the inequality: Derived

[0114]

[0115] Now, by substituting equation (9) and equation (10) into equation (8), we can obtain:

[0116]

[0117] Then according to and matrix inequalities

[0118]

[0119] in, Indicates that the symbol is omitted symmetrically when the matrix is ​​symmetrical. Then according to LQ 2 L≤λ 2 P, 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 * , can be obtained

[0124]

[0125] This is in contrast to t ＊ Therefore, for t∈[t 0 ,t 1 ),inequality Then, assuming that there is satisfy For t∈[t 0 -τ,t k ) situation.

[0126] Furthermore, the following inequality is proved to hold, and its expression is:

[0127]

[0128] Based on the inequality

[0129]

[0130] It can be proved but For the triggering instant t = t k , From the controlled error coupled neural network (7), we can get:

[0131]

[0132] Furthermore, by the matrix inequality

[0133]

[0134] From equation (14), we can deduce

[0135]

[0136] in is the convex hull matrix. In this case, we can deduce

[0137]

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

[0139]

[0140] This is consistent with t * The definition of is contradictory. Therefore, hypothesis (13) is proved to be correct. Finally, it can be deduced that

[0141]

[0142] This further shows

[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] So we can conclude that if there are 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) make the following matrix inequality hold,

[0146]

[0147] LQ 2 L≤λ 2 P, (18)

[0148]

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

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

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

[0152]

[0153] Established, for situation.

[0154] Similar to the method of obtaining the first sufficient condition The definition of , the verification of formula (20) can be transformed into t≥t 0 -τ and for any Proof of establishment.

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

[0156]

[0157]

[0158] Available

[0159]

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

[0161]

[0162] This is in contrast to t ＊ ,t ＊ Therefore, it can be deduced that for t∈[t 0 -τ,t 1 ),have Next, suppose there is Make

[0163]

[0164] First, assume If the assumption is not true, then In this case, according to Available It holds for t∈[t k-1 ,t k ). Then we can deduce that For t∈[t k-1 ,t k ). According to the parameters and the minimum pulse interval θ * , can be obtained

[0165]

[0166] This is similar to Contradictory, so assume Established.

[0167] Furthermore, we prove the following inequality:

[0168]

[0169] It can be understood that for the triggering instant t k , we can deduce similarly

[0170] Similar to the derivation process of obtaining the first sufficient condition, it can be proved that under the assumption that equation (23) is true, It is established. So we can deduce t≥t 0 -τ, further get

[0171]

[0172] The above discussion shows that for the case of η∈(1,+∞), the mean square synchronization between the coupled neural network model (Equation (1)) and the isolated neural network model (Equation (2)) is finally obtained.

[0173] According to the above analysis, we can conclude that if there are 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) make equations (16), (18), (19) and the following matrix inequality hold

[0174]

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

[0176] Based on the above-mentioned first and second sufficient conditions, we 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 express the attraction domain analytically. Therefore, the embodiment of the present invention introduces a polyhedron Approximately measuring the ellipsoid In this case, by finding satisfaction The maximum μ of can maximize the conservative estimate of the attraction domain, where Υ is the symbolic representation of the required quantity during calculation.

[0177] The problem of estimating the maximum μ can be solved by 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. Note that constraint (a) is equivalent to

[0180]

[0181] And constraint (b) is equivalent to

[0182]

[0183] So use Multiplying with constraint (c) before and after gives

[0184]

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

[0186]

[0187] in, In addition, from (d), we can see that LP -1 Q 2 P-1 L≤λ 2 P -1 .

[0188] Given that is a variable that has no benefit in subsequent analysis. Without loss of generality, choose make It is understandable that are variables to be determined, η, ξ i and λ 2 Represents 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] in, Therefore, formula (27) is the first optimization mathematical model.

[0191] Secondly, for the maximum pulse interval of 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 of η∈(1,+∞), the mathematical expression is:

[0194]

[0195] and Multiplying the left and right sides, the constraint (C) can be written as

[0196]

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

[0198]

[0199] in, Therefore, formula (30) is the second optimization mathematical model, and the minimum pulse interval can be obtained by solving the second optimization mathematical model.

[0200] By solving the optimization problem (27) based on linear matrix inequality, the maximum estimate of the impulse control gain K and the attraction domain can be derived. Furthermore, according to the corresponding optimization problems (28) and (30), the maximum allowable impulse interval of η∈(0,1) and the minimum allowable impulse interval of η∈(1,+∞) are given.

[0201] In order to more clearly and intuitively illustrate the embodiments of the present invention, specific examples will be used for detailed description and simulation verification below.

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

[0203]

[0204] And ε=0.2Γ=I 2 , τ(t)=e t / (1+e t ),and q=1,2. In addition, the deception attack signal is And assume that the probability of deception attack ρ = 0.1. The topological structure of the coupled neural network is as follows Figure 2 For the simulation situation without control input state trajectory, please refer to Figure 3 The following is a further analysis of different situations.

[0205] Case 1: Choose η=0.5∈(0,1); set represents the zero matrix; select the polyhedron Υ=co{ξ 1 ,ξ 2},ξ 1 =(0.1,0.2) T ,ξ 2 =(-0.1,0.2) T Based on the proposed optimization problems (27) and (28), by choosing γ = 1.1, λ 2 =0.5,θ * =0.05, we can deduce that the maximum μ = 10.0275. Under this condition, other feasible solutions are as follows

[0206]

[0207] The maximum permissible pulse interval is 0.05. Select the weight matrix Ψ=I 2 , the threshold parameter κ = 0.5. Under the framework of the hybrid event-triggered saturation pulse controller under spoofing attack, the coupled neural network can be synchronized to the target neural network in the mean square sense, such as Figure 4 In addition, the timing of event triggering is as follows Figure 5This indicates that the collaboration between the general event triggering scheme and the mandatory event triggering scheme compensates for the impact of the deception attack, where Figure 5 The vertical axis impulsive effect represents the impulse effect.

[0208] Case 2: Choose η=1.8∈(1,∞), γ=1.1, λ 2 =0.5,θ * =0.24. Considering the synchronous pulse effect, I 2 represents the 2D identity matrix. Based on the optimization problem (30), the minimum allowed pulse interval is 0.231, and several matrices are derived as follows:

[0209]

[0210] Let θ * =0.24,θ * =3, constant κ = 0.9, then Figure 6 It can be seen that mean square synchronization is achieved. Figure 3 , Figure 4 and Figure 6 Description: Figure 3 , Figure 4 and Figure 6 The horizontal axes in the above sub-figures all represent time, and the vertical axes all represent the state of the coupled neural network; Figure 3 , Figure 4 and Figure 6 The horizontal axes in the following sub-figures all represent time, and the vertical axes all represent consistency errors. represents the error of the jth state variable of the ith neural network; Figure 4 The horizontal axis represents time, and the vertical axis represents pulse intensity. Specifically, Figure 3 This is the state trajectory without control input. The upper sub-figure shows the trajectory of the coupled neural network and the target neural network, and the lower sub-figure shows the trajectory of the error coupled neural network. Figure 4 State trajectories with control input and deception attack, where the upper sub-figure shows the trajectories of the coupled neural network and the target neural network, and the lower sub-figure shows the trajectory of the error coupled neural network; Figure 6 The state trajectories of the coupled neural network and the error coupled neural network when there is a pulse disturbance. The upper sub-figure shows the trajectories of the coupled neural network and the target neural network, and the lower sub-figure 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 from the top to the bottom in the graph; Figure 3 , Figure 4 and Figure 6Each lower subgraph of represents the second one from the top in the figure.

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

[0212] Embodiment 2

[0213] Based on the same inventive concept, this embodiment provides an event-triggered pulse control system for a neural network under a fraud attack. The principle of solving the problem is similar to the event-triggered pulse control method for a neural network under a fraud attack provided in Example 1, and the repeated parts will not be repeated.

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

[0215] The model building module is used to build a coupled neural network model with time variation and time lag, determine the synchronization target of the coupled neural network model, obtain the state information of the coupled neural network model, and build an error coupled neural network system model based on the state information;

[0216] 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:

[0217]

[0218] 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;

[0219] A condition acquisition module, used for obtaining a first sufficient condition and a second sufficient condition based on an error coupling neural network system model and a saturation pulse controller;

[0220] 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 to calculate the maximum estimate of the pulse control gain matrix and the attraction domain using the first optimization mathematical model.

[0221] Embodiment 3

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

[0223] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

[0224] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0225] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

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

[0227] Obviously, the above embodiments are merely examples for clear explanation and are not intended to limit the implementation methods. For those skilled 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 the implementation methods here. The obvious changes or modifications derived from these are still within the protection scope of the 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.

Images