Chua's circuit memory event trigger control method for resisting replay attack

Abstract

The invention discloses a Chua's circuit memory event trigger control method for resisting replay attacks, and belongs to the technical field of Chua's circuits and network control. The method comprises the following steps: converting a Chua's circuit model into a Lur'e system mathematical model through a variable replacement technology; an elastic memory event triggering mechanism is provided, and an anti-attack memory controller is designed based on the elastic memory event triggering mechanism; constructing a closed-loop Lur'e system in combination with an elastic memory event triggering mechanism and an anti-attack memory controller, and establishing a criterion for ensuring the stability of the mean square H infinity of the closed-loop Lur'e system; based on the proposed criterion, a collaborative design scheme of the controller gain and the event trigger matrix is given. According to the invention, stable operation of the Chua's circuit under acceptable replay attacks can be effectively ensured, unnecessary data transmission can be reduced, and network communication resources are saved.

Description

Technical Field

[0001] The present invention belongs to the technical fields of Chua's circuit and networked control technology, and particularly relates to a memory event-triggered control method for Chua's circuit to resist replay attacks. Background Art

[0002] Since the model of Chua's circuit was proposed by Leon O. Chua in 1983, Chua's circuit has received extensive attention from scholars at home and abroad. This circuit consists of energy storage elements, Chua's diodes and resistors, and can exhibit complex nonlinear phenomena such as single and double attractors. Maintaining the stability of Chua's circuit is the premise for all its practical applications and is of great significance. Therefore, a large number of control schemes have been proposed in existing research, including feedback control, non-fragile control, sampling control, event-triggered control, etc.

[0003] Among the above control schemes, event-triggered control, as an improvement of sampling control, has gradually received extensive attention because it can significantly reduce unnecessary data transmission while ensuring good control performance. In this control strategy, only when the system state change exceeds a preset threshold, the event generator will send a signal to the controller, thus effectively reducing the update frequency of the controller and saving computing resources and network resources. Currently, a variety of event-triggered mechanisms have been proposed, including continuous event-triggered, switching event-triggered and dynamic event-triggered, etc. It is worth pointing out that up to now, most event-triggered mechanisms are designed based on the difference between the current moment and the last sent data packet. However, if the historical signals released by the event generator can be effectively utilized, it is expected to obtain higher control performance.

[0004] As a networked control scheme, the event generator is vulnerable to network attacks during the process of transmitting data packets. Among them, replay attack is a common and threatening type of attack. The attacker records the information sequence of the sensor and maliciously replaces the real-time data in the transmission channel with the previously intercepted legitimate data, thus disrupting the normal operation of the system. Therefore, how to effectively resist replay attacks has become an urgent problem to be solved, and it is urgently necessary to carry out in-depth research and propose innovative countermeasures.

[0005] After retrieval, there are multiple patents on the control analysis method of the Chua's circuit publicly available. For example, in the patent application titled: A New Criterion for the Chua's Circuit System Based on Fault-Tolerant Synchronous Control (Publication Date: September 6, 2024; Application Number: 202410740807.2), a Chua's circuit system model containing unknown system states and fault signals was constructed on the premise of considering external disturbances. Subsequently, a projection network was used to reduce the dimension of the system state space, and all nodes were divided into two categories: measurable and unmeasurable. An observer was used to observe unknown components, and corresponding controllers were designed to compensate for actuator faults, ultimately achieving the synchronization of the error system.

[0006] Another example is the patent application titled: Adaptive Pinning Control Method and System for Coupled Chua's Circuits with Fixed-Time Synchronization (Publication Date: August 22, 2023; Application Number: 202310492789.6). Based on the constructed Lyapunov function and the principle of fixed-time synchronization, an adaptive pinning control method was proposed to ensure that the established coupled Chua's circuit model achieves synchronization within a fixed time.

[0007] However, the control methods in the above patent applications are all implemented in an absolutely secure communication environment. Nevertheless, in practical engineering applications, it is crucial to consider how to effectively resist unknown cyberattacks. In addition, considering the limited network bandwidth resources, how to reduce unnecessary data transmission to effectively save communication resources is also an issue worthy of attention. Summary of the Invention

[0008] The present invention provides a memory event-triggered control method for Chua's circuit to resist replay attacks, which can effectively ensure the stable operation of Chua's circuit under acceptable replay attacks, and at the same time can reduce unnecessary data transmission and save network communication resources.

[0009] To achieve the above object, the technical solution provided by the present invention is as follows:

[0010] The present invention provides a memory event-triggered control method for Chua's circuit to resist replay attacks, including:

[0011] S1. Through variable substitution technology, transform the Chua's circuit model into a Lur’e system mathematical model;

[0012] S2. Propose an elastic memory event-triggered mechanism and design an anti-attack memory controller based on this;

[0013] S3. Combine the elastic memory event-triggered mechanism and the anti-attack memory controller to construct a closed-loop Lur’e system and establish a criterion to ensure the mean-square H∞ stability of the closed-loop Lur’e system;

[0014] S4. Based on the proposed criterion, a co - design scheme for the controller gain and the event - triggering matrix is given.

[0015] In the prior art, the impacts of spoofing attacks and DOS attacks are usually mainly considered, while the impact of replay attacks is ignored. The present invention provides a chaotic Chua's circuit memory event - triggered control method capable of resisting replay attacks. In particular, an elastic memory event - triggering mechanism is proposed. Compared with the traditional memory - triggering mechanism, the triggering mechanism of the present invention contains terms related to replay attacks. Therefore, it can effectively ensure the stable operation of the chaotic Chua's circuit under acceptable replay attacks. At the same time, by making full use of the attack information, unnecessary data transmission can be further reduced, and the bandwidth utilization rate can be optimized.

[0016] Furthermore, the mathematical model of the Lur’e system is as follows:

[0017]

[0018] where, h(Dψ(t)), u(t), w(t) and φ(t) represent the system state, the nonlinear vector function, the control input, the external disturbance and the measured output respectively; and represent the voltages across the capacitors C0 and C1 respectively; I L represents the current flowing through the inductor L; A, B1, B2, C and D are known system matrices.

[0019] Furthermore, the elastic memory event - triggering mechanism is that the event generator will transmit the system state only when the following triggering condition is satisfied:

[0020] t n+1 h = t n h+min{lh|ξ(ψ(t n h,e s (t),δ1δ2))>0}

[0021] where, h represents the sampling interval, t n h is the instantaneous sequence transmitted at time n, t n+1 h is the instantaneous sequence transmitted at time n + 1, l is a constant, ξ(ψ(t n h,e s (t),δ1δ2)) satisfies the following equation:

[0022]

[0023] where, j represents the number of the most recently transmitted data packets, δ1∈(0,1), δ2∈(0,1) are event - triggering thresholds, Λ>0 is the event - triggering matrix to be solved, e s (t)=ψ(tn h + lh) - ψ(t n-s+1 h), ψ(t r ) represents the signal of malicious replacement, σ s ∈ [0, 1] represents the weight parameter and satisfies

[0024] Furthermore, the anti - attack memory controller is expressed as follows:

[0025]

[0026] where K s is the controller gain, and the Bernoulli variable v(t) ∈ {0, 1} represents the probability of replay attack occurrence.

[0027] Furthermore, the constructed closed - loop Lur’e system is as follows:

[0028]

[0029] Furthermore, the criterion for ensuring the mean - square H∞ stability of the closed - loop Lur’e system is as follows:

[0030] For the given parameters σ s ∈ [0, 1], s ∈ {1, …, j}, v ∈ [0, 1), τ max ∈ [0, ∞), δ1 ∈ (0, 1), δ2 ∈ (0, 1), γ ∈ (0, ∞) and matrices I > 0, Λ > 0, J > 0, K s , if there exist symmetric matrices P > 0, G > 0, M > 0, N > 0, W > 0, and Q1, Q2, Q3 such that the symmetric matrix

[0031] [Π ij 8×8 < 0,

[0032] then under all acceptable replay attacks, the closed - loop Lur’e system is mean - square H∞ stable; where:

[0033] Π 11 = G + Q1 T A + A T Q1 - M, Π 12 = P + D T JWD - Q1 + A T Q2, Π 13 = Q1 T B1 + D T JN, Π 15 = Y T ,​ Π 44 = δ1 / s 2 Λ + Y T + Y - M - M T , Π 45 = -Y T + M, Π 46 = -δ1 / s 2 Λ * I 1×s , Π 55 = -G - M, Π 66 = δ1 / s 2 Λ * I s×s + diag{-σ1Λ, …, -σ s Λ}, Π 67 = -(1 - υ)K s T Q3 * I s×s ,

[0034]

[0035] All other unspecified symbols are zero matrices of appropriate dimensions.

[0036] Furthermore, the design method of the controller gain and the event-triggering matrix is as follows:

[0037] Define Q2 = λ1Q1, Q3 = λ2Q1, and substitute the above three equations into the mean-square H∞ stability criterion inequality in S3 to eliminate the multiple coupling terms in the criterion inequality, obtaining the optimized mean-square H∞ stability criterion;

[0038] Solve the optimized mean-square H∞ stability inequality, and the event-triggering matrix Λ and the controller gain can be obtained

[0039] Adopting the technical solution provided by the present invention, compared with the prior art, the following beneficial effects can be achieved:

[0040] (1) The present invention proposes an elastic memory event-triggering mechanism. By introducing maliciously replaced signals into the elastic memory event-triggering judgment formula, it can not only effectively improve the ability of the Chua's circuit to resist replay attacks, but also further reduce the data transmission frequency and relieve the network burden.

[0041] (2) Based on the elastic memory event-triggering mechanism, the present invention designs an attack-resistant memory controller and constructs a closed-loop Lur’e system. Through the design of the criterion, it can effectively ensure that the closed-loop Lur’e system meets the mean-square H∞ stability requirements. At the same time, through the further optimization of the above mean-square H∞ stability criterion and the solution of the criterion inequality, a co-design scheme for the controller gain and the event-triggering matrix can be obtained. Description of the Drawings

[0042] Figure 1 is a schematic flow chart of the Chua's circuit memory event-triggering control method for resisting replay attacks of the present invention;

[0043] Figure 2 is a schematic diagram of the Chua's circuit model;

[0044] Figure 3 is a networked control architecture diagram of the Chua's circuit in an embodiment of the present invention;

[0045] Figure 4 is a schematic diagram of the chaotic attractor in an embodiment of the present invention;

[0046] Figure 5 is a schematic diagram of the event-triggering interval under the memory event-triggering mechanism in an embodiment of the present invention;

[0047] Figure 6 is the state trajectory of the closed-loop system in an embodiment of the present invention;

[0048] Figure 7 is the evolution trajectory of the control input signal in an embodiment of the present invention;

[0049] Figure 8 is the evolution trajectory of γ(t) in an embodiment of the present invention. Detailed Embodiment

[0050] Combined with Figure 1 shown, the present invention provides a Chua's circuit memory event-triggering control method for resisting replay attacks, including the following steps:

[0051] S1. Through the variable substitution technique, the Chua's circuit model is transformed into a Lur’e system mathematical model;

[0052] S2. An elastic memory event-triggering mechanism is proposed, which contains terms related to replay attacks, so as to make full use of the attack information to further reduce unnecessary data transmission and optimize the bandwidth utilization rate;

[0053] S3. To effectively resist the influence of replay attacks on the system, an attack-resistant memory controller is designed;

[0054] S4. Establish criteria to ensure the mean-square H∞ stability of the closed-loop Lur’e system;

[0055] S5. Based on the proposed criteria, a co-design scheme for the controller gain and the event-triggering matrix is given in the form of linear matrix inequalities.

[0056] According to the control method described in the present invention, the process of constructing the mathematical model of the Lur’e system is as follows:

[0057] Combined with Figure 2 the schematic diagram of the Chua's circuit model in

[0058]

[0059] where R, L, and C i , i = {0, 1} represent the resistor, inductor, and capacitor respectively; I L represents the current flowing through the inductor L; and represent the voltages across the capacitors C0 and C1 respectively; represents the current in the resistor N R ; θ0, θ1, and θ2 are scalars set manually.

[0060] By adopting the variable substitution technique and considering the influence of the external disturbance w(t) on the circuit, the mathematical model of the above Chua's circuit can be expressed as a Lur’e system in the following form:

[0061]

[0062] where, u(t), w(t), and φ(t) represent the system state, the nonlinear vector function, the control input, the external disturbance, and the measured output respectively; A, B1, B2, C, and D are known system matrices, specifically as follows:

[0063] B2 = [0.1 0.1 0.1] T ,

[0064] C = [0.001 0 0], D = [1 0 0] T .

[0065] To facilitate the subsequent stability analysis of the system, it is assumed that the nonlinear vector function h(Dψ(t)) satisfies the following sector condition:

[0066] For there exists a positive definite matrix J such that

[0067] h T(ε)[h(ε)-Jε] ≤ 0。

[0068] As a preferred solution, in order to further reduce the data transmission frequency and relieve the network burden, the present invention proposes an elastic memory event-triggering mechanism. Assuming that the data is vulnerable to replay attacks during the channel transmission, it is stipulated that ψ(t r ) represents the malicious replacement signal, then the memory event-triggering mechanism is expressed as follows:

[0069] t n+1 h = t n h + min{lh | ξ(ψ(t n h, e s (t), δ1δ2)) > 0},

[0070] where h represents the sampling interval, t n h is the instantaneous sequence transmitted at time n, t n+1 h is the instantaneous sequence transmitted at time n + 1, l is a constant, and ξ(ψ(t n h, e s (t), δ1δ2)) satisfies the following equation:

[0071]

[0072] where j represents the number of recently transmitted data packets, δ1 ∈ (0, 1), δ2 ∈ (0, 1) are event-triggering thresholds, Λ > 0 is the event-triggering matrix to be solved, and e s (t) = ψ(t n h + lh) - ψ(t n-s+1 h), σ s ∈ [0, 1] represents the weight parameter and satisfies

[0073] In addition, during the data transmission process, the transmission delay is an important factor that cannot be ignored. Let τ(t n ) represent the transmission delay from the sensor to the actuator. It is assumed that the controller remains unchanged in the interval .

[0074] Define where:

[0075]

[0076] For Define τ(t) = t - t i h - lh. It can be obtained that and where is the transmission delay The upper bound.

[0077] Furthermore, based on the above elastic memory event-triggering mechanism, the present invention designs an anti-attack memory controller to effectively resist the impact of replay attacks on the system. Specifically, let the Bernoulli variable υ(t) ∈ {0, 1} represent the probability of a replay attack occurring, then the proposed anti-attack memory controller can be designed as follows:

[0078]

[0079] where K s is the controller gain;

[0080] Then, combining the elastic memory event-triggering mechanism and the anti-attack memory controller, the closed-loop Lur’e system can be constructed as:

[0081]

[0082] Furthermore, by constructing an appropriate Lyapunov function, a criterion is established to ensure that the closed-loop system is mean-square H∞ stable. Specifically, in some embodiments, the mean-square H∞ stability criterion is as follows:

[0083] For given parameters σ s ∈ [0, 1] (s ∈ 1,..., j), v ∈ [0, 1), τ max ∈ [0, ∞), δ1 ∈ (0, 1), δ2 ∈ (0, 1), γ ∈ (0, ∞) and matrices I > 0, Λ > 0, J > 0, K s (s ∈ {1, 2,..., j}), if there exist matrices P > 0, G > 0, M > 0, N > 0, W > 0, and Q1, Q2, Q3 such that the symmetric matrix:

[0084] [Π ij 8×8 <0 (1)

[0085] Then, under all acceptable replay attacks, the closed-loop Lur’e system is mean-square H∞ stable. Where:

[0086] Π 11 = G + Q1 T A + A T Q1 - M, Π 12 = P + D T JWD - Q1 + A T Q2, Π 13 = Q1 T B1 + D T JN, Π 15 = Y T ,​ Π 44 =δ1 / s 2 Λ + Y T +Y - M - M T , Π 45 =-Y T +M, Π 46 =-δ1 / s 2 Λ * I 1×s , Π 55 =-G - M, Π 67 =-(1 - v)K s T Q3 * I s×s , All other unspecified symbols are zero matrices of appropriate dimensions.

[0087] Next, it is proved that the mean-square H∞ stability of the system can be judged by the above criterion, as follows:

[0088] Construct the following Lyapunov function:

[0089]

[0090] For By taking the derivative of ν(t) and taking the mathematical expectation, we can write:

[0091]

[0092] For the integral term in the above formula, by using the convexity method, the following inequality can be obtained:

[0093]

[0094] From the sector condition, it can be obtained that:

[0095] 0 ≤ -2h(D T ψ(t))N[h(D T ψ(t)) - JD T ψ(t)]

[0096] =-2h T (D T ψ(t))Nh(D T ψ(t)) + 2ψ T (t)D T JNh(Dψ(t)) (4)

[0097] For According to the proposed memory event-triggering mechanism, it can be deduced that:

[0098]

[0099] For the closed-loop Lur’e system, the following equations can be obtained:

[0100]

[0101] Combining equations (1) and (3)-(6), it can be deduced that:

[0102]

[0103] Since is continuous over the entire interval t, then according to (7), it can be concluded that:

[0104]

[0105] Under zero initial conditions, from (8) the following inequality can be obtained:

[0106]

[0107] Thus, it is proved that the system has the given H∞ performance.

[0108] When w(t)=0, by applying condition (1), it can be proved that the closed-loop Lur’e system has mean-square asymptotic stability.

[0109] Furthermore, in the criterion for mean-square H∞ stability of formula (1), there are multiple coupling terms. To facilitate the numerical design of the controller gain and the event-triggering matrix, the present invention further defines Q2 = λ1Q1, Q3 = λ2Q1, λ1 ∈ (0,1), λ2 ∈ (0,1), and substitute the above three equations into the inequality of the criterion for mean-square H∞ stability of formula (1) to eliminate the multiple coupling terms in the criterion inequality, thereby obtaining the optimized criterion for mean-square H∞ stability:

[0110]

[0111] Where:

[0112]

[0113] All other unspecified symbols are 0 matrices of appropriate dimensions.

[0114] Then, solve the optimized inequality for mean-square H∞ stability to obtain the event-triggering matrix Λ and the controller gain K s =(Q1 T ) -1 E s , (s ∈ {0,…,j}).

[0115] Proof: Substitute and Q2 = λ1Q1, Q3 = λ2Q1 into Equation (9), and we can obtain [[Π ij 8×8 < 0, which means that condition (9) can ensure that the closed-loop Lur’e system is mean-square H∞ stable under all acceptable replay attacks.

[0116] It should be noted that all the extra letters introduced in the present invention are intermediate variables without any actual meaning, only for clear mathematical representation.

[0117] To further understand the content of the present invention, the present invention will be described in detail below in conjunction with specific embodiments.

[0118] As Figure 3 shown is the networked control architecture diagram of the Chua's circuit in this embodiment. The memory event-triggered control method for the Chua's circuit specifically includes:

[0119] S1. Through variable substitution technology, transform the Chua's circuit model into a Lur’e system mathematical model;

[0120] S2. Propose an elastic memory event-triggering mechanism and design an anti-attack memory controller based on this;

[0121] S3. Combine the elastic memory event-triggering mechanism and the anti-attack memory controller to construct a closed-loop Lur’e system and establish a criterion to ensure the mean-square H∞ stability of the closed-loop Lur’e system;

[0122] S4. Based on the proposed criterion, give a co-design scheme for the controller gain and the event-triggering matrix.

[0123] Among them, the Lur’e system mathematical model obtained in S1 is as follows:

[0124]

[0125] Among them, h(Dψ(t)), u(t), w(t) and φ(t) respectively represent the system state, the nonlinear vector function, the control input, the external disturbance and the measured output; and respectively represent the voltages across the capacitors C0 and C1; I L represents the current flowing through the inductor L; A, B1, B2, C and D are known system matrices:

[0126] B2 = [0.1 0.1 0.1] T ,

[0127] C = [0.001 0 0], D = [1 0 0]​T 。

[0128] Specifically, in this embodiment, the capacitance C0 = 1 / 9, C1 = 1, the resistance R = 1 / 0.7, θ0 = -0.8, θ1 = -0.5, θ2 = 1. The initial state of the system is set as ψ(t) = [0.25; -0.11; -0.22], and the external disturbance w(t) = 2e -0.8t sin(2πt). Without applying control to the system, the system exhibits a chaotic behavior, as Figure 4 shown.

[0129] First, the effectiveness of the proposed controller scheme will be verified. On the basis of keeping the above parameter values unchanged, assume that the buffer size j = 3, the weight parameters σ1 = 0.5, σ2 = 0.3, σ2 = 0.2, the event-triggering threshold δ1 = δ2 = 0.1; the probability υ of replay attack occurring = 0.1; the upper bound τ max = 0.11; λ1 = 0.190, λ2 = 0.001, J = 1, γ = 1.0422. By using the Matlab toolbox MOSEK to solve the linear matrix inequality (9), the following event-triggering matrix and controller gain can be obtained:

[0130]

[0131] Figure 5 is the event-triggering interval graph under the memory event-triggering mechanism, Figure 6 and Figure 7 respectively give the evolution graphs of the closed-loop system state and the controller input signal. It can be seen that under the action of the designed event-triggering matrix and controller gain, the state trajectory and control input signal of the system finally tend to 0.

[0132] To depict the H∞ performance level of the closed-loop system, the following expression is defined:

[0133]

[0134] Under zero initial conditions, from Figure 8 the evolution trajectory of γ(t) in it can be seen that γ(∞) = 0.0832 (≤ γ min = 1.1899), thus proving the effectiveness of the proposed design scheme.

[0135] Next, it will be proved that the additional introduced terms can further reduce the number of triggers and save network communication resources. Table 1 shows the number of event triggers under different trigger mechanisms. It can be observed that when the trigger thresholds are all 0, the trigger mechanism proposed by the present invention degrades to a periodic sampling mechanism, and the number of triggers at this time is 10,001 times; when δ1 = 0.1 and δ2 = 0, the trigger mechanism changes to a traditional memory trigger mechanism, and the number of triggers is 2,552 times. By comparison, it can be obtained that compared with the periodic sampling mechanism, the data transmission ratio of the anti-attack memory trigger mechanism designed by the present invention is reduced by 94.13%; compared with the traditional memory event trigger mechanism, the data transmission ratio is reduced by 19.64%. This shows that the event trigger mechanism proposed by the present invention can further reduce the number of triggers and save network communication resources.

[0136] Table 1 Comparison of the number of triggers under different trigger mechanisms

[0137]

Claims

1. A Chua's circuit memory event trigger control method for resisting replay attacks, characterized in that: include: S1. Transform the Cai circuit model into the Lur'e system mathematical model through variable substitution technology; S2. Propose a flexible memory event triggering mechanism and design an anti-attack memory controller based on it; S3. Combining the elastic memory event trigger mechanism and the anti-attack memory controller, a closed-loop Lur'e system is constructed, and a criterion for ensuring the mean square H∞ stability of the closed-loop Lur'e system is established; S4. Based on the proposed criterion, a collaborative design scheme of controller gain and event trigger matrix is ​​given.

2. The control method according to claim 1, characterized in that: The mathematical model of the Lur'e system is as follows: in, h(Dψ(t)), u(t), w(t) and φ(t) represent the system state, nonlinear vector function, control input, external disturbance and measurement output respectively; and Represents the voltage across capacitors C0 and C1 respectively; I L represents the current flowing through the inductor L; A, B1, B2, C and D are known system matrices.

3. The control method according to claim 2, characterized in that: The elastic memory event trigger mechanism is that the event generator will transmit the system status only when the following trigger conditions are met: t n+1 h=t n h+min{lh|ξ(ψ(t n h,e s (t),δ1δ2))>0} Where h represents the sampling interval, t n h is the instantaneous sequence transmitted at time n, t n+1 h is the instantaneous sequence transmitted at time n+1, l is a constant, ξ(ψ(t n h,e s (t),δ1δ2)) satisfies the following equation: Where j represents the number of packets transmitted recently, δ1∈(0,1), δ2∈(0,1) are event trigger thresholds, Λ>0 is the event trigger matrix to be solved, and e s (t) = ψ(t n h+lh)-ψ(t n-s+1 h), ψ(t r ) indicates a malicious replacement signal, σ s ∈[0,1] represents the weight parameter and satisfies 4. The control method according to claim 3, characterized in that: The anti-attack memory controller is expressed as follows: Among them, K s is the controller gain, and the Bernoulli variable υ(t)∈{0,1} represents the probability of a replay attack occurring.

5. The control method according to claim 4, characterized in that: The resulting closed-loop Lur'e system is constructed as follows:

6. The control method according to claim 5, characterized in that: The criterion for ensuring the mean square H∞ stability of the closed-loop Lur'e system is as follows: For a given parameter σ s ∈[0,1], s∈{1,…,j}, υ∈[0,1), τ max ∈[0,∞), δ1∈(0,1), δ2∈(0,1)γ∈(0,∞) and the matrix I>0, Λ>0, J>0, K s , if there exists a symmetric matrix P>0, G>0, M>0, N>0, W>0, and Q1, Q2, Q3 make the symmetric matrix [P ij ] 8×8 <0, Then under all acceptable replay attacks, the closed-loop Lur'e system is mean square H∞ stable; where: P 11 =G+Q1 T A+A T Q1-M,∏ 12 =P+D T JWD-Q1+A T Q2,P 13 =Q1 T B1+D T JN, P 15 =Y T , P 33 =-L-L T , P 44 =δ1 / s 2 L+Y T +YMM T ,P 45 =-Y T +M,P 46 =-δ1 / s 2 L*I 1×s , P 55 =-GM,P 66 =δ1 / s 2 L*I s×s +diag{-σ1Λ,…,-σ s Λ},∏ 67 =-(1-υ)K s T Q3*I s×s , All other unspecified symbols are zero matrices of appropriate dimensions.

7. The control method according to claim 6, characterized in that: The controller gain and event trigger matrix are designed as follows: definition Q2=λ1Q1,Q3=λ2Q1,λ1∈(0,1),λ2∈(0,1), and substitute the above three equations into the mean square H∞ stability criterion inequality in S3 to eliminate the multiple coupling terms in the criterion inequality and obtain the optimized mean square H∞ stability criterion; Solve the optimized mean square H∞ stability inequality to obtain the event trigger matrix Λ and controller gain

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