A Control Method for Event-Triggered T-S Fuzzy Systems under Replay Attacks

Abstract

The present invention discloses a control method for a T-S fuzzy system based on an event-triggered mechanism under replay attacks, which includes the following steps: Step S1 introduces an event-triggered mechanism; Step S2 designs the influence mode of replay attacks on transmitted data; Step S3 designs a closed-loop fuzzy control system model; Step S4 constructs a Lyapunov function to obtain sufficient conditions for the asymptotic stability of the closed-loop fuzzy control system; Step S5 formulates and solves linear matrix inequalities to obtain the gains of the parallel distributed compensation controller. The present invention improves resource utilization by adopting an event-triggered mechanism, saves limited bandwidth resources, and ensures the safe and stable operation of the network control system under the influence of replay attacks.

Description

Technical Field

[0001] The present invention belongs to the technical field of cyber - physical system security. Specifically, the present invention relates to a control method for T - S fuzzy systems based on event - triggered under replay attacks. Background Art

[0002] In recent years, with the rapid development of the cyber society, network communication resources have become increasingly precious. When the system operates stably, the controller usually does not need to be updated frequently. A large amount of redundant data generated by high - sampling rates will lead to waste of system - limited resources such as network bandwidth and node energy. Therefore, how to design an effective control strategy to reduce resource consumption and lower the cost of the control system is particularly important. Then, an event - triggered scheme was proposed, that is, the information required by the system can be sent to the next receiver through an event trigger only when the pre - set conditions are met. The pre - set conditions are called event - trigger conditions. That is, data transmission is only allowed when the trigger conditions are satisfied. This method not only reduces the transmission frequency of sampled data but also effectively saves network resources.

[0003] The research on nonlinear network control systems using T - S fuzzy models has received extensive attention. Generally, a T - S fuzzy system is represented by a series of IF - THEN rules. In essence, a nonlinear system is regarded as a fuzzy approximation of multiple local linear fuzzy models. That is to say, theoretically, if we choose a sufficient number of fuzzy rules, this model can achieve arbitrary accuracy. Therefore, T - S fuzzy systems can be analyzed using many traditional linear - system methods and are important tools for nonlinear - system modeling.

[0004] At the same time, the convenience of the data - transmission network has greatly improved the control performance of the system. At the same time, due to the openness of the data - transmission network, the system faces serious security threats. An attacker can easily launch a network attack on the system, such as a replay attack. The attacker launches a replay attack on the network communication channel, causing the system to start repeating the data that has been sent before, resulting in the system controller being unable to complete the control of the stable operation of the system. Therefore, it is extremely necessary to study a control method for T - S fuzzy systems based on event - triggered under replay attacks. Summary of the Invention

[0005] The fundamental starting point of the present invention is to overcome the deficiencies in the prior art and provide a control method for T - S fuzzy systems based on event - triggered under replay attacks. The present invention improves resource utilization by adopting an event - trigger mechanism, saves limited bandwidth, and considers the impact of network attacks on data transmission. By using the Lyapunov - function stability theory, the mean - square exponential stability of the system is ensured.

[0006] To achieve the above object, the present invention intends to adopt the following technical solutions:

[0007] A TS fuzzy system control method based on an event trigger mechanism under a replay attack comprises the following steps:

[0008] Step S1 introduces an event trigger mechanism;

[0009] Step S2 designs the impact of the replay attack on the transmitted data;

[0010] Step S3 designs a closed-loop fuzzy control system model;

[0011] Step S4 constructs a Lyapunov function to obtain sufficient conditions for the asymptotic stability of the closed-loop fuzzy control system;

[0012] Step S5 connects and solves the linear matrix inequality to obtain the parallel distributed compensation controller gain.

[0013] In the above step S1,

[0014] Under the event trigger mechanism, the next trigger time t k+1 h is:

[0015]

[0016] Among them, the threshold parameter δ is a given non-negative scalar, δ∈(0,1), Ω is a positive definite symmetric weight matrix, t k+1 h represents the next triggering time, t k h represents the most recent triggering time, e(t) = x(t k h)-x(t k h+lh) represents the difference between the sampling data at the most recent triggering moment and the sampling data at the current moment, t k h+lh represents the current sampling time, and h represents the sampling period.

[0017] The data transmitted under the influence of the replay attack is expressed as follows:

[0018]

[0019] When θ(t) = 0, it means that the replay attack has not occurred, and when θ(t) = 1, it means that the replay attack has occurred. Where x(t) represents the transmission data after the event trigger mechanism, x(tr(t)) represents the past signal injected by the attacker at time t, and r(t) represents the replayed data is the data transmitted within the previous r(t) seconds.

[0020] The above step S3 includes:

[0021] Step S31 establishes a preliminary TS fuzzy system model system rule Ri :If ψ1(t) is And g (t) Yes but

[0022]

[0023] Where: x(t)∈R n , represents the system state, u(t)∈R n , represents the control output. i=1,2,...,r,r is the number of fuzzy rules, is a fuzzy set. j (t)(j=1,2,...,g) represents the premise variable. ψ(t)=[ψ1(t) … ψ g (t)] T , and assume that ψ(t) is neither given nor a function of x(t), nor does it depend on u(t). i ,B i is a constant matrix of appropriate dimension;

[0024] Step S32 transforms the TS fuzzy system model system into a global fuzzy system model as shown below through single-point fuzzification and defuzzification, namely:

[0025]

[0026] where μ i (ψ(t)) represents the normalized membership function. In M ij In, M ij (ψ j (t)) is ψ j (t); and assuming that for all t>0, Then for all t we have μ i (ψ(t))≥0 and To simplify the calculation, we use μ i Represents μ i (ψ(t));

[0027] Step S33 designs a controller based on the TS fuzzy model by a parallel distributed compensation method to stabilize the TS fuzzy system. The j-th state feedback controller can be designed as follows:

[0028] System Rules j :If ψ1(t) is And... and ψ g (t) Yes

[0029] Then \(u(t)=k\) j \(x1(t)\)

[0030] \(k\) j is the fuzzy control gain matrix. Using the singleton fuzzification and defuzzification, the fuzzified output of the parallel compensation controller can be obtained as:

[0031]

[0032] Based on the above event-triggering mechanism, replay attack model, and the fuzzified output \(u(t)\) of the parallel compensation controller, the following closed-loop fuzzy control system model is obtained:

[0033]

[0034] In the above step S4, for the given positive parameter the time-delay parameter \(d\) m \(>0\), the triggering parameter \(\delta\), the matrix \(K\) j , if there exist \(\Omega, P>0, Q>0, R>0\), free-weight matrices \(T\) ij , \(S\) ij , \(W\) i \(=W\) i T is an arbitrary matrix, and for any \(i, j = 1, 2, \ldots, r\), the following inequality holds, then the system is asymptotically stable:

[0035]

[0036] \(\upsilon\) j \(\Psi\) ij +\(\upsilon\) i \(\Psi\) ji -\(\upsilon\) j \(W\) i -\(\upsilon\) i \(W\) j +\(W\) i +\(W\) j \(<0, (i < j)\).

[0037] In the above step S5, for the given positive parameter the time-delay parameter \(d\) m \(>0\) and the triggering parameter \(\delta\), if there exists a free-weight matrix with appropriate dimensions is an arbitrary matrix, and for any \(i, j = 1, 2, \ldots, r\), the following inequality holds, then the system is asymptotically stable:

[0038]

[0039]

[0040] And the expression for the controller gain is: K j = Y j X -1 .

[0041] Compared with the prior art, the present invention can achieve the following optimization effects:

[0042] The present invention provides a control method and system for a T-S fuzzy system based on event-triggering under replay attacks. A T-S fuzzy system controller is designed by integrating the event-triggering mechanism and network replay attacks. The event-triggering mechanism is adopted to relieve the communication pressure and improve the resource utilization rate, and at the same time, the impact of replay attacks on data transmission is considered. By using the Lyapunov stability theory, sufficient conditions for ensuring the stability of the designed system are obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 FIG. is a specific flowchart of a control method for a T-S fuzzy system based on event-triggering under replay attacks in an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0044] The following describes the present invention in further detail with reference to the drawings and embodiments to help those skilled in the relevant art have a more accurate, intuitive, and in-depth understanding of the strategy design of the present invention.

[0045] The flowchart of the control method for a T-S fuzzy system based on event-triggering under replay attacks provided by the present invention is as Figure 1 shown.

[0046] Step S1 introduces an event-triggering mechanism and designs an event-triggering scheme.

[0047] Under the event-triggering mechanism, the next triggering time t k+1 h is;

[0048]

[0049] where the threshold parameter δ is a given non-negative scalar, δ ∈ (0, 1), Ω is a positive definite symmetric weight matrix, t k+1 h represents the next triggering time, t k h represents the most recent triggering time, e(t) = x(t k h) - x(t k h + lh) represents the difference between the sampled data at the most recent triggering time and the sampled data at the current time, t k h + lh represents the current sampling time, and h represents the sampling period. It can be seen from the above event-triggering scheme that the state of the triggered object x(t kh) and the current state x(t) of the sampling object obtained by the sensor k h + lh), only when the event trigger condition is satisfied:

[0050]

[0051] The switch will close and the current sampled data will be further transmitted. If not satisfied, the current sampled value will be discarded.

[0052] Different from the periodic sampling mechanism that sends all sampled values, the event-triggered mechanism only sends the sampled values that meet the trigger conditions. Therefore, the set of trigger times is a subset of the set of sampling times. Specifically, if the threshold parameter δ is 0 and the trigger condition always holds, the event-triggered mechanism transforms into a periodic sampling mechanism.

[0053] Furthermore, the zero-order hold is event-driven. Assume that an event is generated at time t k h. The data packet passes through the network and experiences a delay of τ k before arriving at the zero-order hold. When the event is triggered at time t k+1 h, it arrives at the zero-order hold after a delay of τ k+1 . For the convenience of analysis, the interval [t k h + τ k , t k+1 h + τ k+1 ) is divided into multiple sub-intervals, that is, the time interval of the zero-order hold is segmented into the following subsets. Let ε k = t k+1 - t k - 1, that is:

[0054]

[0055] where: t k h represents the most recent trigger time, t k+1 h represents the next trigger time, h represents the sampling period, and τ k represents the time delay that the data packet takes from the network to the zero-order hold at time t k h. τ k+1 represents the time delay that the data packet takes from the network to the zero-order hold at time t k+1 h. k h represents the previous trigger time, and τ k represents the time (which is called the time delay) that the sampled data at the previous trigger time takes from the network to reach the zero-order hold. τ k+1 represents the time delay that the sampled data at the most recent trigger time t k+1 h takes from the network to reach the zero-order hold.

[0056] Furthermore, define \(d(t)=t-(t k h + lh)\), where \(d m is the upper bound of the maximum network-induced time delay. Then \(x(t k h)=e(t)+x(t - d(t))\).

[0057] The role of the zero-order hold is to keep the sampled signal value at the \(nT\) moment unchanged until the moment immediately before the \((n + 1)T\) moment during the signal transmission process, keep the sampled value at the \((n + 1)T\) moment until the \((n + 2)T\) moment, and so on, so as to transform a pulse sequence into a continuous stepped signal. Since the value of the continuous stepped signal is a constant within each sampling interval, that is, its first derivative is zero, it is called a zero-order hold. The event trigger determines whether to send the current sampled value according to the event trigger condition. The data packet sent by the event trigger may be subject to a replay attack during communication. The controller updates the control signal according to the received data packet, and the updated control signal will be immediately sent to the zero-order hold. Therefore, the set of update times of the zero-order hold is the same as the set of update times of the controller.

[0058] Step S2 designs the influence mode of the replay attack on the transmitted data.

[0059] The data transmitted under the influence of the replay attack is expressed as follows:

[0060]

[0061] where \(\theta(t)\) represents a Bernoulli variable, which is used to indicate whether a replay attack occurs. When \(\theta(t)=0\), it means that the replay attack does not occur; when \(\theta(t)=1\), it means that the replay attack occurs; where represents the transmitted data after passing through the event trigger mechanism, \(x(t - r(t))\) represents the past signal injected by the attacker recorded at time \(t\), and \(r(t)\) represents that the replayed data is the data transmitted within the previous \(r(t)\) seconds. In addition represents the mathematical expectation of \(\theta(t)\), then

[0062]

[0063]

[0064] Step S3 designs a closed-loop fuzzy control system model.

[0065] Step S31 establishes a preliminary T-S fuzzy control system model, specifically:

[0066] System rule \(R i : If \(\psi_1(t)\) is and... and \(\psi g (t)\) is then

[0067]

[0068] where: \(x(t)\in R\) n , representing the system state, \(u(t)\in R\) n , representing the control output. \(i = 1, 2, \ldots, r\), where \(r\) is the number of fuzzy rules, is a fuzzy set. \(\psi\) j (t) (\(j = 1, 2, \ldots, g\)) represents the premise variables. \(\psi(t)=[\psi_1(t)\ldots\psi\) g (t)] T , and it is assumed that \(\psi(t)\) is neither given nor a functional function of \(x(t)\), and it does not depend on \(u(t)\) either. \(A\) i , \(B\) i are constant matrices with appropriate dimensions.

[0069] Step S32 transforms it into the following global fuzzy system model through singleton fuzzification and defuzzification, that is:

[0070]

[0071] where \(\mu\) i (\(\psi(t)\)) represents the normalized membership function. In \(M\) ij , \(M\) ij (\(\psi\) j (t)) is the membership degree of \(\psi\) j (t); and it is assumed that for all \(t \gt 0\) there is Then for all \(t\) there is \(\mu\) i (\(\psi(t)\)) \(\geq 0\) and holds. For simplicity of calculation, hereinafter \(\mu\) i is used to represent \(\mu\) i (\(\psi(t)\)).

[0072] Step S32 designs a parallel distributed compensation controller based on the T-S fuzzy control system model through the method of parallel distributed compensation to stabilize the T-S fuzzy control system. The \(j\)-th state feedback controller can be designed as the following expression:

[0073] System rule \(R\) j : If \(\psi_1(t)\) is and \(\psi\) g (t) is

[0074] Then \(u(t)=k\) j \(x_1(t)\)

[0075] \(k\) jis the fuzzy control gain matrix. Using singleton fuzzification and defuzzification, the fuzzy output of the parallel distributed compensation controller can be obtained as follows:

[0076]

[0077] Step S33 obtains the following closed-loop fuzzy control system model based on the above event-triggering mechanism, replay attack model, and the fuzzy output u(t) of the parallel distributed compensation controller.

[0078]

[0079] μ i and m j represent membership functions,

[0080] Step S4 constructs a Lyapunov function to obtain the sufficient conditions for the asymptotic stability of the closed-loop fuzzy control system.

[0081] Construct the Lyapunov function as follows:

[0082]

[0083] where: P>0, Q>0, R>0 are all positive definite matrices.

[0084] Calculate the derivative and take the expectation as follows:

[0085] Taking the derivative of V gives:

[0086]

[0087] For taking the expectation gives:

[0088]

[0089]

[0090]

[0091] Λ 2ij =B i K j [x(t - r(t)) + x(t - d(t)) + e(t)]

[0092]

[0093] For the given positive parameter the time-delay parameter d m >0, the triggering parameter δ, the matrix K j , if there exist Ω, P>0, Q>0, R>0, free weight matrices T with appropriate dimensionsij ,S ij , is an arbitrary matrix, and for any i,j=1,2,…,r, the following inequality holds, then the system is asymptotically stable.

[0094] Ψ ij -W i <0;m j -υ j μ j <0;υ i Ψ ii -υ i W i +W i <0;

[0095] υ j Ψ ij +υ i Ψ ji -υ j W i -υ i W j +W i +W j <0,(i<j);

[0096]

[0097]

[0098]

[0099]

[0100]

[0101]

[0102] Step S5 connects and solves the linear matrix inequality to obtain the parallel distributed compensation controller gain.

[0103] definition:

[0104] X=P -1 ,

[0105]

[0106] χ=diag{XXXXXXXXXX}

[0107] For any satisfy Can get So use Replace N 44 and N 55 the -PR in -1 P, thus obtaining and For the equation Ψ ij -W i Multiply <0 on the left by χ and on the right by χ T , the following conclusion can be obtained from the schur complement lemma:

[0108] For the given positive parameter the time-delay parameter d m >0 and the triggering parameter δ, if there exists X>0, a free-weighting matrix with appropriate dimensions is an arbitrary matrix, and for any i, j = 1, 2,..., r, the following inequalities hold, then the system is asymptotically stable.

[0109] m j -υ j μ j <0;

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117] And the expression for the parallel distributed compensation controller gain is obtained as: K j =Y j X -1

[0118] The expression for the j-th parallel distributed compensation controller gain is K j , which decomposes the entire large state space into many fuzzy subspaces, designs corresponding distributed compensation controllers for the decomposed fuzzy subspaces, and the design of the fuzzy controller for the entire system is equivalent to the weighted combination of the local subsystem controller designs.

[0119] The advantages of the present invention are as follows. The present invention designs a method for a T-S fuzzy control system based on an event-triggered mechanism under replay attacks. By analyzing and modeling the system under cyber attacks. And in order to save system resources and reduce network load, an event-triggered strategy is adopted, and a fuzzy controller is designed using the parallel distributed compensation principle, ultimately eliminating the impact of replay attacks on the system. Through the Lyapunov function stability theory, sufficient conditions for the mean-square exponential stability of the system are obtained. Therefore, the control method for the T-S fuzzy system based on the event-triggered mechanism under replay attacks designed by the present invention can reduce the network bandwidth pressure and effectively cope with network security threats while ensuring the stability of the system.

[0120] As described above, the embodiments of the present invention are specifically described in conjunction with the accompanying drawings. Obviously, the specific implementation of the present invention is not limited by the above methods. It should be noted that those skilled in the relevant art can make other improvements without departing from the technical principle of the present invention, and these improvements should be regarded as the protection scope of the present invention.

Claims

1. A control method for a T-S fuzzy system based on an event-triggering mechanism under replay attacks, characterized in that It includes the following steps: Step S1 introduces an event triggering mechanism; Step S2 designs the influence mode of replay attack on transmitted data; Step S3 designs a closed-loop fuzzy control system model; Step S3 includes, Step S31 establishes the system rules of a preliminary T-S fuzzy system model : If is , and is , then Wherein: represents the system state, represents the control output, , is the number of fuzzy rules, is the fuzzy set, represents the premise variable, , and assume that is neither given nor the functional function of, and it does not depend on , is a constant matrix with appropriate dimensions; Step S32 converts the T-S fuzzy system model into the following global fuzzy system model through single-point fuzzification and defuzzification, that is: wherein represents a normalized membership function, , , in , is the membership degree of; and it is assumed that for all t > 0 there is , , then for all t there is and hold. To simplify the calculation, hereinafter is used to represent ; Step S33 designs a T-S fuzzy model-based controller by the method of parallel distributed compensation to stabilize the T-S fuzzy system. The j th state feedback controller is designed as the following expression: System rules : If is , and is , Then For the fuzzy control gain matrix, using single-point fuzzification and defuzzification, the fuzzified output of the parallel compensation controller is obtained as follows: Based on the above event trigger mechanism, replay attack model, and the fuzzification output of the parallel compensation controller , the following closed-loop fuzzy control system model is obtained: ； Step S4 constructs a Lyapunov function to obtain the sufficient conditions for the asymptotic stability of the closed-loop fuzzy control system; Step S5 formulates and solves linear matrix inequalities to obtain the parallel distributed compensation controller gain.

2. The control method of a T-S fuzzy system based on an event-triggered mechanism under replay attack according to claim 1, wherein In step S1, Under the event trigger mechanism, the next trigger time is as follows: Among them, the threshold parameter is a given non - negative scalar, , is a positive definite symmetric weight matrix, represents the next triggering moment, represents the most recent triggering moment, represents the difference between the sampled data at the most recent triggering moment and the sampled data at the current moment, represents the current sampling moment, represents the sampling period.

3. A control method for a T-S fuzzy system based on an event-triggered mechanism under replay attack according to claim 2, characterized in that, The data transmitted considering the influence of replay attack is expressed as follows: Among them, when it means that the replay attack has not occurred, and when it means that the replay attack has occurred, where represents the transmitted data after passing through the event trigger mechanism, represents the past signal injected by the attacker at time t recorded, represents that the replayed data is the data transmitted within the previous seconds.

4. A control method for a T-S fuzzy system based on an event-triggered mechanism under replay attacks according to claim 1, characterized in that, In step S4, for a given positive parameter , the time-delay parameter , the triggering parameter , the matrix , if there exists , a free-weighting matrix with appropriate dimensions, is an arbitrary matrix, and for any , if the following inequality holds, then the system is asymptotically stable: ； ； 。 5. A control method for a T-S fuzzy system based on an event-triggered mechanism under replay attack according to claim 4, characterized in that, In step S5, for a given positive parameter , the time-delay parameter and , the triggering parameter , if there exists , a free-weighting matrix with appropriate dimensions, is an arbitrary matrix, and for any , the following inequality holds, then the system is asymptotically stable: ； ； And the expression for the controller gain is obtained as follows: .

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