2-D system probabilistic security filtering method under relay network spoofing attack

Abstract

A 2-D system probabilistic security filtering method under relay network spoofing attack comprises the four steps of establishing a state space model of a 2-D system, establishing an amplification forwarding relay communication model with spoofing attack, designing a security filter and solving an optimal filter gain matrix. By means of a 2-D mathematical induction method and a stochastic analysis method, the designed filtering method can filter a 2-D system with a given guarantee probability in a complex communication scene of random noise, long-distance transmission and hostile attack. And more schemes are provided for the filtering problem of a complex system in practical engineering while the filtering precision is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of filtering, and specifically relates to a 2-D system probability-preserving secure filtering method under relay network spoofing attacks. Background Art

[0002] Due to the two-way dynamic characteristics that can evolve along two independent directions, 2-D systems exhibit unique advantages in modeling complex processes, such as digital image processing, industrial heat treatment, iterative learning control, etc., and have important research value in practical engineering applications. At the same time, as a core issue in the fields of signal processing and control engineering, filtering technology analyzes dynamic behavior by estimating the system state, which is particularly crucial for 2-D systems. Set-membership filtering, as an important method for solving the problem of system state estimation under the influence of bounded noise, describes the noise by introducing an ellipsoidal constraint, aiming to obtain an ellipsoidal domain containing the system state. However, in practical applications, in addition to bounded noise, random noise also widely exists, and at this time, traditional set-membership filtering techniques are difficult to handle. In addition, due to the unpredictability of equipment failures and signal interference, perfect state estimation with zero error is often difficult to achieve in real engineering. Therefore, a more practical engineering method is usually a probability guarantee strategy. For example, in a missile control system, it is required that the deviation error be limited within a preset range with a given probability. However, currently in the research of 2-D systems, this demand for probability guarantee filtering has not received sufficient attention, and existing set-membership filtering techniques cannot effectively solve this problem.

[0003] At the same time, due to its low cost and high efficiency, network communication has become the main data transmission means to break through geographical restrictions. However, limited by equipment technology, sensors are prone to signal attenuation during long-distance transmission. At the same time, long-distance wireless transmission is more vulnerable to malicious attacks. Among them, spoofing attacks destroy data integrity by injecting false data. Compared with denial-of-service attacks that directly block data transmission, its concealment is stronger, which not only brings security risks but may also cause serious economic losses, posing a great threat to the reliable operation of the system. Existing network filtering technologies mainly focus on simple network-induced phenomena, and in the face of a complex communication environment with both long-distance transmission and malicious attacks, traditional methods are no longer sufficient.

[0004] The differences between the present application and the prior art are as follows:

[0005] Technical comparison with the patent CN116341229A "Recursive Filtering Method for State Saturation Systems Based on High-Speed Networks under the Influence of Spoofing Attacks";

[0006] 1. Patent CN116341229A uses a recursive method to study the filtering problem of a state-saturated system under the influence of spoofing attacks. By using a high-speed network to construct a new measurement model, an optimal recursive filter method is constructed. This study mainly focuses on the filtering problem in a long-distance wireless communication environment. The measurement signal is amplified and forwarded to the filter using a relay communication protocol, greatly alleviating the signal attenuation problem in a complex network environment. Compared with Patent CN116341229A, this study mainly targets 2-D systems. Compared with ordinary 1-D systems, its dynamics evolve along two directions and are suitable for modeling various complex industrial processes, making it more suitable for the requirements of modern industry for long-distance wireless network transmission.

[0007] 2. The recursive filtering method studied in Patent CN116341229A gives results in the sense of expectation under the influence of random noise, but its practical value in actual engineering is not high and it is not intuitive. This study proposes a probability-preserving filtering method that can not only handle random noise but also further study bounded noise. By using an ellipsoidal constraint to describe the filtering error and introducing a probability index to characterize the possibility that the filtering error falls within the ellipsoidal constraint, it solves the problem that the design idea of traditional filtering methods aiming at complete accuracy in random problems leads to unreliable expected results, providing more solutions for filtering problems in actual engineering.

[0008] Based on the above analysis, it is necessary to study the probability-preserving security filtering method technology for 2-D systems under relay network spoofing attacks to solve the state estimation problem of 2-D systems under long-distance wireless transmission and malicious attacks under the requirement of probability guarantee. Summary of the Invention

[0009] Aiming at the probability-preserving estimation problem of 2-D systems in the complex scenario of long-distance wireless transmission with malicious attacks in actual engineering, the present invention proposes a probability-preserving security filtering method for 2-D systems under relay network spoofing attacks. Through four steps: establishing the state space model of the 2-D system, establishing the amplify-and-forward relay communication model with spoofing attacks, designing a security filter, and solving the optimal filter gain matrix, using recursive linear matrix inequalities, 2-D mathematical induction, and stochastic analysis methods, the designed filtering method can achieve filtering of the 2-D system with a given guaranteed probability in complex communication scenarios of random noise, long-distance transmission, and malicious attacks. While ensuring the filtering accuracy, it provides more solutions for the filtering problems of complex systems in actual engineering.

[0010] To achieve the above object, the technical solution adopted by the present invention is:

[0011] The probability-preserving security filtering method for 2-D systems under relay network spoofing attacks includes the following steps:

[0012] S1. Establish the state - space model of the 2 - D system:

[0013] The state - space model includes a state equation and a measurement equation, both of which evolve along two directions. The state equation is a 2 - D difference equation system used to describe the evolution of the internal dynamic variables of the system; the measurement equation is a 2 - D algebraic equation system used to obtain the sampled values of the internal dynamic monitoring of the system by the sensor.

[0014] S2. Construct an amplify - and - forward relay communication model with spoofing attacks:

[0015] Based on step S1, the measured signal is sent to the filter through the amplify - and - forward channel or the direct - transmission channel of the relay node. At the same time, it is considered that spoofing attacks will randomly inject bounded false data during the signal transmission process, thus affecting the signal accuracy.

[0016] S3. Design a security filter:

[0017] Fuse the two signals received by the filter after step S2 and design a probability - guaranteed security filter based on this.

[0018] S4. Solve the optimal filter gain matrix:

[0019] On the premise of meeting the probability requirements, use the recursive linear matrix inequality method to solve the ellipsoidal constraint containing the filtering error, and obtain the optimal filter gain by minimizing the shape matrix of the ellipsoidal constraint, thereby realizing the probability - guaranteed security filtering of the 2 - D system under spoofing attacks in the relay network.

[0020] As a further improvement of the present invention, in step S1, in the state model of the 2 - D linear system, the state equation is:

[0021]

[0022] y j,k =C j,k x j,k +v j,k .

[0023] Where represent the horizontal coordinate and the vertical coordinate respectively, M is an integer representing the finite time domain; and are the system state variable and the observation variable respectively; and are the process noise and the observation noise respectively, both of which are zero - mean Gaussian white noises, and their covariance matrices are and And C j,kis a known time-varying matrix and dimensionally adapted. The initial boundary conditions of the system have the following statistical properties. For

[0024]

[0025] Cov{x 0,k ,x 0,κ} = F 0,k δ(k,κ), Cov{x j,0 ,x 0,k} = F 0,0 δ(j,0)δ(0,k)

[0026] where is a known conformable vector, and F j,0 , F 0,k are known conformable matrices.

[0027] As a further improvement of the present invention, in step S2, for the measurement output y j,k construct an amplify-and-forward relay communication model with spoofing attacks. The specific communication process is as follows:

[0028] The measurement output y j,k reaches the filter through two channels, namely the amplify-and-forward channel of sensor-relay-filter and the direct channel of sensor-filter. Let and respectively represent the transmission signals of the sensor in the sensor-relay and sensor-filter channels, be the transmission signal of the relay node in the relay-filter channel. and The dynamic characteristics of are described as:

[0029]

[0030] where and are transmission amplitude parameters, and are three known channel transmission power parameters; l j,k is a given positive scalar representing the amplification factor of the relay node; is the received signal of the relay node; is zero-mean Gaussian white noise, and their covariance matrices are respectively and and are white random processes describing the fading degree of the signal transmission channel, satisfying:

[0031]

[0032] wherein and are known parameters;

[0033] Under the influence of spoofing attacks, the signals actually received by the relay node and the filter through three channels can be modeled as:

[0034]

[0035] where τ j,k , ξ j,k and ρ j,k are the false data injected by the attacker. Due to the limited attack energy, the false data satisfies norm boundedness and where and are known positive scalars, and the white random processes α j,k , β j,k and γ j,k describe the occurrence of attack events in the channel and follow the following Bernoulli distribution:

[0036]

[0037] Correspondingly, its statistical characteristics are:

[0038]

[0039] As a further improvement of the present invention, in step S3, the measurement signals from the relay forwarding channel and the direct channel are fused, and a corresponding probability-preserving safety filter is designed as follows:

[0040] Based on the hybrid relay network model, the filter finally receives the measurement signals from the relay node's amplify-and-forward and the sensor's direct transmission, that is and Define the augmented vector It can be obtained that

[0041]

[0042] where

[0043]

[0044] and and

[0045] Based on the received augmented signal, a safety filter with the following form is constructed for the augmented system:

[0046]

[0047] Among them

[0048]

[0049] And And are the estimation vectors, is the filter gain matrix to be designed. For the filter, assume its boundary initial conditions are And Among them And are known vector functions and satisfy

[0050] As a further improvement of the present invention, in step S4, under the given probability requirement, the optimal filter gain matrix is calculated using the recursive linear matrix inequality, and the ellipsoidal constraint including the estimation error is minimized. The specific design is as follows:

[0051] S41. For all satisfying Set the random variables And are uncorrelated. In addition, set them to be independent of all initial boundary conditions x j,0 and x 0,k , are all independent;

[0052] S42. Set the initial states of the augmented system and the filter so that the initial value of the error vector satisfies

[0053]

[0054] where P j,0 >0 and P 0,k >0 are known proper-dimensional matrices;

[0055] S43. Combining the definitions of the matrix Ω j,k and the vector in step S3, obtain the matrix

[0056]

[0057] where;

[0058]

[0059] And The matrix and Π j,k are decomposed into

[0060] ​

[0061] where is the d-th eigenvalue of the matrix , and is the corresponding eigenvector, with d = 1, 2,..., n w , similarly is the i-th eigenvalue of the matrix Π j,k , and is the corresponding eigenvector, with i = 1, 2,..., 2n y ;

[0062] S44. Construct a recursive linear matrix inequality to calculate the filter gain matrix. Let P j,0 =(1 - p)R j,0 , P 0,k =(1 - p)R 0,k , where p ∈ (0, 1) is a given probability scalar. For a given sequence of positive scalars If there exist a sequence of positive definite matrices a sequence of matrices where and a sequence of positive scalars and a sequence of non - negative scalars such that the following time - varying recursive linear matrix inequality holds, then the inequality holds for all :

[0063]

[0064] where

[0065]

[0066] and

[0067]

[0068] In addition

[0069]

[0070] Among them, L j,k is an orthogonal factor of the matrix (1 - p)R j,k , that is and and

[0071] S45. Minimize the ellipsoidal constraint containing the estimation error to obtain the optimal filter gain. Under the condition that the recursive linear matrix inequality holds in step S44, solving the following optimization problem can obtain the minimum ellipsoidal constraint and the corresponding optimal filter gain:

[0072]

[0073] subject to the conditions in step S44;

[0074] Furthermore, complete the setting of the gain parameters of the optimal filter.

[0075] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0076] The present invention provides a method and system for guaranteed-probability secure filtering of a 2-D system under relay network spoofing attacks. When establishing the state-space model of the 2-D system, the proposed method can characterize the dynamics of the system from two direction variables and consider the influence of random Gaussian process noise, so it is more accurate and reliable. When establishing the amplify-and-forward relay communication model with spoofing attacks, the probability of attack occurrence, the bound of false data, and the transmission attenuation rate considered for each channel are different, which can make the constructed model more general. When designing the secure filter, the signals relayed by the relay and the direct signals of the sensor are fused, enabling the filter to utilize more information and improving the reliability of the filter. In addition, when solving the optimal filter gain matrix, the present invention uses an ellipsoidal constraint to describe the estimation error and, with the help of the recursive linear matrix inequality method, minimizes the ellipsoidal constraint to obtain the optimal filter gain under the premise of ensuring the probability requirement, thus ensuring the filtering accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] Figure 1 is the flowchart of the steps of the method for guaranteed-probability secure filtering of a 2-D system under relay network spoofing attacks of the present invention;

[0078] Figure 2 is the framework diagram of the filtering technology in the present invention;

[0079] Figure 3 is the schematic diagram of the estimation effect of the test example of the present invention Figure 1 ;

[0080] Figure 4 is the schematic diagram of the estimation effect of the test example of the present invention Figure 2 ;

[0081] Figure 5 is the schematic diagram of the estimation effect of the test example of the present invention Figure 3 ;

[0082] Figure 6 is the schematic diagram of the estimation effect of the test example of the present invention Figure 4。 Detailed implementation manners

[0083] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners:

[0084] Embodiment

[0085] The method for guaranteed-probability secure filtering of a 2-D system under relay network spoofing attacks, as Figure 1 shown, includes the following steps:

[0086] Step S1: Establish a state-space model of the 2-D system.

[0087] The state of the 2-D system evolves along two directions, and its state equation is:

[0088]

[0089] y j,k = C j,k x j,k + v j,k .

[0090] Wherein respectively represent the horizontal coordinate and the vertical coordinate, and M is an integer representing a finite time domain; and are the system state variable and the observation variable respectively; and are the process noise and the observation noise respectively, and both are zero-mean Gaussian white noises, and their covariance matrices are and And C j,k is a known time-varying matrix and is dimensionally adapted. The initial boundary conditions of the system have the following statistical characteristics (for ):

[0091]

[0092] Cov{x 0,k , x 0,κ} = F 0,k δ(k, κ), Cov{x j,0 , x 0,k} = F 0,0 δ(j, 0)δ(0, k)

[0093] Wherein is a known vector of appropriate dimension, and F j,0 , F 0,k are known matrices of appropriate dimension.

[0094] Step S2: Establish an amplify-and-forward relay communication model with spoofing attacks.

[0095] As Figure 2 shown for the measurement output y j,k Construct an amplify-and-forward relay communication model with spoofing attacks. The specific communication process is as follows:

[0096] The measurement output y j,k Reaches the filter through two channels, namely the amplify-and-forward channel of sensor-relay-filter and the direct channel of sensor-filter. Let and Represent the transmitted signals of the sensor in the sensor-relay and sensor-filter channels respectively, Be the transmitted signal of the relay node in the relay-filter channel. And The dynamic characteristics of are described as:

[0097]

[0098] Where and Are the transmission amplitude parameters, and Are three known channel transmission power parameters; l j,k Is a given positive scalar representing the amplification factor of the relay node; Is the received signal of the relay node; Is zero-mean Gaussian white noise, and its covariance matrices are respectively and and Are white random processes describing the fading degree of the signal transmission channel, and they satisfy:

[0099]

[0100] Where and Are known parameters.

[0101] Under the influence of spoofing attacks, the signals actually received by the relay node and the filter through three channels can be modeled as:

[0102]

[0103] Where τ j,k , ξ j,k and ρ j,k Are the false data injected by the attacker. Due to the limited attack energy, the false data satisfies the norm boundedness and Where and are known positive scalars. The white random processes α j,k , β j,k and γ j,k describe the occurrence of attack events in the channel and follow the following Bernoulli distribution:

[0104]

[0105] Correspondingly, its statistical characteristics are:

[0106]

[0107] Step S3: Design a secure filter.

[0108] Fuse the measurement signals from the relay forwarding channel and the direct channel, and design the corresponding probability-preserving secure filter based on this. The specific design is as follows:

[0109] Based on the hybrid relay network model, the filter can finally receive the measurement signals from the relay node's amplify-and-forward and the sensor's direct transmission (i.e., and ). Define the augmented vector to obtain

[0110]

[0111] where

[0112]

[0113] and and

[0114] Based on the received augmented signal, we construct a secure filter with the following form for the augmented system:

[0115]

[0116] where

[0117]

[0118] and and is the estimation vector, is the filter gain matrix to be designed. For the filter, assume its boundary initial conditions are and where and are known vector functions and satisfy

[0119] Step S4: Solve for the optimal filter gain matrix.

[0120] First, for all that satisfy Set the random variables and to be uncorrelated. Additionally, set them to be independent of all initial boundary conditions \(x\) j,0 and as well.

[0121] Then set the initial states of the augmented system and the filter such that the error vector has an initial value that satisfies

[0122]

[0123] where \(P\) j,0 > 0 and \(P\) 0,k > 0 are known well - dimensioned matrices.

[0124] Next, combine with the definitions of the matrix \(\Omega\) j,k and the vector in Step S3 to obtain the matrix

[0125]

[0126] where

[0127]

[0128] And The matrix and \(\Pi\) j,k can be decomposed as

[0129]

[0130] where is the \(d\) - th eigenvalue of the matrix , is its corresponding eigenvector, \(d = 1,2,\cdots,n\) w . Similarly, is the \(i\) - th eigenvalue of the matrix \(\Pi\) j,k , is its corresponding eigenvector, \(i = 1,2,\cdots,2n\) y .

[0131] Subsequently, construct a recursive linear matrix inequality to calculate the filter gain matrix. Let \(P\) j,0 =(1 - p)R j,0 , \(P\) 0,k =(1 - p)R 0,k , where \(p\in(0,1)\) is a given probability scalar. For a given sequence of positive scalars If there exists a sequence of positive definite matrices A sequence of matrices where and a sequence of positive scalars and a sequence of non - negative scalars such that the following time - varying recursive linear matrix inequality holds, then the inequality For all holds:

[0132]

[0133] where

[0134]

[0135] and

[0136]

[0137] In addition

[0138]

[0139] Among them, L j,k is an orthogonal factor of the matrix (1 - p)R j,k i.e., and and

[0140] Finally, minimize the ellipsoidal constraint containing the estimation error to obtain the optimal filter gain. Under the condition that the recursive linear matrix inequality holds in step S44, solving the following optimization problem can obtain the minimum ellipsoidal constraint and the corresponding optimal filter gain:

[0141]

[0142] subject to the conditions in step S44.

[0143] Furthermore, complete the setting of the gain parameters of the optimal filter.

[0144] Test example

[0145] To verify the effectiveness of the method proposed in the present invention, the following test experiment is specifically made. During the experiment: Use the Darboux equation commonly used in actual dynamic processes such as air drying, water flow heating, and gas absorption for verification, and its partial differential equation form is as follows:

[0146]

[0147] where s ∈ [0, S] represents the spatial position, t ∈ [0, T] represents the time, us,t is the variable to be estimated, S and T are known positive constants, ∈ s,t represents the input function, θ (0) , θ (1) , θ (2) and are known real coefficients.

[0148] By defining a new variable When , the above equation can be transformed into the following system of first-order partial differential equations

[0149]

[0150] Select an appropriate step size Δ s and Δ t , and perform a forward difference approximation on the above formula. The original partial differential equation can be expressed as the following discrete-time 2-D system type:

[0151] x j+1,k+1 = A (1) x j,k+1 + A (2) x j+1,k

[0152] where

[0153]

[0154] In fact, environmental noise will affect the system due to physical defects. At the same time, time-varying parameters can better describe the changes in the operating state. Therefore, we set Δs = 0.2, Δt = 0.2, θ (0) j,k = 0.3 + 0.5cos(0.8j), θ (1) j,k = -1, θ (2) j,k = -1, and finally the system matrix

[0155]

[0156] The environmental noise w j,k is zero-mean Gaussian white noise with covariance In the measurement output, the measurement matrix C j,k = [0.35 0.45 -0.1sin(5j)], and the covariance of the noise v j,k is selected as In this example, take M = 30.

[0157] A hybrid relay network affected by spoofing attacks is adopted. The relevant parameters of each channel of this relay network are set as:

[0158]

[0159] Noise per channel and All obey zero-mean Gaussian distribution, and their covariances are The expected value of the spoofing attack is set to The bounded false data injected by the attack is set to τ j,k =ξ j,k =ρ j,k =0.1sin(5(j+k)), its boundary value is easy to obtain Given the initial conditions of the augmented system and filter:

[0160]

[0161] Let P j,0 =P 0,k =0.5I2. Take probability p=0.8 and slack variable According to the filtering method proposed in the present invention, the optimal filter gain is recursively calculated using MATLAB software and compared with the actual state trajectory.

[0162] Figure 3 and Figure 4 Shows the state component and its corresponding estimated value of the changing trends, among which and Respectively represent the state x j,k The first and second components of . It can be seen that even if the system state trajectory shows a divergent trend, the filter can still achieve close tracking. This shows that the designed security filter can effectively estimate the 2-D system with relay network deception attack under the premise of guaranteeing probability p.

[0163] Figure 5 and Figure 6 Shows the estimated error state e j,k The amount and The Monte Carlo technique is used to perform 50 independent simulation experiments and take the average value. The convergence trend further verifies the effectiveness of the proposed method.

[0164] In summary, the method of the present invention discloses a 2-D system probability-preserving secure filtering method under relay network spoofing attacks. For the complex scenario of long-distance wireless transmission with malicious attacks, a relay network architecture with an amplify-and-forward strategy is constructed, and a scenario with spoofing attacks is considered to construct a secure filter. A criterion in the form of a recursive linear matrix inequality is derived by means of 2-D mathematical induction. This criterion can not only ensure that the estimated value is distributed in the ellipsoidal region of the system state neighborhood with a preset probability, but also obtain the optimal filter gain parameter by minimizing the trace value of the ellipsoidal shape matrix, thereby achieving the optimal estimation effect.

[0165] The above description is only a preferred embodiment of the present invention and does not impose any other form of limitation on the present invention. Any modification or equivalent change made based on the technical essence of the present invention still falls within the scope claimed by the present invention.

Claims

1. Probability-preserving secure filtering method for 2-D systems under relay network spoofing attacks, characterized in that It includes the following steps: S1. Establish the state-space model of the 2-D system: The state-space model includes a state equation and a measurement equation, both of which evolve along two directions. The state equation is a 2-D difference equation system used to describe the evolution of the internal dynamic variables of the system; the measurement equation is a 2-D algebraic equation system used to obtain the sampling values of the sensors for the internal dynamic monitoring of the system. S2. Construct an amplify-and-forward relay communication model with spoofing attacks: Based on step S1, the measurement signal is sent to the filter through the relay node amplify-and-forward channel or the direct channel. At the same time, it is considered that spoofing attacks will randomly inject bounded false data during the signal transmission process, thus affecting the signal accuracy. S3. Design a secure filter: Fuse the two signals received by the filter after step S2 and design a probability-preserving secure filter based on this. S4. Solve the optimal filter gain matrix: On the premise of meeting the probability requirements, use the recursive linear matrix inequality method to solve the ellipsoidal constraint containing the filtering error, and obtain the optimal filter gain by minimizing the shape matrix of the ellipsoidal constraint, so as to realize the probability-preserving secure filtering of the 2-D system under spoofing attacks in the relay network.

2. The method for guaranteed-probability secure filtering of a 2-D system under relay network spoofing attacks according to claim 1, wherein In step S1, in the state model of the 2-D linear system, the state equation is: where represent the horizontal and vertical coordinates respectively, and M is an integer representing a finite time domain; and are the system state variable and the observation variable respectively; and are the process noise and the observation noise respectively, both of which are zero-mean Gaussian white noises, and their covariance matrices are and and C j,k are known time-varying matrices and dimensionally compatible. The initial boundary conditions of the system have the following statistical characteristics, for wherein is a known dimension-appropriate vector, and F j,0 F 0,k is a known dimension-appropriate matrix.

3. The method for guaranteed-probability secure filtering of a 2-D system under relay network spoofing attacks according to claim 1, wherein In the step S2, for the measurement output y j,k Construct an amplify-and-forward relay communication model with spoofing attacks. The specific communication process is as follows: Measured output y j,k It reaches the filter through two channels, namely the amplify-and-forward channel of sensor-relay-filter and the direct channel of sensor-filter. Let and represent the transmitted signals of the sensor in the sensor-relay and sensor-filter channels respectively, is the transmitted signal of the relay node in the relay-filter channel. and The dynamic characteristics of are described as: where and are the transmission amplitude parameters, and are three known channel transmission power parameters; l j,k is a given positive scalar representing the amplification factor of the relay node; is the signal received by the relay node; is zero-mean Gaussian white noise, and its covariance matrices are respectively and and is a white random process describing the fading degree of the signal transmission channel and satisfies: wherein and are known parameters; Under the influence of spoofing attacks, the signals actually received by the relay node and the filter through three channels can be modeled as: where τ j,k , ξ j,k and ρ j,k are the false data injected by the attacker. Due to the limited attack energy, the false data satisfies the norm boundedness and where and are known positive scalars, and the white random processes α j,k , β j,k and γ j,k describe the occurrence of attack events in the channel and follow the following Bernoulli distribution: Correspondingly, its statistical characteristics are:

4. The method for guaranteed probability secure filtering of a 2-D system under relay network spoofing attacks according to claim 1, wherein In step S3, fuse the measurement signals from the relay forwarding channel and the direct channel and design the corresponding probability-preserving secure filter based on this. The specific design is as follows: Based on the hybrid relay network model, the filter finally receives the measurement signals amplified and forwarded by the relay node and directly sent by the sensor, that is and Define the augmented vector It can be obtained that where and and Based on the received augmented signal, construct a secure filter of the following form for the augmented system: where and is the estimated vector, is the filter gain matrix to be designed. For the filter, assume its boundary initial conditions are and where and are known vector functions and satisfy 5. The method for guaranteed probability secure filtering of a 2-D system under relay network spoofing attacks according to claim 1, wherein In step S4, under the given probability requirements, use the recursive linear matrix inequality to calculate the optimal filter gain matrix and minimize the ellipsoidal constraint containing the estimation error. The specific design is as follows: S41. For all that satisfy Set random variables and are uncorrelated. In addition, set them to be independent of all initial boundary conditions x j,0 and x 0,k , all independent; S42. Set the initial states of the augmented system and the filter such that the initial value of the error vector satisfies where P j,0 > 0 and P 0,k > 0 are known dimension-adaptive matrices; S43. Combine with the definitions of matrix Ω j,k and vector in step S3 to obtain the matrix where; And matrix and Π j,k are decomposed into where is the d-th eigenvalue of the matrix , and is the corresponding eigenvector, with d = 1, 2, ..., n w . Similarly is the i-th eigenvalue of the matrix Π j,k , and is the corresponding eigenvector, with i = 1, 2, ..., 2n y ; S44. Construct a recursive linear matrix inequality to calculate the filter gain matrix. Let P in step S42 j,0 =(1 - p)R j,0 , P 0,k =(1 - p)R 0,k , where p ∈ (0, 1) is a given probability scalar. For a given positive scalar sequence If there exist a sequence of positive definite matrices a sequence of matrices where and a sequence of positive scalars and a sequence of non - negative scalars such that the following time - varying recursive linear matrix inequality holds, then the inequality holds for all : where and and also Among these, L j,k is an orthogonal factor of the matrix (1 - p)R j,k , that is and and S45. Minimize the ellipsoidal constraint containing the estimation error and obtain the optimal filter gain. Under the condition that the recursive linear matrix inequality holds in step S44, solving the following optimization problem can obtain the minimum ellipsoidal constraint and the corresponding optimal filter gain: subject to the conditions in step S44; Furthermore, complete the setting of the gain parameters of the optimal filter.

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