A distributed probability preserving estimation method for wireless sensor networks under a deception attack

Abstract

The application discloses a wireless sensor network distributed guaranteed probability estimation method under a deception attack, and specifically comprises the following steps: a nonlinear system mathematical model under the condition that innovation is deceived is established; a neural network-based distributed state estimator structure model is established according to the nonlinear system mathematical model under the condition that innovation is deceived; a neural network weight matrix adaptive updating strategy is designed; a distributed guaranteed probability state estimator gain is designed; and estimator parameters are optimized to achieve local optimal estimation performance; the application comprehensively considers the nonlinear characteristics and the influence of malicious attacks existing in actual engineering, can be widely applied to military monitoring systems and environmental monitoring systems, estimates the state of a monitoring or monitoring object, is more in line with engineering application, and is high in practicality.

Description

I. TECHNICAL FIELD

[0001] The present application relates to a system state estimation method, in particular to a wireless sensor network distributed guaranteed probability estimation method under deception attack. II. BACKGROUND

[0002] At present, the most typical distributed state estimation is still Kalman filter and H∞ method. However, the performance of Kalman filter depends largely on whether the noise meets the Gaussian assumption, and H∞ method is only applicable to energy-bounded disturbance. However, in many practical engineering scenarios, the Gaussian assumption and the energy-bounded assumption cannot fully reflect the real noise characteristics, which makes the existing technology greatly limited in application. In addition, for the form of deception attack, it is generally assumed that the false signal is known and linear, but this assumption is often unrealistic in actual engineering application scenarios, so the existing method cannot well cope with real network attacks. At the same time, in engineering practice, the guaranteed probability design principle has the advantages of high flexibility and has been widely used in various engineering fields. III. SUMMARY

[0003] In view of the above problems, the purpose of the present application is to solve the technical problems of the prior art under the deception attack form.

[0004] The present application provides a wireless sensor network distributed guaranteed probability estimation method under deception attack, and the specific steps are as follows:

[0005] Step 1, establish a nonlinear system mathematical model under the condition that the innovation is deceived;

[0006] Step 2, establish a neural network-based distributed state estimator structure model according to the nonlinear system mathematical model under the condition that the innovation is deceived;

[0007] Step 3, design a neural network weight matrix adaptive update strategy;

[0008] Step 4, design a distributed guaranteed probability state estimator gain;

[0009] Step 5, optimize the estimator parameters to achieve local optimal estimation performance;

[0010] Optionally, the nonlinear system mathematical model under the condition that the innovation is deceived includes:

[0011]

[0012] wherein the system is established on the time domain [0, T], represents the state of the system at the current time k, represents the measurement output of the system at the current time k; Bk is one of known real matrices with appropriate dimensions, E i,k is the second one of known real matrices with appropriate dimensions; f(x k ): is one of smooth nonlinear functions, g(x k ): is the second one of smooth nonlinear functions; and are process noise and measurement noise, respectively, satisfying the following constraint set:

[0013]

[0014] where V k > 0 is one of known matrices with appropriate dimensions, U k > 0 is the second one of known matrices with appropriate dimensions;

[0015] By using the Taylor series expansion method, the nonlinear terms f(x k ) and g i (x k ) in the system are expressed as

[0016]

[0017] where, represents the error caused by ignoring high-order terms in the Taylor series expansion of the nonlinear function f(x k ), represents the error caused by ignoring high-order terms in the Taylor series expansion of the nonlinear function g(x k ), is one of known matrices, is the second one of known matrices, is one of unknown matrices, is the second one of unknown matrices, which are used to describe the modeling errors in matrices Φ i,k and Ψ i,k , respectively, satisfying ||Δ 1i ||≤1 and ||Δ 2i ||≤1; Φ i,k and Ψ i,k are obtained by

[0018]

[0019] Optionally, a neural network-based distributed state estimator structure model is established according to the nonlinear system mathematical model under the condition of new information being deceived, including:

[0020]

[0021] in, It is the state estimate of node i, z i,k The output estimate is The latest news; F i,k G i,k and H ij,k These are the three parameters to be designed in the estimator; W is the weight matrix of the neural network. ij The estimated value, W ij W is the neighbor node of node i. j Set the ideal weight matrix; define the output estimate. And new information It is the fake news sent by node j under a spoofing attack, symbol The definition is as follows:

[0022]

[0023] Where, α k It is a random sequence that follows a Bernoulli distribution and is used to describe whether a network attack has occurred, satisfying the following conditions: Prob{·} represents the probability of an event occurring; It is the attacker based on the new information z obtained j,k The generated deception signal is injected into the original information, and the deception signal has the following form:

[0024]

[0025] Where, χ(·): It is an unknown nonlinear function defined on a compact set, and the following equation is used to approximate the unknown nonlinear function χ(z) using a neural network. j,k Specific methods:

[0026] χ(z j,k ) = W j φ(z j,k )+δ j,k (8)

[0027] Among them, W j It is the ideal weight matrix of the neural network, φ(·) is the activation function, and δ j,k The estimation error satisfies the following conditions:

[0028] ||W j || F ≤∈ 1j ,||φ(·)||≤∈2,||δ j,k ||≤∈ 3j (9)

[0029] where ∈ 1j is one of the known positive scalars, ∈2is one of the known positive scalars, ∈ 3j is one of the known positive scalars;

[0030] Definition is the estimation error of the state of the observed object in the wireless sensor network, is the update error of the neural network, then the initial condition and satisfies

[0031]

[0032] where Q ij,0 is one of the known positive definite real matrices, P0is one of the known positive definite real matrices;

[0033] The one-step estimation error of the system state is obtained by

[0034]

[0035] Intermediate definition:

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046] The one-step estimation error is expressed according to the intermediate definition:

[0047]

[0048] where, When has θ ij = 0, then is a sparse matrix, expressed as

[0049]

[0050] wherein Furthermore, for two block matrices of appropriate dimensions, one A = [A ij ] N×N and the other B = [B ij ] N×N , defined as

[0051] Optionally, the neural network weight matrix adaptive updating strategy is designed, including:

[0052]

[0053] wherein the cost function of the neural network is defined as

[0054]

[0055] defined as

[0056]

[0057] is one of the self-tuning scalar parameters, is the second self-tuning scalar parameter, and the specific method of designing the parameters and is as follows:

[0058] Given a matrix sequence {Q ij,k} k∈[0,T] ; if there exists a positive scalar sequence parameters are obtained by solving the following linear matrix inequality:

[0059]

[0060] wherein, M ij,k is the factorization of the matrix , i.e. If the linear matrix inequality is established, then the following inequality is established

[0061]

[0062] 5. The wireless sensor network distributed guaranteed probability estimation method under the fraud attack according to claim 1, characterized in that a distributed guaranteed probability state estimator gain is designed, including:

[0063] 1. The state estimation error defined at each node needs to satisfy the ellipsoidal constraint with a probability p as follows:

[0064]

[0065] Define the system state estimation error Then the constraint can also be equivalently expressed as

[0066]

[0067] where, is a pre-specified matrix, p is a pre-specified scalar satisfying 0 < p < 1, is an ellipsoid in , and represents the probability of the event happening, is defined as follows:

[0068]

[0069] where, represents the center of the ellipsoid, Y > 0 is a positive definite matrix of appropriate dimension to depict the shape of the ellipsoid.

[0070] A sufficient condition for the state estimation error to converge to an allowed ellipsoidal region with a pre-specified probability is established:

[0071] Given the three gain matrices F i,k , G i,k and H ij,k of the estimator, a sufficient condition for the update error of the neural network to be bounded in the weighted Frobenius norm sense is established under the premise that there exists a positive definite matrix sequence {P k} k∈[0,T-1] with a factorized form, if there exists a sequence of non-negative definite scalars satisfying the following recursive matrix inequality:

[0072]

[0073] where,

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] Then the following inequality holds:

[0081]

[0082] 2. The method for solving the estimator gain parameters is as follows:

[0083] Given a pre-specified \(0 < p < 1\) and a series of positive definite matrices If there exists a series of non-negative scalars Satisfying

[0084]

[0085] Then the estimator parameters \(F\) i,k , \(G\) i,k and \(H\) ij,k Are obtained by solving this linear matrix inequality;

[0086] 3. The following algorithm is given to iteratively obtain the required estimator parameters \(\{F\) i,k , \(G\) i,k , \(H\) ij,k \}:

[0087] Input: Topological information The initial value \(x\) i of \(x\) i,0 , the initial value \(W\) ij of \(W\) ij,0 , The initial value of The initial value \(Q\) ij of \(Q\)[[ID=​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

[0090] iii. If k < T, return ii to continue execution, otherwise exit.

[0091] Optionally, the estimator parameters are optimized to achieve a locally optimal estimation performance, including:

[0092] Among the obtained estimator parameter sets, the following three optimization problems are proposed from different performance requirements respectively to seek locally optimal estimator parameters:

[0093] 1. Minimize the trace of matrix Q ij,k to ensure that the weighted Frobenius norm of the neural network weight matrix estimation error is minimized, and then the following Conjecture 1 is obtained:

[0094] Under the premise that the sufficient condition for the boundedness of the update error of the neural network in the weighted Frobenius norm is established, when the following minimization problem has a feasible solution:

[0095]

[0096] subjectto (14)

[0097] The minimum value of the trace of the matrix sequence {Q ij,k} k∈[0,T] can be achieved.

[0098] 2. Minimize the trace of matrix to ensure the optimal estimation performance under the constraint of a pre-specified fixed probability;

[0099] For simplicity of expression, let then the following Conjecture 2 is obtained:

[0100] Given the probability p, the matrix is bounded in the weighted Frobenius norm under the premise that the sufficient condition is established, and based on Conjecture 1, when the following minimization problem has a feasible solution:

[0101]

[0102] subjectto (19)

[0103] The minimum value of the trace of the matrix sequence can be achieved;

[0104] 3. Minimize q k under the fixed constraint to ensure that the probability performance indicator at each time has a lower bound;

[0105] Assuming that p is time-varying, p kTo define the probabilistic performance index at time k, define Therefore, we have the following corollary 3:

[0106] Given a probability p, and assuming the update error of the neural network is bounded in the sense of the weighted Frobenius norm, based on Corollary 1, when the following minimization problem has a feasible solution:

[0107]

[0108]

[0109] Able to ensure p k There is a lower realm.

[0110] The system considered in this invention is described by nonlinear difference equations, which have greater versatility and can more comprehensively and realistically reflect engineering applications. Secondly, a new transient performance index is defined for the system at each time step, enabling an appropriate description of the system dynamics within a finite time interval of interest. This application proposes a probabilistic design method that avoids the adverse effects of unnecessary strict constraints in practical applications and achieves the expected performance with satisfactory probability, providing greater flexibility. Finally, a joint estimation algorithm is provided, enabling simultaneous estimation of the system state and the innovations of the attack.

[0111] The present invention will now be further described with reference to the accompanying drawings. IV. Description of the attached drawings

[0112] Figure 1 This is a flowchart of a distributed probability-preserving estimation method for wireless sensor networks under spoofing attacks, provided in an embodiment of this application.

[0113] Figure 2 This is the system state x provided in the embodiments of this application. k The first component and estimated state The first component

[0114] Figure 3 This is the system state x provided in the embodiments of this application. k The second component and estimated state The second component

[0115] Figure 4 The estimation error provided in the embodiments of this application

[0116] Figure 5 The estimation error provided in the embodiments of this application V. DETAILED DESCRIPTION

[0117] The application will be described in detail below with reference to the accompanying drawings and examples.

[0118] As Figure 1 shown, the application includes the following steps:

[0119] Step 1, establish a nonlinear system mathematical model under the condition of innovation deception attack, which can describe:

[0120]

[0121] Where, the system is established on the time domain [0, T], represents the state of the system at the current time k, represents the measurement output of the system at the current time k; B k is one of known real matrices with appropriate dimensions, E i,k is the second one of known real matrices with appropriate dimensions; f(x k ): is one of smooth nonlinear functions, used to describe the nonlinear properties in the observation object, such as unmanned aerial vehicles, g(x k ): is the second one of smooth nonlinear functions, used to describe the nonlinear properties in the sensor system; and are process noise and measurement noise respectively, with the following assumptions:

[0122] Assumption 1, v k and μ k satisfy the following constraint set:

[0123]

[0124] Where, V k > 0 is one of known matrices with appropriate dimensions, U k > 0 is the second one of known matrices with appropriate dimensions.

[0125] Using Taylor series expansion method, the nonlinear terms f(x k ) and g i (x k ) in the system are expressed as

[0126]

[0127]

[0128] Where, Represents a nonlinear function f(x) k The error caused by neglecting higher-order terms in Taylor series expansion. Represents the nonlinear function g(x) k The error caused by neglecting higher-order terms in Taylor series expansion. It is one of the known matrices. It is the second known matrix. It is one of the unknown matrices. The second unknown matrix is ​​used to describe the uncertainty of matrix Φ. i,k and Ψ i,k The modeling error in the model satisfies ||Δ 1i ||≤1 and||Δ 2i ||≤1;Φ i,k and Ψ i,k Obtained from the following formula

[0129]

[0130] Step 2: Establish a distributed state estimator structure model based on neural networks based on the mathematical model of the nonlinear system under the condition of information being deceived.

[0131]

[0132] in, It is the state estimate of node i, z i,k The output estimate is The latest news; F i,k G i,k and H ij,k These are the three parameters to be designed in the estimator; W is the weight matrix of the neural network. ij The estimated value, W ij W is the neighbor node of node i. j Set the ideal weight matrix. Define the output estimate. And new information It is the fake news sent by node j under a spoofing attack, symbol The definition is as follows:

[0133]

[0134] Where, α k It is a random sequence that follows a Bernoulli distribution and is used to describe whether a network attack has occurred, satisfying the following conditions: Prob{·} represents the probability of an event occurring; It is the attacker based on the new information z obtained j,kThe generated deception signal is injected into the original information, and the deception signal has the following form:

[0135]

[0136] Where, χ(·): It is an unknown nonlinear function defined on a compact set, and the following equation is used to approximate the unknown nonlinear function χ(z) using a neural network. j,k Specific methods:

[0137] χ(z j,k ) = W j φ(z j,k )+δ j,k (9)

[0138] Among them, W j It is the ideal weight matrix of the neural network, φ(·) is the activation function, and δ j,k It is an estimation error, based on the following assumptions:

[0139] Assumption 2, W j ,φ(·),δ j,k The following conditions must be met respectively:

[0140] ||W j || F ≤∈ 1j ,||φ(·)||≤∈2,||δ j,k ||≤∈ 3j (10)

[0141] Where, ∈ 1j It is one of the known positive scalars, ∈2 is the second known positive scalar, ∈ 3j It is the third known positive scalar.

[0142] definition To account for the estimation error of the state of objects observed by wireless sensor networks, For the update error of the neural network, we have the following assumptions:

[0143] Assumption 3, Initial Conditions and satisfy

[0144]

[0145]

[0146] Among them, Q ij,0 P0 is one of the known positive definite real matrices, and P1 is another known positive definite real matrix. Therefore, the one-step estimation error of the system state can be obtained from the following equation.

[0147]

[0148] For more compact notation, we define

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157]

[0158]

[0159] Thus, the one-step estimation error can be written more simply as

[0160]

[0161] where When θ ij = 0, then is a sparse matrix, denoted as

[0162]

[0163] where In addition, for two block matrices of appropriate dimensions, one A = [A ij ] N×N and the other B = [B ij ] N×N A, is defined as

[0164] Step 3, design the adaptive update strategy of neural network weight matrix:

[0165]

[0166] where the cost function of the neural network is defined as

[0167]

[0168] Definition

[0169]

[0170] is one of the self-tuning scalar parameters, is the second self-tuning scalar parameter, design parameter and The specific method is:

[0171] Given a sequence of matrices {Q ij,k} k∈[0,T] If there exists a sequence of positive scalars parameters By solving the following linear matrix inequality:

[0172]

[0173] where, M ij,k is the factorization of the matrix , i.e. If the linear matrix inequality is true, then the following inequality is true

[0174]

[0175] That is, the matrix is bounded in the sense of weighted Frobenius norm. The specific content of the proof of the above method is:

[0176] By using the S-procedure and the Schur Complement Equivalence lemma, the linear matrix inequality (14) is obtained:

[0177] Lemma 1, S-procedure lemma

[0178] Define a series of quadratic functions about the variable κ0(·), κ1(·), …, κ ι (·), where If there exists a sequence of non-negative scalars {∈1, ∈2, …, ∈ ι} satisfying Then the following conclusions are obtained:

[0179] κ1(a)≤0,…,κ ι (a)≤0→κ0(a)≤0

[0180] Lemma 2, Schur Complement Equivalence Lemma

[0181] For matrices where and then the matrix inequality is equivalent to

[0182]

[0183] The proof is done by mathematical induction as follows:

[0184] i. According to assumption 3, at time k = 0, we have ii. Assume that at time k, the inequality holds.

[0185] iii. It is to be shown that at time k + 1, the inequality also holds under the given conditions:

[0186] From (13), we have

[0187]

[0188] Define the function as follows:

[0189]

[0190] where A (ι) denotes the i-th row of matrix A, then the inequality

[0191] can be equivalently expressed as:

[0192]

[0193] Therefore, consider we have

[0194]

[0195] where r ij,k satisfies

[0196]

[0197] Define a vector π ij,k as follows:

[0198]

[0199] Then, equation (16) can be rewritten as​

[0200]

[0201] Inequality Can be rewritten as

[0202]

[0203] Where,

[0204] Similarly, based on assumption 2, we have

[0205]

[0206] This inequality can be equivalently described as

[0207]

[0208] Where,

[0209] In addition, it is easy to deduce that

[0210]

[0211] Can be equivalently represented as

[0212]

[0213] And can be further represented as

[0214]

[0215] Therefore, using Lemma 1 (S-procedure Lemma), we know that if there exist positive definite scalars And Satisfying the following inequalities

[0216]

[0217] Then inequality (26) also holds.

[0218] Finally, according to Lemma 2 (Schur Complement Equivalence Lemma), we can get that inequality (28) holds is equivalent to inequality (14) holds.

[0219] Thus far, the design of the neural network weight matrix adaptive update strategy is completed.

[0220] Step 4, the specific process of designing the distributed probability preserving state estimator gain is:

[0221] 1) First, the state estimation error defined at each node needs to satisfy the following ellipsoidal constraint under a probability p:

[0222]

[0223] Define the system state estimation error Then this constraint can also be equivalently expressed as

[0224]

[0225] where, is a pre-specified matrix, p is a pre-specified scalar satisfying 0 < p < 1, is an ellipsoid within,​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

[0238] Then the following inequality holds:

[0239]

[0240] where, denotes the mathematical expectation, and thus, a sufficient condition is obtained for the system state estimation error to converge to an allowable ellipsoidal region with a pre-specified probability. The specific content of the above method is proved as follows:

[0241] By using the S-procedure and the Schur Complement Equivalence Lemma, the mathematical induction method is used for proof:

[0242] i. It is easy to know from Assumption 3 that

[0243]

[0244] ii. It is assumed that the following inequality holds at time k > 0

[0245]

[0246] iii. It is proved that under the given condition, the inequality (39) also holds at time k + 1:

[0247] From the inequality (41), it can be found that there is a vector satisfying

[0248]

[0249] Let and The above formula can be written as

[0250]

[0251] Then, formula (12) can be rewritten as

[0252] where,

[0253]

[0254]

[0255]

[0256] Under the condition of step 3, there is

[0257] ​

[0258] where the vector r ij,k satisfies

[0259] Definition

[0260]

[0261] then

[0262]

[0263] Thus we have

[0264]

[0265] where

[0266] Define a new vector ζ k is

[0267]

[0268] where The system state estimation error is defined as

[0269]

[0270] where

[0271]

[0272]

[0273]

[0274]

[0275]

[0276]

[0277]

[0278]

[0279]

[0280]

[0281]

[0282] Given that the following conditions hold

[0283]

[0284]

[0285]

[0286]

[0287]

[0288] ζ k Equivalently, we have

[0289]

[0290]

[0291]

[0292]

[0293]

[0294] In addition, we have 1i 2i

[0295]

[0296]

[0297] The above inequality can be equivalently expressed as

[0298]

[0299]

[0300] By Lemma 2 (Schur Complement Equivalence Lemma), we know that the inequality (32) holds if and only if the following inequalities hold

[0301]

[0302] Consider the statistical property of From the above inequality, we have

[0303]

[0304] Consider (33) and (46), the inequality (48) is equivalent to

[0305] ​​

[0306] According to Lemma 1 (S-procedure Lemma), we have

[0307]

[0308] or equivalently expressed as

[0309]

[0310] So far, the sufficient condition that the system state estimation error converges to an allowable ellipsoidal region with a pre-specified probability has been proved.

[0311] 2) The method to solve the estimator gain parameters is as follows:

[0312] Given a pre-specified 0 < p < 1 and a series of positive definite matrices If there exists a series of non-negative scalars satisfying

[0313]

[0314] then the estimator parameters F i,k , G i,k and H ij,k can be obtained by solving the linear matrix inequality. To prove the above algorithm, Lemma 3 and Lemma 4 are given as follows:

[0315] Lemma 3, given a random variable v with proper dimension, it is in an ellipsoidal domain as follows

[0316]

[0317] where a and Y are defined in step 3. If the following inequality holds for any given 0 < p < 1

[0318]

[0319] then there is

[0320]

[0321] Using Lemma 3, the following lemma is easily obtained

[0322] Lemma 4, if then the following inequality holds

[0323]

[0324] where the matrix P k is defined as

[0325] The specific method to prove the above method is to use Lemma 4 and the conclusion obtained in 1) and substitute them into Then the linear matrix inequality method for solving the estimator gain parameter can be proved.

[0326] 3) Give the following algorithm to iteratively obtain the required estimator parameters {F i,k , G i,k , H ij,k}: <00011​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​If the trace is such that the weighted Frobenius norm of the estimation error of the neural network weight matrix is ​​minimized, then Corollary 1 can be obtained as follows:

[0335] The matrix described in step 3 Given sufficient conditions for boundedness in the sense of the weighted Frobenius norm, the following minimization problems have feasible solutions:

[0336]

[0337] subjectto(14)

[0338] Able to achieve matrix sequence {Q ij,k} k∈[0,T] The minimum value of the trace.

[0339] 2) Minimize the matrix The trace is used to guarantee optimal estimation performance under pre-specified fixed probability constraints.

[0340] To simplify the expression, remember Therefore, we can derive Corollary 2 as follows:

[0341] Given probability p, the matrix described in step 3 Given that the boundedness in the sense of the weighted Frobenius norm holds, based on Corollary 1, when the following minimization problems have feasible solutions:

[0342]

[0343] subjectto(32)

[0344] Able to achieve matrix sequence The minimum value of the trace.

[0345] 3) Minimize fixed q under constraints k This ensures that the probability performance index has a lower bound at every moment.

[0346] Assume p is time-varying, p k To define the probabilistic performance index at time k, define Therefore, we have the following corollary 3:

[0347] Given probability p, the matrix described in step 3 Given that the boundedness in the sense of the weighted Frobenius norm holds, based on Corollary 1, when the following minimization problems have feasible solutions:

[0348]

[0349]

[0350] p k has a lower bound.

[0351] The system considered in the present application is described by a nonlinear difference equation, which has stronger generality and can more comprehensively and realistically reflect engineering applications. Secondly, a new transient performance index is defined for the system at each time, which can appropriately describe the system dynamics within a limited time interval of interest. The present application proposes a probabilistic design method, which can avoid the adverse effects of unnecessary strict constraint conditions in practical applications and achieve the expected performance with satisfactory probability, providing more flexibility. Finally, a joint estimation algorithm is provided, which can simultaneously estimate the system state and the new information attacked.

Claims

1. A method for distributed probability of preservation estimation of wireless sensor network under a deception attack, characterized in that, The specific steps are: Step 1, establishing a nonlinear system mathematical model under the condition of new information being attacked by deception; Comprising: where the system is defined on the time domain [0, T], represents the state of the system at the current time k, represents the measurement output of the system at the current time k; B k is one of the known real matrices of appropriate dimension, E i,k is the second one of the known real matrices of appropriate dimension; f(x k ): is one of the smooth nonlinear functions, g(x k ): is the second one of the smooth nonlinear functions; and are the process noise and the measurement noise, respectively, satisfying the following set of constraints: wherein V k >0 is one of the known matrices with appropriate dimensions, U k >0 is two of the known matrices with appropriate dimensions; Using the Taylor series expansion method, the nonlinear terms f(x k ) and g i (x k ) in the system are expressed as wherein, represents the error caused by neglecting the high order terms in the Taylor series expansion of the nonlinear function f(x k ), represents the error caused by neglecting the high order terms in the Taylor series expansion of the nonlinear function g(x k ), is one of the known matrices, is the other known matrix, is one of the unknown matrices, is the other unknown matrix, which are used to describe the modeling errors in matrices Φ i,k and Ψ i,k , respectively, satisfying ||Δ 1i ||≤1 and ||Δ 2i ||≤1; Φ i,k and Ψ i,k are obtained by Step 2, establishing a neural network-based distributed state estimator structure model according to the nonlinear system mathematical model under the condition of new information being attacked by deception; Step 3, designing a neural network weight matrix adaptive updating strategy; comprising: Wherein, the cost function of the neural network is defined as Step 4, designing a distributed guaranteed probability state estimator gain; defined as is one of the self-tuning scalar parameters, is the second self-tuning scalar parameter, design parameter and The specific method is: Given a sequence of matrices {Q ij,k} k∈[0,T] ; if there exists a sequence of positive scalars parameters by solving the following linear matrix inequality: where M ij,k is a factorization of the matrix i.e. If the linear matrix inequality holds, then the following inequality holds Step 5, optimizing the estimator parameters to achieve a locally optimal estimation performance. According to the nonlinear system mathematical model under the condition of new information being attacked by deception, a neural network-based distributed state estimator structure model is established, comprising:

2. The method of claim 1, wherein, The one-step estimation error of the system state is obtained by the following formula in, It is the state estimate of node i, z i,k The output estimate is The latest news; F i,k G i,k and These are the three parameters to be designed in the estimator; W is the weight matrix of the neural network. ij The estimated value, W ij W is the neighbor node of node i. j Set the ideal weight matrix; define the output estimate. And new information It is the fake news sent by node j under a spoofing attack, symbol The definition is as follows: where α k is a random sequence obeying Bernoulli distribution, which is used to describe the occurrence of network attacks, satisfying Prob{·} represents the probability of an event; is the deception signal generated by the attacker according to the new information z j,k obtained, which is injected into the original new information, and has the following form: Where, χ(·): It is an unknown nonlinear function defined on a compact set, and the following equation is used to approximate the unknown nonlinear function χ(z) using a neural network. j,k Specific methods: where W j is the ideal weight matrix of the neural network, φ(·) is the activation function, δ j,k is the estimation error, respectively satisfy the following conditions: ||W j || F ≤∈ 1j ,||φ(·)||≤∈2,||δ j,k ||≤∈ 3j (9) wherein ∈ 1j is a known positive scalar, ∈2is a known positive scalar two, ∈ 3j is a known positive scalar three; Definitions is the estimation error of the state of the observed object of the wireless sensor network, is the update error of the neural network, then the initial condition and is satisfied. where Q ij,0 is one of the known positive definite real matrices, P0is the other known positive definite real matrix. Intermediate definition: The one-step estimation error is expressed according to the intermediate definition as: Designing a distributed guaranteed probability state estimator gain, comprising: where When θ ij = 0, then is a sparse matrix, represented as wherein Furthermore, for one of the two block matrices A = [A ij ] N×N and B = [B ij ] N×N , defined as 3. The method of claim 1, wherein, 1, define the state estimation error at each node to satisfy the ellipsoid constraint under the probability p as follows: Establish a sufficient condition for the state estimation error to converge to an allowed ellipsoid region with a pre-specified probability: Defining system state estimation error The constraint can also be equivalently expressed as wherein is a pre-specified matrix, p is a pre-specified scalar satisfying 0 < p < 1, is an ellipsoid within represents the probability of an event occurring, is defined as follows: wherein, represents the center of the ellipsoid, Y > 0 is a positive definite matrix of appropriate dimension to depict the shape of the ellipsoid; Wherein, Three gain matrices F, G and H of a given estimator i,k i,k ij,k Under the premise that a sufficient condition for the update error of a neural network to be bounded in the weighted Frobenius norm sense holds, for a sequence of positive definite matrices {P k} k∈[0,T-1] , there is a factorization form, if there exists a sequence of non-negative definite scalars satisfying the following recursive matrix inequality:​​ Then the following inequality holds: 2, the method for solving the estimator gain parameters is as follows: i. Let k = 0; Given a pre-specified 0 < p < 1 and a sequence of positive definite matrices If there exists a sequence of non-negative scalars satisfying The estimator parameters F i,k , G i,k and H ij,k are obtained by solving this linear matrix inequality.

3. The following algorithm is given to iteratively obtain the desired estimator parameters {F i,k , G i,k , H ij,k}: Input: Topology information x i Initial value of x i,0 W ij Initial value of W ij,0 , Initial value of Q ij Initial value of Q ij,0 P k Initial value of P0; one of performance constraints {Q ij,k} k>0 Two of performance constraints p ; activation function φ(·); other parameters: ∈ 1i ,∈2,∈ 3i U k V k ; maximum step T; iii. If k < T, return ii to continue execution, otherwise exit. ii. Solve equation (27) to obtain the estimator gain i,k ,G i,k ,H ij,k ; obtain the one-step estimate from equation (5) i,k+1 ; solve equation (14) to obtain the updated parameters from equation (13) Let k = k + 1; Optimizing the estimator parameters to achieve a locally optimal estimation performance, comprising:

4. The method of claim 1, wherein, In the obtained estimator parameter set, the following three optimization problems are proposed from different performance requirements to seek locally optimal estimator parameters: Under the premise that the sufficient condition for the update error of the neural network to be bounded in the weighted Frobenius norm is established, when the following minimization problem has a feasible solution:

1. Minimize the trace of matrix Q ij,k The following corollary 1 can be derived from the above theorem 1: Given the probability p, under the premise that the sufficient condition for the update error of the neural network to be bounded in the weighted Frobenius norm is established, based on the inference 1, when the following minimization problem has a feasible solution: The minimum of the trace of the matrix sequence {Q ij,k} k∈[0,T]} 2. Minimization of the trace of the matrix to guarantee the optimal estimation performance under the pre-specified fixed probability constraint; For simplicity of expression, let Then, the following corollary 2 can be drawn: Given a probability p, the matrix described in step 3 Under the premise that the sufficient condition for being bounded in the sense of the weighted Frobenius norm holds, based on inference 1, when the following minimization problem has a feasible solution: The minimum of the trace of the matrix sequence can be achieved.

3. Minimizing the fix q under constraints k to ensure that the probability performance indicator has a lower bound at every instant. Assume p is time-varying, p k is the probability performance indicator at time k, define Then, we have the following corollary 3: ​ p k has a lower bound.

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