Differential privacy fusion filtering method for asynchronous sampling positioning system in low-reliability environment

Abstract

The invention discloses a differential privacy fusion filtering method of an asynchronous sampling positioning system in a low-reliability environment. The method comprises the following steps: 1, establishing a dynamic model of an asynchronous sampling nonlinear positioning system based on sensor network communication; 2, converting the asynchronous sampling nonlinear positioning system dynamic model into a synchronous sampling nonlinear positioning system dynamic model through a zero-order holding strategy; 3, designing a differential privacy distributed fusion filter; 4, calculating one-step prediction and one-step prediction error covariance; 5, deriving a local filter gain according to the one-step prediction error covariance; 6, substituting the local filter gain and the one-step prediction into the step 3 to obtain fusion filtering; and 7, calculating a local filtering error covariance upper bound. According to the method, the problem that the privacy protection and filtering performance of the positioning system cannot be considered cooperatively in the existing filtering method is solved, and meanwhile, the influence of randomly generated nonlinearity of the sensor on the fusion filtering effect is discussed, so that the estimation precision of the problem is improved.

Description

Technical Field

[0001] The present invention relates to a distributed fusion filtering method, and more particularly to a differential privacy fusion filtering method for an asynchronous sampling non-linear positioning system with randomly occurring sensor non-linearity in a low-reliability environment. Background Art

[0002] With the rapid development of wireless communication technology, sensor networks have been widely used in the field of positioning. As a new generation of communication technology, 6G brings new opportunities to the positioning system with its characteristics of ultra-high speed, low latency, and high bandwidth. However, at the same time, the high complexity and variability of 6G technology also make the positioning system vulnerable to signal interference in a low-reliability environment, resulting in a decrease in positioning accuracy. On the other hand, asynchronous sampling is relatively flexible in the way of signal acquisition and processing, which can well balance the relationship between system performance and resource conservation, and has broad application prospects in the positioning system. Therefore, it is of practical significance to study the distributed filtering problem of an asynchronous sampling positioning system in a low-reliability environment based on sensor network communication.

[0003] The problem of privacy protection is another research hotspot in the positioning system. Under the 6G network architecture, data interaction will be more frequent, and a large amount of location-related information is continuously exchanged through the sensor network, increasing the risk of information leakage. Eavesdroppers may illegally obtain the location privacy of users by intercepting and analyzing the data. Therefore, how to achieve privacy protection while ensuring the filtering accuracy of the positioning system is an urgent problem to be solved. At present, differential privacy technology, with its simple operation characteristics, realizes privacy protection through a certain random mechanism, which has attracted research interest. Summary of the Invention

[0004] In order to solve the problem that the existing fusion filtering methods cannot consider privacy protection and filtering performance in a low-reliability environment, the present invention provides a differential privacy fusion filtering method for an asynchronous sampling positioning system in a low-reliability environment.

[0005] The object of the present invention is achieved by the following technical solutions:

[0006] A differential privacy fusion filtering method for an asynchronous sampling positioning system in a low-reliability environment, comprising the following steps:

[0007] Step 1: Establish a dynamic model of an asynchronous sampling non-linear positioning system based on sensor network communication:

[0008] x(t k+1 ) = f(x(t k )) + B(t k )ω(t k )

[0009] y i (sk ) = λ i (s k )C i (s k )x(s k ) + (1 - λ i (s k ))g i (x(s k )) + ν i (s k )

[0010] In the formula, t k is the state update time of the asynchronous sampling non - linear positioning system; x(t k ) is the state vector of the asynchronous sampling non - linear positioning system at time t k ; x(t k+1 ) is the state vector of the asynchronous sampling non - linear positioning system at time t k+1 ; f(x(t k )) is the non - linear function based on the asynchronous sampling non - linear positioning system at time t k ; ω(t k ) is the process noise with a mean of 0 and a covariance of Q(t k ) at time t k ; B(t k ) is the coefficient matrix of the process noise at time t k ; s k is the measurement sampling time of the asynchronous sampling non - linear positioning system; i is the label of the sensor node, i = 1, 2, …, N, where N is the number of sensor nodes; y i (s k ) is the measurement output of the i - th sensor node in the asynchronous sampling non - linear positioning system at s k ; C i (s k ) is the measurement matrix of the i - th sensor at s k ; x(s k ) is the state vector of the asynchronous sampling non - linear positioning system at s k ; g i (x(s k )) is the non - linear function based on the asynchronous sampling non - linear positioning system of the i - th sensor node at s k ; ν i (s k ) is the measurement noise of the i - th sensor node with a mean of 0 and a covariance of R k (s i ) at s k ; λ i (s k ) is used to describe the i - th sensor node at sk Random occurrence of sensor nonlinearity at a moment

[0011] Step 2: Through the zero-order hold strategy, transform the dynamic model of the asynchronous sampling nonlinear positioning system based on sensor network communication in Step 1 into a synchronous sampling nonlinear positioning system dynamic model. The specific steps are as follows:

[0012] Introduce the time-varying delay ρ k , and obtain the measurement output model of the synchronous sampling nonlinear positioning system as:

[0013]

[0014] In the formula, z i (t k ) is the measurement output of the i-th sensor node in the synchronous sampling nonlinear positioning system at time t k ; is the measurement output of the i-th sensor node in the asynchronous sampling nonlinear positioning system at ; ρ k = mod(k, b) is the time-varying delay at time t k , b≥2 is a positive integer, and mod(k, b) is the modulo operation, representing the remainder of k divided by b;

[0015] Step 3: For the dynamic model of the synchronous sampling nonlinear positioning system in Step 2, design a differential privacy distributed fusion filter. The specific steps are as follows:

[0016] Step 3-1: During the data communication process of the sensor network, to prevent the leakage of privacy information and ensure the security of data transmission, before the sensor node j sends its measurement output y j (s k ) to the sensor node i, the measurement output y j (s k ) is processed by a random mechanism:

[0017]

[0018] In the formula, is the set of neighbor nodes of the sensor node i; is the random algorithm; is the measurement information transmitted from the j-th sensor node to the i-th sensor node at time s k ; y j (s k ) is the measurement output of the j-th sensor node in the asynchronous sampling nonlinear positioning system at time s k ; θ j (s k ) is at s kInject y at the moment j (s k ) of zero-mean Gaussian white noise;

[0019] Step 32. To ensure that the random algorithm achieves ∈-δ differential privacy under the given adjacency relationship, the covariance of the injected Gaussian white noise satisfies:

[0020]

[0021] where,

[0022]

[0023] In the formula, Y 2 (∈, δ) and (Φ -1 (δ)) 2 are the squares of Δ2(y j (s k ), Y(∈, δ) and Φ -1 (δ) respectively; I is the identity matrix with appropriate dimension; ∈ and δ are positive scalars, representing the privacy protection level and the relaxation term respectively; Φ -1 (δ) is the reciprocal of Φ(δ); max{a, b} represents taking the maximum value between a and b; and are positive scalars; H is a diagonal matrix with elements 0 or 1; C j (s k ) is the measurement matrix of the j-th sensor at time s k ; π is the pi; e is the natural constant; "∫" represents the integral symbol; ||·|| represents the norm of "·";

[0024] Step 33. When the sensor node i receives , according to the zero-input compensation strategy, obtain the compensated measurement at time t k :

[0025]

[0026] In the formula, is the compensated measurement at time t k , β(t k ) is the auxiliary variable at time t k ; is the measurement information transmitted from the j-th sensor node to the i-th sensor node at time t k ;

[0027] Step 34. Based on the obtained information, construct a local distributed filter:

[0028]

[0029] Among them,

[0030]

[0031] In the formula, "∑" is the summation symbol; is the filtering of the i-th sensor node at time t k+1 ; is the one-step prediction of the i-th sensor node at the t-th k time; is the one-step prediction of the i-th sensor node at time; is the one-step prediction of the j-th sensor node at t k time; z i (t k+1 ) is the measurement output of the i-th sensor node in the synchronous sampling nonlinear positioning system at time t k+1 ; is the compensated measurement at time t k+1 ; K i (t k+1 ) and G ij (t k+1 ) are the local filter gains of the i-th sensor node at time t k+1 ; h ij is the connection coefficient between sensor node i and sensor node j; is the expectation of the random variable ; is used to describe the randomly occurring sensor nonlinear phenomenon of the i-th sensor node at time; is the expectation of the random variable λ j (t k+1 ) ; λ j (t k+1 ) is used to describe the randomly occurring sensor nonlinear phenomenon of the j-th sensor node at time t k+1 ; is the measurement matrix of the i-th sensor node at time; C j (t k+1 ) is the measurement matrix of the j-th sensor node at time t k+1 ; is the nonlinear function prediction form of the i-th sensor node at time based on the synchronous sampling nonlinear positioning system; is the nonlinear function prediction form of the j-th sensor node at time t k based on the synchronous sampling nonlinear positioning system; ρk+1 is the time-varying time delay at time t k+1 ; β(t k+1 ) is the auxiliary variable at time t k+1 ;

[0032] Step 35: Obtain the differential privacy fusion filter according to the inverse covariance intersection fusion criterion:

[0033]

[0034] wherein is the fusion filtering at time t k ; is the filtering of the i-th sensor node at time t k ; is the upper bound of the local filtering error covariance of the i-th sensor node at time t k ; and are respectively and inverses; w i is a positive scalar;

[0035] Step 4: Use the second-order central difference method to calculate the one-step prediction k and the one-step prediction error covariance P (t i |t k+1 |t k ) of the i-th sensor node at time t

[0036]

[0037] wherein

[0038]

[0039] wherein, n x is the dimension of the state vector of the asynchronous sampling nonlinear positioning system; h is the step size of the second-order Stirling interpolation formula; is the -th column of the matrix ; k is the upper bound of the local filtering error covariance of the i-th sensor node at time t ; that is is the transpose of and are respectively and transposes;

[0040] Step 5: According to P in Step 4i (t k+1 |t k ), the local filter gains K k+1 (t i (t k+1 ) and G ij (t k+1 ) are derived as follows:

[0041]

[0042] where,

[0043]

[0044] in the formula, tr{·} represents taking the trace of "·"; is the element does not belong to the symbol; and respectively and the transpose of C j (t k+1 ); and are respectively and transpose; and are respectively the inverses of Φ i (t k+1 ) and Ω ij (t k+1 ); P j (t k+1 |t k ) is the one-step prediction error covariance of the jth sensor node at time t k ; is the one-step prediction error covariance of the ith sensor node at time; and are respectively and squared; tr{P j (t k+1 |t k ), and are respectively P j (t k+1 |t k ), and trace; is the measurement noise covariance of the ith sensor node at time; R j (t k+1) is the measurement noise covariance of the j-th sensor node at time t k+1 ; is the Gaussian noise covariance injected at time t k+1 ; θ m3 and are positive scalars; and are respectively the reciprocals of θ2, θ3 and , where m1 = 1, 2, …, 5, m2 = 1, 2, …, 11, m3 = 1, 2, 3, r > j; m4 = 1, 3, 4;

[0045] Step Six: Substitute the local filter gains K i (t k+1 ) and G ij (t k+1 ) calculated in Step Five, and the one-step prediction calculated in Step Four into Step Three to obtain the fusion filter k+1 at time t Judge whether t k+1 reaches the total duration M. If t k+1 < M, then execute Step Seven; otherwise, end the operation;

[0046] Step Seven: According to K i (t k+1 ) and G ij (t k+1 ) obtained in Step Five, and P i (t k+1 |t k ) obtained in Step Four, calculate the upper bound of the local filtering error covariance of the i-th sensor node at time t k+1 ; Let t k = t k+1 , and execute Step Three until t k+1 = M.

[0047] Compared with the prior art, the present invention has the following advantages:

[0048] 1. The present invention simultaneously considers the influence of privacy protection and randomly occurring sensor nonlinearity on the filtering performance, and designs a local filtering algorithm through a minimum upper bound optimization mechanism. In particular, during the information interaction between sensor nodes, the differential privacy mechanism is used to achieve privacy protection.

[0049] 2. The present invention uses the central difference Kalman filtering idea based on second-order Stirling interpolation to perform state estimation on an asynchronous sampling non-linear positioning system. Compared with the extended Kalman filter, this method avoids the calculation of the Jacobian matrix, has a wider application range, and can be effectively applied to non-linear processing. At the same time, the present invention uses the distributed inverse covariance intersection fusion criterion to fuse the local filtering results, which not only reduces the computational burden but also further improves the estimation accuracy, showing obvious advantages compared with the traditional covariance intersection fusion method.

[0050] 3. The present invention solves the problem that the existing filtering methods cannot jointly consider the privacy protection and filtering performance of the positioning system, and at the same time explores the influence of randomly occurring sensor non-linearity on the fusion filtering effect, thereby improving the estimation accuracy of such problems. It can be seen from the simulation diagram that as the occurrence probability of randomly occurring sensor non-linearity decreases, the mean square error of the fusion filtering also decreases. When the occurrence probability changes from 0.8 to 0.7, the mean square error decreases by about 49%; when the occurrence probability changes from 0.8 to 0.5, the mean square error decreases by about 60%; when the occurrence probability changes from 0.8 to 0.2, the mean square error decreases by about 80%. Description of the Drawings

[0051] Figure 1 is the flowchart of the differential privacy fusion filtering algorithm;

[0052] Figure 2 is the actual trajectory x1(t k ) of the first state component of the asynchronous sampling non-linear positioning system and its fusion filtering tracking trajectory

[0053] Figure 3 is the actual trajectory x2(t k ) of the second state component of the asynchronous sampling non-linear positioning system and its fusion filtering tracking trajectory

[0054] Figure 4 is the comparison diagram of the cumulative error squares of four local filters and the fusion filter;

[0055] Figure 5 is the comparison diagram of the cumulative error squares of the fusion filter and the covariance intersection fusion filter;

[0056] Figure 6 is the comparison diagram of the cumulative error squares of the fusion filtering under different occurrence probabilities of randomly occurring sensor non-linearity. Detailed Embodiment

[0057] The technical solution of the present invention will be further described below in conjunction with the accompanying drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

[0058] The present invention provides a differential privacy fusion filtering method for an asynchronous sampling positioning system in a low-reliability environment, as Figure 1 shown. The method includes the following steps:

[0059] Step 1: Establish a dynamic model of an asynchronous sampling nonlinear positioning system based on sensor network communication.

[0060] In this step, the established dynamic model of the asynchronous sampling nonlinear positioning system based on sensor network communication is:

[0061]

[0062] In the formula, t k is the state update time of the asynchronous sampling nonlinear positioning system; x(t k ) is the state vector of the asynchronous sampling nonlinear positioning system at time t k ; x(t k+1 ) is the state vector of the asynchronous sampling nonlinear positioning system at time t k+1 ; f(x(t k )) is the nonlinear function based on the asynchronous sampling nonlinear positioning system at time t k ; ω(t k ) is the process noise with a mean of 0 and a covariance of Q(t k ) at time t k ; B(t k ) is the coefficient matrix of the process noise at time t k ; s k is the measurement sampling time of the asynchronous sampling nonlinear positioning system; i is the label of the sensor node, i = 1, 2,..., N, and N is the number of sensor nodes; y i (s k ) is the measurement output of the i-th sensor node in the asynchronous sampling nonlinear positioning system at time s k ; C i (s k ) is the measurement matrix of the i-th sensor at time s k ; x(s k ) is the state vector of the asynchronous sampling nonlinear positioning system at time s k ; g i (x(s k )) is the i-th sensor node at time s kNonlinear function of the asynchronous sampling-based nonlinear positioning system at time ν i (s k ) The measurement noise of the i-th sensor node at time s with mean 0 and covariance R k at time s; λ i (s k ) is used to describe the randomly occurring sensor nonlinear phenomenon of the i-th sensor node at time s i (s k ) k At this step, for any two adjacent state update times t

[0063] and t k and t k+1 satisfy t k+1 -t k =ΔT, where ΔT is a positive scalar representing the state update period; for any two adjacent measurement sampling times s k and s k+1 satisfy s k+1 -s k =bΔT, where b≥2 is a positive integer and bΔT represents the measurement sampling period; the initial state update time t0 and the initial measurement sampling time s0 satisfy t0 = s0

[0064] At this step, the nonlinear function g i (x(s k )) satisfies g i (0)=0, and for any n x -dimensional vectors χ1 and χ2, there is where is a positive scalar

[0065] At this step, λ i (s k ) is a random variable following a Bernoulli distribution with values of 0 or 1, satisfying is the expectation of the random variable λ i (s k ), is the mathematical expectation, and Prob{·} represents the probability of the occurrence of the event "·"

[0066] Step 2: By using the zero-order hold strategy, transform the dynamic model of the asynchronous sampling-based nonlinear positioning system based on sensor network communication in Step 1 into a synchronous sampling-based nonlinear positioning system dynamic model. The specific steps are as follows:

[0067] Introduce the time-varying time delay ρ k , and obtain the measurement output model of the synchronous sampling-based nonlinear positioning system as:

[0068]

[0069] where zi (t k ) is the measurement output of the i-th sensor node in the synchronous sampling non-linear positioning system at time t k ; is the measurement output of the i-th sensor node in the asynchronous sampling non-linear positioning system at time; ρ k = mod(k, b) is the time-varying time delay at time t k , and mod(k, b) is a modulo operation, representing the remainder of k divided by b.

[0070] In this step, ρ k ∈ {0, 1, 2, …, b - 1} is the periodic time delay.

[0071] Step 3: For the dynamic model of the synchronous sampling non-linear positioning system in Step 2, design a differential privacy distributed fusion filter. The specific steps are as follows:

[0072] Step 3-1: During the data communication process in the sensor network, to prevent the leakage of privacy information and ensure the security of data transmission, before sensor node j sends its measurement output y j (s k ) to sensor node i, the measurement output y j (s k ) is processed using a random mechanism:

[0073]

[0074] where “∈” is the element belongs to symbol, is the set of neighbor nodes of sensor node i; is a random algorithm; is the measurement information transmitted from the j-th sensor node to the i-th sensor node at time s k ; y j (s k ) is the measurement output of the j-th sensor node in the asynchronous sampling non-linear positioning system at time s k ; θ j (s k ) is the zero-mean Gaussian white noise injected into y k at time s j (s k ), and its covariance is to be designed.

[0075] Step 3-2: To ensure that the random algorithm achieves ∈-δ differential privacy under the given adjacency relationship, the covariance of the injected Gaussian white noise satisfies:

[0076]

[0077] Among them,

[0078]

[0079] In the formula, Y 2 (∈, δ) and (Φ -1 (δ)) 2 are respectively the squares of Δ2(y j (s k )); I is the identity matrix with appropriate dimensions; ∈ and δ are positive scalars, representing the privacy protection level and the relaxation term respectively; Φ -1 (δ) is the reciprocal of Φ(δ); -1 is the arithmetic square root of (Φ (δ)) -1 + 2∈; max{a, b} represents taking the maximum value between a and b; 2 and and are positive scalars; H is a diagonal matrix with elements 0 or 1; C j (s k ) is the measurement matrix of the j-th sensor at time s k ; is the arithmetic square root of 2π; π is the pi; e is the natural constant; "∫" represents the integral symbol; ||·|| represents the norm of "·".

[0080] In this step, the definition of ∈-δ differential privacy is: for given positive scalars ∈ and δ, if for the random algorithm and any two adjacent elements a and [a]', there is

[0081]

[0082] holds, then the random algorithm satisfies ∈-δ differential privacy, where [a]' represents the adjacent element of a; and respectively represent applying the random algorithm to a and [a]'; M j is 's subset; is the observation domain of the random algorithm ; represents arbitrary; represents included in.

[0083] In this step, the adjacency relationship between the state vector x(t k ) and its adjacent element [x(t k )]' is expressed as satisfies (I - H)x(t k ) = (I - H)[x(t k )]′。

[0084] Step Three: When sensor node i receives , according to the zero - input compensation strategy, obtain the compensated measurement at time t k :

[0085]

[0086] where is the compensated measurement at time t k , β(t k ) is the auxiliary variable at time t k , satisfying s t is the sampling time of the sensor node; is the measurement information transmitted from the j - th sensor node to the i - th sensor node at time t k .

[0087] Step Four: Based on the obtained information, construct a local distributed filter:

[0088]

[0089] where

[0090]

[0091] in the formula, "∑" is the summation symbol; is the filtering of the i - th sensor node at time t k+1 ; is the one - step prediction of the i - th sensor node at the t - th k time; is the one - step prediction of the i - th sensor node at time; is the one - step prediction of the j - th sensor node at time t k ; z i (t k+1 ) is the measurement output of the i - th sensor node in the synchronous sampling non - linear positioning system at time t k+1 ; is the compensated measurement at time t k+1 ; K i (t k+1 ) and G ij (t k+1 ) are the local filter gains of the i - th sensor node at time t k+1 ; h ijis the connection coefficient between sensor node i and sensor node j; is a random variable expectation; used to describe the randomly occurring sensor nonlinear phenomenon of the ith sensor node at moment; is the random variable λ j (t k+1 ) expectation; λ j (t k+1 ) used to describe the randomly occurring sensor nonlinear phenomenon of the jth sensor node at t k+1 moment; is the measurement matrix of the ith sensor node at moment; C j (t k+1 ) is the measurement matrix of the jth sensor node at t k+1 moment; is the nonlinear function prediction form of the synchronous sampling nonlinear positioning system based on the ith sensor node at moment; is the nonlinear function prediction form of the synchronous sampling nonlinear positioning system based on the jth sensor node at t k moment; ρ k+1 is the time-varying time delay at t k+1 moment; β(t k+1 ) is the auxiliary variable at t k+1 moment.

[0092] Step 35. According to the inverse covariance intersection fusion criterion, obtain the differential privacy fusion filter:

[0093]

[0094] In the formula, the superscript "-1" represents the inverse of the matrix; is the fusion filtering at t k moment; is the filtering of the ith sensor node at t k moment; is the upper bound of the local filtering error covariance of the ith sensor node at t k moment; and are respectively and inverse; w i is a positive scalar.

[0095] In this step, w i can be determined by solving the following optimization problem:

[0096]

[0097] In the formula, represents minimizing the objective function by selecting the decision variable w that satisfies the constraint conditions i s.t. represents subject to. s.t. means subject to.

[0098] Step 4: Use the second-order central difference method to calculate the one-step prediction k at time t of the i-th sensor node and the one-step prediction error covariance P i (t k+1 |t k ).

[0099] In this step, calculate the one-step prediction and the one-step prediction error covariance P i (t k+1 |t k ) according to the following formula:

[0100]

[0101] Where,

[0102]

[0103] In the formula, n x is the dimension of the state vector of the asynchronous sampling nonlinear positioning system; the superscript "T" represents taking the transpose of the matrix; h is the step size of the second-order Stirling interpolation formula; is the th column of the matrix; is the upper bound of the local filtering error covariance at time t k of the i-th sensor node Cholesky decomposition factor, that is is transpose; and are respectively and transpose.

[0104] Step 5: Derive the local filter gains K i (t k+1 |t k ) and G k+1 i (t (t k+1 ) and G ij (t k+1 ) at time t according to P

[0105] In this step, by minimizing t k+1Upper bound of the local filtering error covariance at a moment Take the trace of it to obtain the local filter gain K i (t k+1 ) and G ij (t k+1 ):

[0106]

[0107] Among them,

[0108]

[0109] In the formula, tr{·} represents taking the trace of "·"; is an element not belonging to the symbol; and respectively and the transpose of C j (t k+1 ); and are respectively and 's transpose; and are respectively the inverses of Φ i (t k+1 ) and Ω ij (t k+1 ); P j (t k+1 |t k ) is the one-step prediction error covariance of the jth sensor node at time t k ; is the one-step prediction error covariance of the ith sensor node at time; and are respectively and 's square; tr{P j (t k+1 |t k )}, and are respectively P j (t k+1 |t k ), and 's trace; is the measurement noise covariance of the ith sensor node at time; R j (t k+1 ) is the measurement noise covariance of the jth sensor node at time t k+1Measurement noise covariance at a moment; is the Gaussian noise covariance injected at time t k+1 ; θ m3 and are positive scalars; and are respectively the reciprocals of θ2, θ3, and m1 = 1, 2, …, 5, m2 = 1, 2, …, 11, m3 = 1, 2, 3, r > j; m4 = 1, 3, 4;

[0110] Step 6. Substitute the local filter gain K i (t k+1 ) and G ij (t k+1 ) calculated in Step 5, and the one-step prediction calculated in Step 4 into Step 3 to obtain the fusion filter k+1 at time t Judge whether t k+1 reaches the total duration M. If t k+1 < M, then execute Step 7; otherwise, end the operation.

[0111] Step 7. According to K i (t k+1 ) and G ij (t k+1 ) obtained in Step 5, and P i (t k+1 |t k ) obtained in Step 4, calculate the upper bound of the local filtering error covariance of the i-th sensor node at time t k+1 ; Let t k = t k+1 , and execute Step 3 until t k+1 = M.

[0112] In this step, the upper bound of the local filtering error covariance can be calculated by the following formula:

[0113]

[0114] where

[0115] τ8 = α2 + α3 + β(t k+1 )(η1 + η2)

[0116]

[0117] In the formula, is the upper bound of the local filtering error covariance of the i-th sensor node at time t k+1 ; is the square of h ij ; and are the transposes of K i (t k+1 ) and G ij (t k+1 ), respectively; is a positive scalar; is the reciprocal of ξ jr,2 ;

[0118] In this step, the upper bound of the local filtering error covariance of each sensor node is derived such that holds, where is the local filtering error covariance of the i-th sensor node at time t k+1 , is the local filtering error of the i-th sensor node at time t k+1 , is the mathematical expectation. Next, by optimizing the trace of , the local filter gains K k+1 (t i ) and G k+1 (t ij ) at time t k+1 are derived.

[0119] Example:

[0120] This example uses an asynchronous sampling nonlinear positioning system measured by a four-node sensor network for simulation. Its dynamic model parameters and sensor network topology can be expressed as:

[0121]

[0122] C1(s k ) = [1.2 1.3], C2(s k ) = [0.4 0.7]

[0123] C3(s k ) = [3.2 0.8], g1(x(s k )) = g3(x(s k )) = 0.01sin(x1(s k )x2(s k ))

[0124] C4(s k ) = [2.4 0.8], g2(x(sk )) = g4(x(s k )) = 0.01sin(x1(s k )x2(s k ))

[0125] Ε = {(1,3),(1,4),(2,1),(2,3),(3,2),(3,4),(4,1),(4,2)}

[0126] (i,j) ∈ E, h ij = 1; h ij = 0(i,j = 1,2,3,4)

[0127] Wherein, x(t k ) = [x1(t k )x2(t k )] T ; sin(·) and cos(·) represent the sine function and cosine function of “·” respectively; Ε is the edge set of the sensor network; (i,j) is an ordered pair, indicating the information interaction between sensor node i and sensor node j. If (i,j) ∈ E, it means that sensor node i can receive the information of sensor node j, otherwise it cannot.

[0128] The initial value and other parameters are set as:

[0129]

[0130] Q(t k ) = 0.5, δ = 0.1

[0131] R1(s k ) = 0.07, T = 1, b = 2,

[0132] R2(s k ) = 0.8, θ m3 (m3 = 1,2,3) = 0.6

[0133] R3(s k ) = 0.6, ∈ = 0.15

[0134] R4(s k ) = 0.4,

[0135] The present invention uses the cumulative squared error to verify the superiority of the proposed differential privacy fusion filtering. The specific calculation formula is:

[0136]

[0137] Among them, AES i (t k ) and AES i (t k+1 ) respectively represent the cumulative squared error of the i-th local filter at time t k and at time t k+1 ; AES ICI (t k ) and AES ICI (t k+1 ) respectively represent the cumulative squared error of the inverse covariance intersection fusion filter at time t k and at time t k+1 ; and respectively represent and squared. In addition, the average mean square error is introduced to measure the influence of the occurrence probability of different randomly occurring sensor non-linearities on the performance of the fusion filter, and its calculation formula is MSE' ICI = AES ICI (M) / M, where M = 100 is the total running time in the present invention.

[0138] Effects of the differential privacy fusion filter:

[0139] Figure 2 and Figure 3 respectively depict the actual trajectory and the fusion filter tracking trajectory of the asynchronous sampling non-linear positioning system. From Figure 2 and Figure 3 , the effectiveness of the proposed fusion filter algorithm can be seen. Figure 4 gives the comparison of the cumulative squared errors of the local filter and the inverse covariance intersection fusion filter, Figure 5 depicts the comparison of the cumulative squared errors of the inverse covariance intersection fusion filter and the covariance intersection fusion filter, where AES CI (t k ) is the cumulative squared error of the covariance intersection fusion filter at time t k . From Figure 4 and Figure 5 , it can be seen that AES ICI (t k ) < AES i (t k ) (i = 1, 2, 3, 4), AES ICI (t k ) < AES CI (t k ), which indicates that the estimation effect of the inverse covariance intersection fusion filter is better than that of the local filter and the covariance intersection fusion filter.

[0140] Figure 6 The cumulative squared error of the inverse covariance intersection fusion filter is given under the occurrence probabilities of different randomly occurring sensor non - linearities. It can be seen from the figure that as the occurrence probability of the randomly occurring sensor non - linearity decreases, the AES ICI (t k ) of the inverse covariance intersection fusion filter also decreases, which indicates that when the occurrence probability decreases, the filtering performance of the inverse covariance intersection fusion filter gets better. Specifically, when the occurrence probability takes values of 0.8, 0.7, 0.5, and 0.2, the average mean - square error MSE′ ICI of the fusion filtering is 2.5505, 1.3080, 1.0204, and 0.5095 respectively. In other words, when the occurrence probability changes from 0.8 to 0.7, MSE′ ICI decreases by about 49%; when the occurrence probability changes from 0.8 to 0.5, MSE′ ICI decreases by about 60%; when the occurrence probability changes from 0.8 to 0.2, MSE′ ICI decreases by about 80%.

[0141] In summary, the differential privacy fusion filtering method of the asynchronous sampling positioning system in a low - reliability environment proposed by the present invention can also effectively estimate the system state when there are both randomly occurring sensor non - linearities and eavesdroppers.

Claims

1. A differential privacy fusion filtering method for an asynchronous sampling positioning system in a low-reliability environment, characterized in that The method includes the following steps: Step 1, establish a dynamic model of an asynchronous sampling non-linear positioning system based on sensor network communication: x(t k+1 ) = f(x(t k )) + B(t k )ω(t k ) y i (s k ) = λ i (s k )C i (s k )x(s k ) + (1 - λ i (s k ))g i (x(s k )) + ν i (s k ) where t k is the state update time of the asynchronous sampling nonlinear positioning system; x(t k ) is the state vector of the asynchronous sampling nonlinear positioning system at time t k ; x(t k+1 ) is the state vector of the asynchronous sampling nonlinear positioning system at time t k+1 ; f(x(t k )) is the nonlinear function based on the asynchronous sampling nonlinear positioning system at time t k ; ω(t k ) is the process noise with a mean of 0 and a covariance of Q(t k ) at time t k ; B(t k ) is the coefficient matrix of the process noise at time t k ; s k is the measurement sampling time of the asynchronous sampling nonlinear positioning system; i is the label of the sensor node, i = 1, 2, …, N, where N is the number of sensor nodes; y i (s k ) is the measurement output of the i-th sensor node in the asynchronous sampling nonlinear positioning system at time s k ; C i (s k ) is the measurement matrix of the i-th sensor at time s k ; x(s k ) is the state vector of the asynchronous sampling nonlinear positioning system at time s k ; g i (x(s k )) is the nonlinear function based on the asynchronous sampling nonlinear positioning system of the i-th sensor node at time s k ; ν i (s k ) is the measurement noise of the i-th sensor node with a mean of 0 and a covariance of R i (s k ) at time s i ; λ k (s k ) is used to describe the randomly occurring sensor nonlinear phenomenon of the i-th sensor node at time s k . Step 2, through the zero-order hold strategy, transform the dynamic model of the asynchronous sampling non-linear positioning system in Step 1 into a synchronous sampling non-linear positioning system dynamic model. The specific steps are as follows: Introduce the time-varying time delay ρ k , and the measurement output model of the synchronous sampling non-linear positioning system is obtained as follows: where z i (t k ) is the measurement output of the i-th sensor node in the synchronous sampling non-linear positioning system at time t k ; is the measurement output of the i-th sensor node in the asynchronous sampling non-linear positioning system at ; ρ k = mod(k, b) is the time-varying time delay at time t k , b≥2 is a positive integer, and mod(k, b) is the modulo operation, representing the remainder of k divided by b; Step 3, for the synchronous sampling non-linear positioning system dynamic model in Step 2, design a differential privacy distributed fusion filter. The specific steps are as follows: Step 3.

1. During the data communication process of the sensor network, to prevent the leakage of privacy information and ensure the security of data transmission, before the sensor node j sends its measured output y j (s k ) to the sensor node i, the measured output y j (s k ) is processed using a random mechanism: In the formula, is the set of neighbor nodes of sensor node i; is a random algorithm; is the measurement information transmitted from the j-th sensor node to the i-th sensor node at time s k ; y j (s k ) is the measurement output of the j-th sensor node in the asynchronous sampling nonlinear positioning system at time s k ; θ j (s k ) is the zero-mean Gaussian white noise injected into y k (s j ) at time s k ; Step 32: To ensure that the random algorithm achieves differential privacy under the given adjacency relationship, the covariance of the injected Gaussian white noise satisfies: Among them, wherein, Y 2 (∈,δ) and (Φ -1 (δ)) 2 are respectively the squares of Δ2(y j (s k )); Y(∈,δ) and Φ -1 (δ); I is an identity matrix with appropriate dimension; ∈ and δ are positive scalars, representing the privacy protection level and the relaxation term respectively; Φ -1 (δ) is the reciprocal of Φ(δ); max{a, b} represents taking the maximum value between a and b; and are positive scalars; H is a diagonal matrix with elements 0 or 1; C j (s k ) is the measurement matrix of the jth sensor at time s k ; π is the pi; e is the natural constant; "∫" represents the integral symbol; ||·|| represents the norm of "·"; Step 33. When sensor node i receives , according to the zero-input compensation strategy, obtain the compensated measurement at time t k : wherein, is the compensation measurement at time t k , and β(t k ) is the auxiliary variable at time t k ; is the measurement information transmitted from the j-th sensor node to the i-th sensor node at time t k ; Step 3-4, construct a local distributed filter based on the obtained information: Among them, In the formula, "∑" is the summation symbol; is the filtering of the i-th sensor node at time t k+1 ; is the one-step prediction of the i-th sensor node at the t-th k time; is the one-step prediction of the i-th sensor node at time; is the one-step prediction of the j-th sensor node at time t k ; z i (t k+1 ) is the measurement output of the i-th sensor node in the synchronous sampling non-linear positioning system at time t k+1 ; is the compensated measurement at time t k+1 ; K i (t k+1 ) and G ij (t k+1 ) are the local filter gains of the i-th sensor node at time t k+1 ; h ij is the connection coefficient between sensor node i and sensor node j; is the expectation of the random variable ; is used to describe the randomly occurring sensor non-linearity phenomenon of the i-th sensor node at time; is the expectation of the random variable λ j (t k+1 ) ; λ j (t k+1 ) is used to describe the randomly occurring sensor non-linearity phenomenon of the j-th sensor node at time t k+1 ; is the measurement matrix of the i-th sensor node at time; C j (t k+1 ) is the measurement matrix of the j-th sensor node at time t k+1 ; is the non-linear function prediction form of the i-th sensor node at time based on the synchronous sampling non-linear positioning system; is the non-linear function prediction form of the j-th sensor node at time t k based on the synchronous sampling non-linear positioning system; ρ k+1 is the time-varying time delay at time t k+1 ; β(t k+1 ) is the auxiliary variable at time t k+1 ; Step 3-5, obtain a differential privacy fusion filter according to the inverse covariance intersection fusion criterion: Wherein, is the fusion filtering at time t k ; is the filtering of the i-th sensor node at time t k ; is the upper bound of the local filtering error covariance of the i-th sensor node at time t k ; and are respectively and inverses; w i is a positive scalar; Step 4: Use the second-order central difference method to calculate the one-step prediction of the i-th sensor node at time t k and the one-step prediction error covariance P (t i (t k+1 |t k ): Among them, where n x is the dimension of the state vector of the asynchronous sampling non-linear positioning system; h is the step size of the second-order Stirling interpolation formula; is the th column of the matrix is the upper bound of the local filtering error covariance of the i-th sensor node at time t k , that is, is the Cholesky decomposition factor of , namely is the transpose of and are respectively and the transposes of Step Five: Based on P in Step Four i (t k+1 |t k ), derive the local filter gains K k+1 at time t i (t k+1 ) and G ij (t k+1 ): Among them, In the formula, tr{·} represents taking the trace of "·"; " " is the symbol for an element not belonging to; and respectively and C j (t k+1 )'s transpose; and are respectively and )'s transpose; and are respectively the inverses of Φ i (t k+1 ) and Ω ij (t k+1 ) ; P j (t k+1 |t k ) is the one-step prediction error covariance of the j-th sensor node at time t k ; is the one-step prediction error covariance of the i-th sensor node at time; and are respectively and )'s square; tr{P j (t k+1 |t k )}, and are respectively P j (t k+1 |t k ), and )'s trace; is the measurement noise covariance of the i-th sensor node at time; R j (t k+1 ) is the measurement noise covariance of the j-th sensor node at time t k+1 ; is the Gaussian noise covariance injected at time t k+1 ; θ m3 as well as are positive scalars; as well as are respectively θ2, θ3 as well as )'s reciprocals, m1 = 1, 2,..., 5, m2 = 1, 2,..., 11, m3 = 1, 2, 3, r, r > j; m4 = 1, 3, 4; Step 6: Substitute the local filter gains K i (t k+1 ) and G ij (t k+1 ) calculated in Step 5, and the one-step prediction calculated in Step 4 into Step 3 to obtain the fusion filter k+1 at time t Judge whether t k+1 reaches the total duration M. If t k+1 < M, then execute Step 7; otherwise, end the operation. Step 7: According to the K obtained in Step 5 i (t k+1 ), G ij (t k+1 ), and P i (t k+1 |t k ) obtained in Step 4, calculate the upper bound of the local filtering error covariance of the i-th sensor node at time t k+1 At time Let t k = t k+1 , execute Step 3 until t k+1 = M.

2. The differential privacy fusion filtering method of the asynchronous sampling positioning system in a low-reliability environment according to claim 1, characterized in that In the first step, any two adjacent state update times t k and t k+1 satisfy t k+1 - t k = ΔT, where ΔT is a positive scalar representing the state update period; any two adjacent measurement sampling times s k and s k+1 satisfy s k+1 - s k = bΔT, where b ≥ 2 is a positive integer and bΔT represents the measurement sampling period; the initial state update time t0 and the initial measurement sampling time s0 satisfy t0 = s0.

3. The differential privacy fusion filtering method for the asynchronous sampling positioning system in a low-reliability environment according to claim 1, wherein In the first step, the non-linear function g i (x(s k )) satisfies g i (0) = 0, and for any n x -dimensional vectors χ1 and χ2, there is is a positive scalar.

4. The differential privacy fusion filtering method for the asynchronous sampling positioning system in a low-reliability environment according to claim 1, wherein In the first step, λ i (s k ) is a random variable following a Bernoulli distribution with values of 0 or 1, satisfying is the expectation of the random variable λ i (s k ), is the mathematical expectation, and Prob{·} represents the occurrence probability of the event "·".

5. The differential privacy fusion filtering method of the asynchronous sampling positioning system in a low-reliability environment according to claim 1, wherein In the second step, ρ k ∈ {0, 1, 2, …, b - 1} is the periodic time delay.

6. The differential privacy fusion filtering method for the asynchronous sampling positioning system in a low-reliability environment according to claim 1, wherein In step 32, the definition of ∈-δ differential privacy is: for given positive scalars ∈ and δ, if for a random algorithm and any two adjacent elements a and [a]', the following holds If it holds, the random algorithm satisfies ∈-δ differential privacy, where [a]′ represents the adjacent element of a; and respectively represent applying the random algorithm to a and [a]′; M j is a subset of; is the observation domain of the random algorithm ; " " means arbitrary; " " means included in; State vector \(x(t k ) and the adjacency relationship of its adjacent element \([x(t k )]'\) are expressed as Satisfy (I - H)x(t k )=(I - H)[x(t k )]'.

7. The differential privacy fusion filtering method of the asynchronous sampling positioning system in a low-reliability environment according to claim 1, characterized in that In step 33, β(t k ) satisfies s t is the sampling time of the sensor node.

8. The differential privacy fusion filtering method for the asynchronous sampling positioning system in a low-reliability environment according to claim 1, characterized in that In step 35, w i is determined by solving the following optimization problem: In the formula, denotes minimizing the objective function by selecting the decision variable w that satisfies the constraint conditions i where s.t. means subject to.

9. The differential privacy fusion filtering method for the asynchronous sampling positioning system in a low-reliability environment according to claim 1, wherein In the seventh step, the upper bound of the local filtering error covariance is calculated by the following formula: Among them, τ = α2 + α3 + β(t k+1 )(η1 + η2) In the formula, is the upper bound of the local filtering error covariance of the i-th sensor node at time t k+1 ; is the square of h ij ; and are the transposes of K i (t k+1 ) and G ij (t k+1 ), respectively; is a positive scalar; is the reciprocal of ξ jr,2 .

10. The differential privacy fusion filtering method of the asynchronous sampling positioning system in a low-reliability environment according to claim 1, characterized in that In step seven, the upper bound of the local filtering error covariance of each sensor node is derived such that holds, where is the local filtering error covariance of the i-th sensor node at time t k+1 , is the local filtering error of the i-th sensor node at time t k+1 , is the mathematical expectation. By optimizing the trace of , the local filter gains K k+1 (t i ) and G k+1 (t ij ) at time t k+1 are derived.

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