A Distributed Receding Horizon Estimation Method for Moving Targets with Privacy Protection

Abstract

The present invention belongs to the technical field of tracking filtering, and specifically relates to a distributed moving target receding horizon estimation method with privacy protection. The method of the present invention mainly calculates the mean value of the arrival cost in the receding horizon estimation by using a consensus protocol with privacy protection. Specifically, the state of each network node is randomly divided into two parts: visible and private, and only the visible part is transmitted outward to protect the node privacy. Then, the local estimator is calculated by minimizing the cost function of the receding horizon estimation, and the arrival cost function is updated according to the local estimator. It is verified that the method can effectively realize the state estimation in a distributed system and can protect the privacy of the initial state of the node.

Description

Technical Field

[0001] The present invention belongs to the technical field of target tracking, and specifically relates to a distributed moving target receding horizon estimation method with privacy protection. Background Art

[0002] In the past few decades, significant research interest has been generated in multi-sensor network systems. As one of the important concerns, distributed estimation has been widely studied in various applications such as robot formation control, environmental monitoring, and spacecraft navigation. Receding horizon estimation is a filtering technique based on the idea of estimating the system state by minimizing a cost function defined over a fixed-size sliding window. Extending it to a distributed system can achieve better robustness, flexibility, and reliability. However, in traditional distributed systems, the interaction information between nodes may cause the leakage of privacy information. Therefore, a distributed state estimation method with privacy protection is particularly important. Summary of the Invention

[0003] To address the above problems, the present invention proposes a distributed moving target receding horizon estimation method with privacy protection, which realizes the privacy protection of the initial state of each node while ensuring the performance of receding horizon estimation remains unchanged.

[0004] The technical solution of the present invention is as Figure 1 shown:

[0005] The present invention is applicable to the scenario of tracking a moving target through a linear sensor network, and the target motion is modeled by a linear model with a state transition matrix

[0006] x t+1 = Fx t + w t

[0007] where x t = [p x , v x , p y , v y T represents the motion state of the target (p x , p y represent the position coordinates of the target, v x , v y represent the speed of the target), and F represents the state transition matrix

[0008]

[0009] where T s is the tracking time interval. The interference noise w t ​Subject to a uniform random distribution on [-0.5, 0.5]. The entire sensor network consists of nodes, and the observation model of node i is

[0010] y t i = C i x t + v t i

[0011] where y t i represents the sensor measurement value, and C i is the observation matrix

[0012]

[0013] Observation noise is subject to a uniform random distribution on [-10, 10]. The weighting matrices Q and R i are respectively equal to w t and the inverse covariance matrix.

[0014] In a distributed sensor network, each node can only use the local information it can obtain to perform a rolling horizon estimation of the system state. Rolling horizon estimation is a filtering technique whose idea is to estimate the system state by minimizing a cost function defined on a fixed-size sliding window. The cost function is as follows:

[0015]

[0016] where N represents the window length of the rolling horizon, represents the estimated value of the target state at time k by sensor i at time t. is the arrival cost function, which characterizes the influence of the estimation results at past times on the current time estimation and plays a key role in the behavior and performance of the rolling horizon estimation. Therefore, we use the average arrival cost function of the entire network to replace the arrival cost function of each node itself.

[0017]

[0018]

[0019]

[0020] represents the predicted value of the target state at time t - N by sensor i, is a positive definite weight matrix that quantifies the confidence of the predicted value Taking the mean of the arrival cost function is equivalent to taking the mean of the parameters and to find the mean value, so for any time t, any node i in the network performs the following steps:

[0021] (1) Obtain the observation information y t i

[0022] (2) Obtain the average arrival cost function of the entire network through a privacy-preserving consensus algorithm.

[0023] a) Initialize the cost function parameters Each network node arbitrarily decomposes its own parameter into and such that it satisfies Decompose the parameter arbitrarily into and such that it satisfies

[0024] b) Let and represent the and at the l-th iteration respectively (1 < l < L, L is the total number of iterations). Send and to its neighbors and receive and

[0025] c) Update

[0026]

[0027]

[0028]

[0029]

[0030] where π is the consensus weight matrix, to enable the network to converge to the mean value, κ i is the consensus weight between, satisfying 0 < κ i < π i,i .

[0031] d) Repeat steps b - c for a total of L times to obtain

[0032] (3) Calculate the local estimator by minimizing the cost function. Let Denotes the average arrival cost function, obtained from Step 2 and characterize

[0033]

[0034] At this time,

[0035]

[0036] By setting the derivative equal to 0, we can obtain where:

[0037]

[0038]

[0039]

[0040] Δ i = F T QF+(C i ) T R i C i

[0041] (4) Calculate the predicted state and the corresponding weight matrix The update rule is as follows:

[0042]

[0043]

[0044] where α and S are adjustable parameters, and should satisfy 0 < α < 1, S is a positive definite matrix. The other terms are as follows:

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051] The beneficial effects of the present invention are as follows: The method of the present invention mainly calculates the mean value of the arrival cost in the receding horizon estimation by using a consistency protocol with privacy protection. Specifically, the state of each network node is randomly divided into two parts: visible and private, and only the visible part is transmitted outward to protect the node privacy. Then, the local estimator is calculated by minimizing the cost function of the receding horizon estimation, and the arrival cost function is updated according to the local estimator. It has been verified that this method can effectively achieve state estimation in a distributed system and can protect the privacy of the initial state of the nodes. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flowchart of the method of the present invention.

[0053] Figure 2 It is a schematic diagram of consistency based on state decomposition.

[0054] Figure 3 It is a topology diagram of a wireless sensor network.

[0055] Figure 4 It is a curve graph of the root mean square error of position varying with time. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0056] The technical solution of the present invention has been described in detail in the section of the invention content, and will not be repeated here. Next, the feasibility of the present invention will be analyzed in combination with simulation examples.

[0057] Analysis of average consistency and privacy:

[0058] For the traditional average consistency algorithm

[0059]

[0060] It can be written as

[0061] ξ t,l = Πξ t,l-1 = Π l ξ t,0

[0062] Among them, Π is a square matrix composed of π i,j If there is an edge between node i and node j, then π i,j = π j,i > 0, otherwise π i,j = 0, and By default, the entire network is connected. At this time, the square matrix Π is a doubly stochastic initial matrix, which satisfies

[0063]

[0064] Among them Therefore

[0065]

[0066] After multiple rounds of iteration, each network node will converge to the mean value of all in the network.

[0067] The average consensus algorithm with privacy protection decomposes all nodes into a public part and a private part As Figure 2 shown, the public part replaces the part of the original node in the network, and the private part is only connected to its public part. The consensus weight is set to κ i , 0 < κ i < π i,i . The decomposed network remains connected, and its consensus weight matrix still maintains the double-stochastic and initial characteristics. Therefore, after normal consensus protocol iteration, the decomposed network will still converge to the mean value of the network. Also, because the mean value of the entire network is the same as that before decomposition, so each network node and will converge to the mean value of all in the original network. This method will not change the result of network convergence, but since what is transmitted in the network are all randomly generated and κ i which is only visible to node i itself, it can effectively prevent the initial information of the node from being stolen by external parties.

[0068] Estimation error analysis:

[0069] Let Because is not the minimum solution, there is

[0070]

[0071] From simple inequalities, we can get

[0072]

[0073]

[0074]

[0075] where W t = [(w t-N ) T ,…(w t-1 ) T T , From the upper and lower bound inequality​

[0076]

[0077] It can be obtained that

[0078]

[0079]

[0080] Let

[0081]

[0082] where \(p\) i is the eigenvector corresponding to the eigenvalue 1 of the consistency weight matrix \(\Pi\). Because

[0083]

[0084] Also, from and the update formula of \(\cdots\) it is easy to obtain

[0085]

[0086] Therefore, it can be proved that

[0087] \(\upsilon\) t (\(e\) t-N ) \(\leq \alpha\upsilon\) t-1 (\(e\) t-N-1 ) + \(\rho\)

[0088]

[0089] So

[0090]

[0091] That is

[0092]

[0093] It can be proved that there exists \(\gamma\) such that

[0094]

[0095] Therefore, there exists \(K\) such that the estimation error

[0096] Simulation experiment:

[0097] 20 network nodes are randomly distributed in the area \([0, 600]\times[0, 600]\), among which 5 have the ability to sense external information, and the other 15 are only used as communication nodes. The entire network topology is as shown in Figure 3As shown. Let the window length N of the rolling horizon estimation be 5, the total process time be 100 + N, and the number of iterations K of the consensus protocol be 1000. After 10 Monte Carlo simulations, the root mean square error of the position is as Figure 4 shown. It can be seen that the root mean square error of the position quickly stabilizes at a low level, indicating that this method ensures accuracy on the basis of providing scalability and privacy.

Claims

1. A distributed receding horizon estimation method for moving targets with privacy protection, which is used for the scenario of tracking moving targets through a linear sensor network. A linear model with a state transition matrix is established as follows: x t+1 = Fx t + w t where x t = [p x , v x , p y , v y T represents the motion state of the target, p x , p y represents the position coordinates of the target, v x , v y represents the speed of the target, and F represents the state transition matrix:​ where T s is the tracking time interval, and the interference noise w t obeys a uniform random distribution on [-0.5, 0.5]. The entire sensor network consists of nodes, and the observation model of node i is y t i = C i x t + v t i where y t i represents the sensor measurement value, and C i is the observation matrix: Observation noise obeys a uniform random distribution on [-10, 10]; define the weighting matrices Q and R i equal to w respectively t and the inverse covariance matrix; In the distributed sensor network, each node uses the local information it can obtain to perform receding horizon estimation on the system state. The receding horizon estimation estimates the system state by minimizing a cost function defined on a sliding window of a fixed size. The cost function is as follows: where N represents the window length of the rolling time domain, represents the estimated value of the target state at time k by sensor i at time t, is the arrival cost function: Denote the predicted value of the target state at time t-N by sensor i, is a positive definite weight matrix that quantifies the confidence of the predicted value ; Use the average arrival cost function of the entire network to replace the arrival cost function of each node itself, and take the mean of the arrival cost function, which is equivalent to taking the mean of the parameters and Take the mean; The specific method is that for any node i in the network at any time t, the following steps are executed: (1) Obtain observation information (2) Obtain the average arrival cost function of the entire network through a privacy-protected consensus algorithm. Specifically: a. Initialize the cost function parameters Each network node decomposes its own parameters arbitrarily into and such that it satisfies Decompose the parameters arbitrarily into and such that it satisfies b. Let and represent the and at the l-th iteration respectively, where 1 < l < L and L is the total number of iterations. Send and to its neighbors and receive and c. Update using the following formula where π is the consistency weight matrix, to enable the network to converge to the mean, κ i is the consistency weight between, satisfying 0 < κ i < π i,i ; d. Repeat steps b to c for a total of L times to obtain (3) Calculate the local estimator by minimizing the cost function, and let denote the average arrival cost function, which is obtained from step (2) and characterize: At this time, By setting equal to zero, we can obtain where: Δ i = F T QF + (C i ) T R i C i (4) Calculate the predicted state of the moving target and the corresponding weight matrix The update rule is as follows: where α and S are adjustable parameters, and should satisfy 0 < α < 1. S is a positive definite matrix, and other terms are shown as follows:

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