CN110516198B - Distributed nonlinear Kalman filtering method - Google Patents

Distributed nonlinear Kalman filtering method Download PDF

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CN110516198B
CN110516198B CN201910645592.5A CN201910645592A CN110516198B CN 110516198 B CN110516198 B CN 110516198B CN 201910645592 A CN201910645592 A CN 201910645592A CN 110516198 B CN110516198 B CN 110516198B
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夏威
任媛媛
孙美秋
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University of Electronic Science and Technology of China
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Abstract

The invention belongs to the field of signal processing, in particular to a distributed nonlinear Kalman filtering method, which describes the real state of a target by using a posterior distribution function, and approximates the posterior distribution function by using a function distributed by an exponential family; firstly, the invention obtains the intermediate state estimation by optimizing the backward KL divergence between the intermediate posterior distribution approximate function and the real posterior distribution function of each node; then, the final state estimation result of each node is calculated by optimizing the convex combination of the forward KL divergence between the final posterior distribution approximation function of each node and the intermediate posterior distribution approximation function of the neighboring node. The invention has high convergence speed, adopts a mode that a plurality of nodes work simultaneously, can effectively save the calculation cost and greatly improve the calculation efficiency, and is widely applied to the fields of positioning, target tracking and the like of a distributed infinite sensor and a nonlinear power system.

Description

Distributed nonlinear Kalman filtering method
Technical Field
The invention belongs to the field of signal processing, relates to a target tracking problem in the field of signal processing, particularly relates to a target tracking problem on a distributed wireless sensor network, and particularly relates to a distributed nonlinear Kalman filtering method.
Background
Because the Kalman filtering is very suitable for real-time signal processing and large state space models, the Kalman filtering has wide application in the fields of communication systems, power systems, aerospace, environmental pollution control, industrial control, radar signal processing and the like since the time comes, and a plurality of achievements are obtained. In recent years, with the explosive growth of sequences and streaming data, kalman filters have been widely used in machine learning.
Kalman filtering is an algorithm for optimally estimating the state of a system by inputting observation data through the system by using a linear system state equation. However, the typical kalman filtering is only suitable for linear signal models, and in real life, more nonlinear environments are often encountered. At present, the research on the general nonlinear filtering problem is quite active, and the common nonlinear filtering is Extended Kalman Filtering (EKF), insensitive kalman filtering (UKF) and the like; however, these algorithms perform linear approximation processing on the nonlinear kalman filter system, and then perform processing by using the method of the linear kalman filter system.
Common centralized computing needs to consume a large amount of time and resources, and a distributed method decomposes problems into a plurality of small parts which are distributed to a plurality of computers for processing, so that the computing cost can be saved, and the computing efficiency is greatly improved. In recent years, cooperative flooding processing over distributed wireless sensor networks has become an effective data processing technique; the processing mode greatly improves the expandability and the flexibility of the network and is widely applied to the fields of environment monitoring, disaster relief management, parameter estimation, target tracking and the like. However, most of the existing distributed non-linear kalman filtering is based on extended kalman or insensitive kalman filtering, and linear approximation processing is performed on the non-linear kalman when the state estimation of each node is calculated.
Disclosure of Invention
The invention aims to provide a distributed nonlinear Kalman filtering method, which adopts a distribution function q (x) of an index family to approximate real posterior distribution, measures the difference between the distribution function q (x) and the real posterior distribution by using KL divergence, optimizes the KL divergence to be minimum, and ensures that the q (x) is closest to the real posterior distribution p (x | y).
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a distributed nonlinear Kalman filtering method comprises the following steps:
step 1, aiming at a node k, according to a posterior distribution function of the t-1 moment of the node k
Figure BDA0002133243790000021
The prior distribution function at time t is calculated with the state transition equation>
Figure BDA0002133243790000022
Step 2, setting a sampling suggestion distribution function at the time t
Figure BDA0002133243790000023
Sampling according to the proposed distribution function such that each particle is independently co-distributed to the proposed distribution:
Figure BDA0002133243790000024
wherein S is the number of input sampling particles,
Figure BDA0002133243790000025
a state vector representing the s-th particle sampled at time t by node k;
and 3, step 3: and calculating the particle weight of the node k according to the following formula, and performing normalization processing:
Figure BDA0002133243790000026
Figure BDA0002133243790000027
wherein the content of the first and second substances,
Figure BDA0002133243790000028
represents the weight of the s-th sampled particle at time t of node k, W k,t Representing the sum of the weights of all the sampled particles at time t, y, of node k l,t Representing the observed value of the neighbor node l of the node k at the time t; />
Figure BDA0002133243790000029
Representing neighbor node l pairs &>
Figure BDA00021332437900000210
Prediction of the observed value; />
Figure BDA00021332437900000211
Representing node k to a particle s state vector>
Figure BDA00021332437900000212
Predicting;
and 4, step 4: calculating an intermediate posterior distribution approximation function based on the particle weights calculated by the above equation
Figure BDA00021332437900000213
In (d) is based on the mean value>
Figure BDA00021332437900000214
And covariance matrix>
Figure BDA00021332437900000215
Figure BDA00021332437900000216
Figure BDA00021332437900000217
To obtain
Figure BDA00021332437900000218
The distribution function of (a) is:
Figure BDA00021332437900000219
and 5: in a distributed network, the intermediate posterior distribution approximation function obtained by each node
Figure BDA00021332437900000220
Diffusion is performed according to an intermediate posterior distribution approximation function of each neighbor node of node k>
Figure BDA00021332437900000221
Calculating to obtain the final state estimation q of the node k k (x k,t )：
Figure BDA0002133243790000031
Wherein, a l,k The weight of the neighbor node l to the current node k is satisfied
Figure BDA0002133243790000032
And calculate q k (x k,t ) Mean value of (a) k,t Sum covariance matrix sigma k,t ：
Figure BDA0002133243790000033
Figure BDA0002133243790000034
μ k,t As a state estimation vector for node k.
The invention has the beneficial effects that:
the nonlinear distributed Kalman filtering method provided by the invention has the following advantages:
1. the algorithm provided by the invention is suitable for a nonlinear environment, and has a wider application range compared with a classical linear Kalman filtering algorithm;
2. the distributed algorithm provided by the invention only needs to obtain the observation information of the neighbor nodes at each node, allows each node to process simultaneously, does not need to send all the observation information to the fusion center for processing, requires less communication energy and has higher operation efficiency;
3. the distributed algorithm provided by the invention has stronger robustness compared with a corresponding centralized algorithm. For the centralized type, when the processing center has a problem, the whole system can not work normally, and the distributed algorithm can effectively avoid the risk of crash of the whole system when the fusion center has a problem;
4. when the intermediate state estimation of each node is calculated, the backward KL divergence between the approximate distribution and the real posterior distribution is directly optimized based on the unbiased algorithm directly optimized by the Monte Carlo technology, and the linear approximation of a nonlinear function is not needed;
5. the method is realized by diffusing the intermediate state estimation result of each node in the neighborhood. When the number of nodes in the topological structure of the distributed network and the number of neighbor nodes corresponding to each node are large, the method provided by the invention has smaller steady-state error fluctuation than that of a corresponding centralized method.
Drawings
Fig. 1 is a schematic flow diagram of each node in the distributed nonlinear kalman filtering method according to the present invention.
Fig. 2 shows a distributed network topology (taking 10 nodes in the network as an example) adopted in the embodiment.
Fig. 3 is a graph of the tracking result of a node in a centralized monte carlo experiment according to the embodiment of the present invention.
Fig. 4 is a diagram comparing MSE to centralized location in accordance with the present invention.
Fig. 5 is a diagram comparing MSE with centralized speed in the example of the invention.
Detailed Description
The invention is further illustrated by the following figures and examples.
In the present invention, consider that
Figure BDA0002133243790000041
Distributed network architecture of nodes, each node being treated with a distribution function of an exponential family>
Figure BDA0002133243790000042
To approximate the a posteriori distribution function &'s for the node>
Figure BDA0002133243790000043
It is called as middleAn approximation function of the posterior distribution, the invention takes>
Figure BDA0002133243790000044
The difference between the two is measured by KL divergence as a Gaussian function; defining the backward KL divergence expression for each node as:
Figure BDA0002133243790000045
where k denotes the kth node of the overall topology, N k A neighborhood network (including itself) representing node k; x is a radical of a fluorine atom k,t Representing the state vector at the time t of the kth node; y is k,t Representing observation data of the node k at the time t; suppose k's neighbor nodes are respectively used
Figure BDA0002133243790000046
Indicates, then>
Figure BDA0002133243790000047
Representing that all the neighbor nodes of k obtain an observation data set at the moment t:
Figure 4
y 1:t represents a set of observation information from time 1 to t; />
Figure BDA0002133243790000049
Set for representing observation information of all neighbor nodes of node k at 1-t
Figure 5
It is assumed that the observed information of each neighboring node is independent of each other, i.e.
Figure BDA00021332437900000411
Are independent of each other;
Figure BDA00021332437900000412
the expression node k is obtained according to the accumulated observation information of all the neighbor nodes at the time from 1 to tEstimating the state of the last posteriori; />
Optimizing the backward KL divergence of each node to minimize the backward KL divergence to obtain an intermediate posterior distribution approximation function closest to the true posterior distribution in the sense of the backward KL divergence
Figure BDA00021332437900000413
Because of the selected->
Figure BDA00021332437900000414
The function being Gaussian distributed, by pairs
Figure BDA00021332437900000415
And &>
Figure BDA00021332437900000416
Performing moment matching to obtain: />
Figure BDA00021332437900000417
E represents expectation; calculated to be->
Figure BDA00021332437900000418
I.e. get>
Figure BDA00021332437900000419
Mean and covariance matrix of (a), i.e. can be determined>
Figure BDA00021332437900000420
A function.
The present embodiment provides a distributed non-linear kalman filtering method, a flow of which is shown in fig. 1; in this embodiment, a state estimation result is obtained by calculating each node in the distributed network;
the method comprises the following specific steps:
step 1, aiming at the node k, according to the posterior distribution function of t-1 of the node k
Figure BDA0002133243790000051
The prior distribution function at time t is calculated with the state transition equation>
Figure BDA0002133243790000052
Figure BDA0002133243790000053
Representing prior state estimation obtained by a node k according to observation information of all neighbor nodes (including the node k) at 1-t-1 moment;
step 2, setting a sampling suggestion distribution function at the time t
Figure BDA0002133243790000054
Sampling according to the proposed distribution function such that each particle is independently identically distributed to the proposed distribution:
Figure BDA0002133243790000055
wherein S is the number of input sampling particles,
Figure BDA0002133243790000056
a state vector representing the s-th particle sampled by node k at time t;
and 3, step 3: and calculating the particle weight of the node k according to the following formula, and performing normalization processing:
Figure BDA0002133243790000057
Figure BDA0002133243790000058
wherein the content of the first and second substances,
Figure BDA0002133243790000059
representing the weight, W, of the s-th sampled particle at time t of node k k,t Represents the sum of the weights of all the sampled particles at time t of node k, y l,t Representing the observed value of the neighbor node l of the node k at the time t; />
Figure BDA00021332437900000510
Represents neighbor node/pair->
Figure BDA00021332437900000511
Predicting the observation value by calculating an observation equation; />
Figure BDA00021332437900000512
Represents the status vector ≥ of node k for particle s>
Figure BDA00021332437900000513
The prediction is obtained by calculating a state transition equation;
and 4, step 4: particle weight calculation from the above equation
Figure BDA00021332437900000514
Figure BDA00021332437900000515
Therefore, the temperature of the molten steel is controlled,
Figure BDA00021332437900000516
due to being ^ based>
Figure BDA00021332437900000517
A Gaussian distribution function, i.e. get->
Figure BDA00021332437900000518
The distribution function of (c) is as follows:
Figure BDA00021332437900000519
wherein the content of the first and second substances,
Figure BDA00021332437900000520
/>
and 5: in a distributed network, each willIntermediate posterior distribution approximation function obtained by node
Figure BDA00021332437900000521
Performing diffusion according to the intermediate posterior distribution approximation function of each neighbor node of the node k>
Figure BDA00021332437900000522
Correcting the state estimation of the node k to obtain the final state estimation q of the node k k (x k,t )；
Defining the forward KL divergence between the final state estimation of the node k and the intermediate posterior distribution approximation function of the neighbor nodes as:
Figure BDA0002133243790000061
wherein, a l,k The weight of the neighbor node l to the current node k is satisfied
Figure BDA0002133243790000062
Minimizing the forward KL divergence, we get:
Figure BDA0002133243790000063
because in which>
Figure BDA0002133243790000064
All obey a Gaussian distribution, so q k (x k,t ) Q, also following a Gaussian distribution, calculated by k (x k,t ) Mean value μ k,t Sum covariance matrix Σ k,t
Figure BDA0002133243790000065
Figure BDA0002133243790000066
Resulting in a final state estimate: q. q of k (x k,t )～N(μ k,t ,Σ k,t )，q k (x k,t ) The mean value is the state estimation vector of the node;
step 6: and carrying out time iteration to obtain a state estimation result at the next moment.
Simulation conditions
Simulation experiment: the method provided by the invention is used for target tracking of a distributed network, and is compared With a centralized method provided by an article 'Nonlinear Kal-man Filtering With university Minimization'. The topological structure of the whole distributed network is 10 nodes, the topological structure is shown in fig. 2, one sensor at each node respectively tracks a target, and the observed noise power of each node is the same; the state space model of the object motion is as follows:
x t ＝F t x t-1 +w t ,w t ～N(0,Q t )
y t ＝h(x t )+v t ,v t ～N(0,R t )
wherein, F t Represents the state transition matrix, h (x) t ) Representing an observation function, w t Is state noise, v t Is the observation noise, w t And v t Mean value is zero and covariance matrix is Q t And R t White gaussian noise of (1):
Figure BDA0002133243790000067
Figure BDA0002133243790000068
Figure BDA0002133243790000071
wherein s represents position information of the sensor; in the present invention, a constant measurement rate is adopted, let Δ t =1,σ CV ＝10 -2 (ii) a Covariance matrix of observed noise is
Figure BDA0002133243790000072
σ R =20. The state vector is four-dimensional: />
Figure BDA0002133243790000073
Position information including two directions, the second dimension and the fourth dimension representing velocity information of the two directions; the number of particles was sampled 500 times, iterated 300 times, and subjected to 100 monte carlo experiments. The simulation results of the method of the invention and the centralized nonlinear Kalman filtering are shown in FIGS. 3, 4 and 5.
FIG. 3 shows that the distributed nonlinear Kalman filtering method provided by the invention can effectively track the target. Fig. 4 and 5 show that compared With the centralized method (labeled as "centralized" in the figure) proposed by "Nonlinear Kalman Filtering With conversion Minimization", the convergence speed of the MSE curve for position and speed estimation is the same, but the steady-state error fluctuation of the method proposed by the present invention is smaller, and the steady-state performance of the method is even slightly better than that of the centralized algorithm.
While the invention has been described with reference to specific embodiments, any feature disclosed in this specification may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps.

Claims (1)

1. A distributed nonlinear Kalman filtering method comprises the following steps:
step 1, aiming at a node k, according to a posterior distribution function of the t-1 moment of the node k
Figure FDA0002133243780000011
The prior distribution function at time t is calculated with the state transition equation>
Figure FDA0002133243780000012
Step 2, setting a sampling suggestion distribution function at the time t
Figure FDA0002133243780000013
Sampling according to the proposed distribution function such that each particle is independently co-distributed to the proposed distribution:
Figure FDA0002133243780000014
wherein S is the number of input sampling particles,
Figure FDA0002133243780000015
a state vector representing the s-th particle sampled at time t by node k;
and step 3: and calculating the particle weight of the node k according to the following formula, and performing normalization processing:
Figure FDA0002133243780000016
Figure FDA0002133243780000017
wherein the content of the first and second substances,
Figure FDA0002133243780000018
representing the weight, W, of the s-th sampled particle at time t of node k k,t Representing the sum of the weights of all the sampled particles at time t, y, of node k l,t Representing the observed value of the neighbor node l of the node k at the time t; />
Figure FDA0002133243780000019
Represents a neighbor node l pair
Figure FDA00021332437800000110
Prediction of the observed value; />
Figure FDA00021332437800000111
Representing node k versus a particle s state vector->
Figure FDA00021332437800000112
Predicting;
and 4, step 4: calculating the intermediate posterior distribution approximation function according to the particle weights obtained by the above formula
Figure FDA00021332437800000113
Is based on the mean value->
Figure FDA00021332437800000114
And covariance matrix ≥>
Figure FDA00021332437800000115
Figure FDA00021332437800000116
Figure FDA00021332437800000117
To obtain
Figure FDA00021332437800000118
The distribution function of (a) is:
Figure FDA00021332437800000119
and 5: in a distributed network, the intermediate posterior distribution approximation function obtained by each node
Figure FDA00021332437800000120
Performing diffusion according to the intermediate posterior distribution approximation function of each neighbor node of the node k>
Figure FDA00021332437800000121
Calculating to obtain the final state estimation q of the node k k (x k,t )：
Figure FDA0002133243780000021
Wherein, a l,k The weight of the neighbor node l to the current node k is satisfied
Figure FDA0002133243780000022
And calculate q k (x k,t ) Mean value of (a) k,t Sum covariance matrix Σ k,t ：/>
Figure FDA0002133243780000023
Figure FDA0002133243780000024
μ k,t As a state estimation vector for node k.
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