https://orcid.org/0000-0003-4264-2346
Figures
Abstract
In nonlinear multisensor system, abrupt state changes and unknown variance of measurement noise are very common, which challenges the majority of the previously developed models for precisely known multisensor fusion techniques. In terms of this issue, an adaptive cubature information filter (CIF) is proposed by embedding strong tracking filter (STF) and variational Bayesian (VB) method, and it is extended to multi-sensor fusion under the decentralized fusion framework with feedback. Specifically, the new algorithms use an equivalent description of STF, which avoid the problem of solving Jacobian matrix during determining strong trace fading factor and solve the interdependent problem of combination of STF and VB. Meanwhile, A simple and efficient method for evaluating global fading factor is developed by introducing a parameter variable named fading vector. The analysis shows that compared with the traditional information filter, this filter can effectively reduce the data transmission from the local sensor to the fusion center and decrease the computational burden of the fusion center. Therefore, it can quickly return to the normal error range and has higher estimation accuracy in response to abrupt state changes. Finally, the performance of the developed algorithms is evaluated through a target tracking problem.
Citation: Guan B, Tang X (2020) Multisensor decentralized nonlinear fusion using adaptive cubature information filter. PLoS ONE 15(11): e0241517. https://doi.org/10.1371/journal.pone.0241517
Editor: Qichun Zhang, University of Bradford, UNITED KINGDOM
Received: July 17, 2020; Accepted: October 15, 2020; Published: November 5, 2020
Copyright: © 2020 Guan, Tang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: This work is all supported by the National Natural Science Foundation of China (Nos.61403218 to BLG, 61503336 and 61602141 to XFT). There was no additional external funding received for this study.
Competing interests: The authors have declared that no competing interests exist.
Introduction
For function and purpose of data fusion, multisensor data fusion is defined as the combination of data from a variety of sources (including sensors, or data from the network reporting) that provides inferences of a quality far superior to any single sensor or single information source [1]. Multisensor fusion techniques have long been a fascinating focus of research attracting constant attention from a variety of practical applications. These include mobile robot navigation, surveillance, air-traffic control and intelligent vehicle operations [2–5].
It is worth noting that the models of practical systems are usually nonlinear. Therefore, it is basic and important to design high performance nonlinear filter before fusion estimation. A number of suboptimal algorithms have been developed to solve the nonlinear filtering problem such as extended Kalman filter (EKF) [2], unscented Kalman filter (UKF) [6], particles filter (PF) [3], and Minimum entropy filter (MEF) [7,8]. Recently, a relatively new nonlinear filtering algorithm named cubature Kalman filter (CKF) is proposed in [9]. As a Gaussian approximation of Bayesian filter, CKF shows better filtering estimation capability than existing Gaussian filters. On the other hand, Cubature Information Filter (CIF), as the information form of CKF, is also used to make up for the deficiencies of CKF algorithm [10–12].
A limitation in aforementioned methods is that the models of system must be exact and known. However, in many nonlinear filtering or fusion problems considered in practice, the models of system are usually uncertain. These uncertainties include the sudden change of state and unknown variance of measurement noise. Strong tracking filter (STF) is a powerful tool to cope with the first case [13]. For the second case, variational Bayesian method is a good choice [14]. Naturally, a novel approach is proposed by combining the STF with the VB method [15].
Unfortunately, this method is only suitable for single sensor (i.e., local sensor). Moreover, an interdependent problem of some parameters arises in iterative calculation. The motivation of this paper is to investigate decentralized fusion problem of nonlinear multisensor system with modeling uncertainty. The key is to solve interdependent problem of combination of STF and VB, and derive efficient algorithm for computing global fading factor.
Brief review of cubature information filter
In this section, the information form of CKF is briefly reviewed. The nonlinear filtering problems with additive process and measurement noise can be modeled using some discrete-time difference equations
(1)
(2)
where x(k) ∈ ℜn×1 and z(k) ∈ ℜm×1 are the state of the system and the measurement at time stepk, respectively; f(⋅) and h(⋅)are some known nonlinear functions; w(k) ∈ ℜn×1 and v(k) ∈ ℜm×1 are the uncorrelated zero-mean Gaussian white noises with covariances
where R(k) is an unknown diagonal covariance matrix.
x(0) is the initial target state with mean x0 and variance P0, and independent of w(k) and v(k).
The cubature information filter (CIF) is a modified version of CKF. In CIF, the information states and the inverse of the information matrix are the main two parameters that should be deal with. The details of CIF can be summarized as follows [16–18]:
Time update
The predicted covariance and the predicted state have equivalent information forms [16–18]:
(3)
where
(4)
(5)
(6)
where S(k − 1|k − 1) is the square root factor of P(k − 1|k − 1). Point set {ξi}is defined in [9].
Measurement update
With respect to ŷ (k|k − 1) and Y(k|k − 1), their updated expressions are given by
(7)
Here, the contribution of information state i(k) and the corresponding information matrix I(k) are given in [16–18]
(8)
(9)
where
(10)
is H(k) is pseudo measurement matrix. γ(k) is measurement innovation. The predicted measurement
and cross-covariance Pxz(k|k − 1) are given in [16–18]
(11)
where S(k|k − 1) is the square root factor of P(k|k − 1).
Finally, the state vector and the covariance matrix can be therefore obtained by
(12)
where En is the n × n identity matrix.
Adaptive CIF using strong tracking filter and variational bayesian
However, CIF has bad performance in practical system with modeling uncertainty. In this article, we consider two cases of uncertainty. One is the sudden change of state; the other is unknown variance of measurement noise. In view of those, we introduce strong tracking filter technology and variational Bayesian technology to improve CIF, and propose an adaptive CIF (ACIF-STF-VB).
Strong tracking filter
The strong tracking filter (STF) based on the orthogonal theory can reduce adaptively estimate bias and thus has ability to track abrupt changes in nonlinear systems [13]. Therefore, for the CIF, a modified state prediction error covariance with the fading factor λ(k) is given in [16,17]
(13)
The time-varying suboptimal fading factor λ(k) can be calculated as follows:
(14)
where, c(k) = tr[N(k)]/tr[M(k)], and N(k) and M(k) is given in [16,17]
(15)
where,
(16)
where, the forgetting factor φ should be 0 < φ ≤ 1. The softening factor τ ≥ 1plays a role of smoothing the state estimation. It is worth noting that some Jacobian matrixes are not computed.
In order to guarantee the normal execution of the algorithm, we need to compute pseudo measurement matrix H(k) and innovation γ(k). Those parameters can be evaluated as follows:
Variational bayesian approximation
Clearly, the fading factor λ(k) cannot be determined if the variance R(k) is unknown. But VB can help to estimate R(k) online. VB approximates the optimal posteriori distribution of state and measurement noise by a free form distribution of the factorization. Based upon VB, estimation formula of variance R(k) are suggested in [14,15,18]:
- Evaluate parameters predict of variance R(k)
(17) where “•” is the point operation in Matlab. ρ,α(k), β(k) are all m-dimensional column vectors and ρi ⊂ (0,1)(i = 1,⋯,m)
- Iteration initialization, set t = 0 and the number of iterations be N1. In practical application, N1 is usually no more than 3., and
(18)
- Compute the covariance of measurement noise
(19) where operation diag(*) means that diagonal entries of matrix ‘*’ form a column vector.
- Estimate the updated state
and its error covariance Pt+1(k|k) by Eqs (3)–(12).
- Update parameter βt+1(k)
(20)
- t = t+1, repeat step 2)-6) until t = N1.
- Complete the iteration, and
(21)
ACIF-STF-VB
In this subsection, a novel adaptive filter is presented by embedding the fading factor and Variational Bayesian estimation of R(k), which is called ACIF-STF-VB. The procedure of the ACIF-STF-VB is described as follows:
- Evaluate predicted state
by Eq (4).
- Estimate
according to Eqs (17)–(19).
- Calculate H(k) and γ(k) in terms of step a)-c) in subsection of strong tracking filter.
- Compute λ(k), Y(k|k − 1) and
by Eqs (3), (13)–(16).
- Iteratively estimate
and Pt(k|k) by using Eqs (7)–(12).
- If t < N1, update parameter βt+1(k) by Eq (20), let t = t + 1 and go back to ii); else go to vii).
- Iteration completed, and
Obviously, estimating variance R(k) and determining the fading factor λ(k) are interdependent in ACIF-STF-VB. There are many differences between ACIF-STF-VB with VB-ACSTIF presented in [15]. 1) All Jacobian matrixes are avoided to calculate in ACIF-STF-VB. 2) Interdependent problem on the computation of H(k) and λ(k) is solved by introduce original error covariance Pori(k|k − 1). 3) In VB-ACSTIF, only iteration estimate is used to evaluate λ(k). In ACIF-STF-VB, both
and λ(k) are corrected in each iteration. In a word, ACIF-STF-VB has better performance than VB-ACSTIF.
ACIF-STF-VB in multisensor decentralized fusion
In this section, we specifically consider a decentralized fusion framework with feedback. As shown in Fig 1, in this configuration, each local sensor has its own information processor. Compared with processing characteristics of Kalman filters, the decentralized fusion structure is easier to fuse information from multiple sensors, simply by adding information contributions to the information vector and associated matrix.
Suppose the different sensors used for state estimation be given by
(22)
where, zs(k) ∈ Rm×1 is a measurement vector from the sensor s (s = 1,2,⋯, N); hs(⋅) is the corresponding nonlinear functions; vs(k) ∈ Rm×1 is the zero-mean Gaussian white noises with diagonal covariances Rs(k).
Suppose that information contributions (is(k) and Is(k)) of each sensor are available at time k. Then, the updated information vector and information matrix in fusion center can be obtained as follows
(23)
(24)
where
(25)
Therefore, the key for fusion estimation is to compute global fading factorλg(k). Clearly, λg(k) cannot be simply formulated using local fading factor λs(k). In order to reveal the relationship between two fading factors, we will rewrite the calculation process of λs(k). Define a five-dimensional parameter vector (This parameter is named as fading vector) of the sensor s as follows:
(26)
where
Combining Eq (26) with Eqs (14)–(23), it obtains
(27)
where, cs(k) = qs(k)/ps,4(k), and
(28)
where
(29)
That is to say, λs(k) can be determined by fading vector ps(k) in terms of Eqs (27)–(29). Similarly, we can define pg(k) as a global fading vector. Then, we can contain the following theorem 1.
Theorem 1 For decentralized fusion using strong tracking filter, if all local fading vector ps(k)are available at time k, then the global fading vector pg(k)can be calculate in form
(30)
According to the definition of pg(k), We have
(31)
where,
Second component of pg(k) is
(32)
Third components of pg(k)is
(33)
Fourth component of pg(k) is
(34)
where
is used in above equation. This is because there is a feedback from fusion center to local sensor.
Fifth component of pg(k) is
(35)
Finally, we discuss first component.
So, holds if k = 1.
That is to say, holds if k = 2.
Assume that k = l (l > 2), , then
(38)
In a word, holds for arbitrarily k.
In summary, always holds. So, the proof of theorem 1 is completed.
According to theorem 1 and combining with Eqs (27)–(29), it’s easy to yield
(39)
where, cg(k) = qg(k)/pg,4(k), and
(40)
where
(41)
In this section, we develop an effective method for computing global fading factor pg(k) by introducing fading vectors ps(k). Obviously, the proposed method reduces the computational demands of fusion center, but also reduces the burden of communication transmission.
Numerical examples
To test and verify the performance of ACIF-STF-VB and the decentralized fusion algorithm (called DF-ACIF-STF-VB), we consider the problem of tracking a target in two-dimensional space.
The target moves as a CV model, which is given by
where x(k) is the state of the aircraft and
. x(k) and y(k) are the coordinates for the position of the aircraft.
and
are the velocity in the corresponding axis. ΔT is the sampling time, and ΔT = 1s. In this section, the simulation time is 100s. w(k − 1) is a zero-mean Gaussian noise vector with covariance Q, and
where q(k) is power spectral density and q(k) = 0.15m2⋅s−3.
Consider a nonlinear target tracking system composed of three radars and the nonlinear measurement model is expressed as
where (xs,i, ys,i) is the position of the radar i. Suppose the position axis of three radars are (0m, 0m), (-2000m, 0m) and (2000m, 0m). vi(k) is an additive zero-mean Gaussian noise vector with variance
. σr,i = 5m and σb,i = 0.1°.
We assume Ri(0) = diag[(81m)2, (0.3°)2] is known so that the algorithms can normally work. For the STF and VB method in examples, the corresponding parameters are φ = 0.95, τ = 3; N1 = 2, ρ = [1 − e−6, 1 − e−6]T, α(0) = [1,1]T, β(0) = [100,0.05]T.
Initial state and error covariance matrix are set
For comparison purposes, we use the square error of position (SEP(k)), square error of velocity (SEV(k)), mean square error of position (MSEP) and mean square error of velocity (MSEV) defined in [19].
Example 1
This example is used to validate the proposed ACIF-STF-VB compared with VB-ACIF in [15]. In this case, an abrupt state change happens at k = 15s. We set x(k) = 1.1x(k), y(k) = 1.1y(k). For simplicity, we choose the radar fixed at the origin. We compare the two algorithms by the square error of position and the square error of velocity. The simulation results are illustrated by Figs 2 and 3.
From Figs 2 and 3, the results show: 1) When the target is in normal motion within 0~15s, both of the algorithms can track the target well. 2) As an abrupt state change take place at 15s, both of the algorithms have large tracking errors simultaneously. 3) After a short time of adjustment, the ACIF-ST-VB algorithm can track the target quickly, and the tracking error curve of the response tends to converge. In contrast, the tracking error of VB-ACIF algorithm is increasing and the curve diverges gradually after the target state mutation, which means it can no longer track the target.
The mean square error of two methods is given by Table 1. The results show that ACIF-STF-VB has better accuracy than VB-ACIF. Obviously, the strong tracking fading factor improves the accuracy and stability of the algorithm.
Example 2
In this example, we compare the estimation accuracy of DF-ACIF-STF-VB and ACIF-STF-VB. Then, the results are given by Figs 4 and 5 and Table 2.
From Figs 4 and 5, it is obvious that DF-ACIF-STF-VB has better estimation accuracy than ACIF-STF-VB. In addition, Table 2 also shows that mean square error of the former is lower than that of the latter. Specifically, the estimation accuracy of MSEP and MSEV of DF-ACIF-STF-VB increased by 45.45% and 24.57% respectively. These results are consistent with the general conclusion of multisensor fusion theory, this is, the fusion results should be superior to the single sensor results. All these results verify the effectiveness of the decentralized fusion algorithm. In other words, the calculation method of the global factor is effective.
Conclusions
This article investigates the problem of multisensor nonlinear fusion with sudden change of state and unknown variance of measurement noise. In order to deal with these uncertainties, strong tracking filter (STF) and variational Bayesian (VB) technology are adopted in cubature information filter (CIF), a novel adaptive CIF is proposed, which is called ACIF-STF-VB. Compared with existing method in [15], ACIF-STF-VB does not require the evaluation of Jacobians and solves the interdependent problem on the computation of some parameters. Subsequently, proposed algorithm is easy to extend for multisensor state estimation in decentralized fusion framework with feedback. We define a parameter named fading vector, which is convenient to evaluate global fading factor and reduce the computational burden of fusion center. The experimental results demonstrate the effectiveness of the proposed algorithms. It will be an interesting future research topic to consider the fusion system with correlated noises.
Supporting information
S1 Fig. Information filter for decentralized fusion.
https://doi.org/10.1371/journal.pone.0241517.s001
(TIF)
S1 Table. Mean square error of two algorithms.
https://doi.org/10.1371/journal.pone.0241517.s006
(PDF)
S2 Table. Mean square error of two algorithms.
https://doi.org/10.1371/journal.pone.0241517.s007
(PDF)
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