IMM fusion estimation with multiple asynchronous sensors

doi:10.1016/j.sigpro.2014.02.019

Signal Processing

Volume 102, September 2014, Pages 46-57

https://doi.org/10.1016/j.sigpro.2014.02.019 Get rights and content

Highlights

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An asynchronous IMM fusion estimation algorithm is proposed for stochastic multi-model systems with multiple asynchronous sensors, whose sampling rates and initial sampling time instants are arbitrary.
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The complex noise correlations are calculated explicitly and the model transitions at the asynchronous sampling time instants in the fusion interval are considered.
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An equivalent recursive form is derived to reduce the computational complexity of the proposed algorithm.
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The proposed algorithm avoids the counter-intuitive performance degradation phenomenon of the sequential IMM filtering approach.

Abstract

This paper presents an asynchronous IMM fusion estimation algorithm for stochastic multi-model systems with multiple asynchronous sensors. Sampling rates of the sensors we considered are arbitrary as well as initial sampling time instants. Asynchronous measurements collected in each filtering interval are sorted in time sequence, and transformed to the fusion time instant as an equivalent measurement. Then, the equivalent measurement is used to update elemental filters in IMM, taking into account the correlation between the equivalent measurement noise and the process noise. Model transitions at asynchronous sampling time instants in the fusion interval are considered. Elemental filters are both re-initialized and updated conditioned on the model transition sequence in the fusion filtering interval. The fused estimate and covariance are obtained by combining model-sequence conditioned estimates and covariances with probabilities of corresponding model sequences. In addition, an equivalent recursive form is derived to reduce the computational complexity of the proposed algorithm. The proposed algorithm avoids the counter-intuitive performance degradation phenomenon of the sequential IMM filtering approach. Finally, simulation results are given to illustrate the feasibility and effectiveness of the proposed algorithm.

Introduction

In the past few decades, multiple model estimation has been well researched to handle the estimation problem for systems with both structural and parametric uncertainties. The basic idea of multiple model estimation is to run a bank of elemental filters simultaneously, each based on a candidate model of the system, and generate the overall estimate by combining the results of these elemental filters according to their probabilistic weights. Compared with the direct two-stage strategy where single model estimation is performed after a mode decision process, multiple model estimation accomplishes joint estimation and decision with the potential of obtaining the globally optimal solution.

As one of the most cost-effective approach for multiple model estimation, IMM was first introduced by Blom in [1], [2]. The superiority of IMM to other multiple model approaches, such as autonomous MM (AMM) and generalized pseudo-Bayesian algorithms of order n (GPBn), stems from a special input mixing step introduced at the beginning of each elemental filter in each cycle. It is this effective input mixing process that leads to an improvement in performance while without significantly increasing the computational requirement. Traditional IMM considers discrete-time hybrid system with Gaussian measurement noises and discrete-time model transitions. Mori et al. in [3] proposed a novel IMM extrapolation algorithm, where system models as well as their transitions were all formulated in continuous time. Huang et al. in [4] studied the state estimation problem of semi-Markovian switching system with non-Gaussian measurement noise. A maximum likelihood solution was provided by combining the IMM filter with the expectation–maximization algorithm. To analyze the performance of IMM, Sean et al. in [5] proposed an algorithm to compute the means of the likelihood functions and mean-squared estimation errors of IMM based on the means and cross-covariances of each Kalman filter residual in IMM, and the cross-covariances of each two Kalman filter residuals. The works of Bar-Shalom et al. in [6], [7] made IMM a prevailing approach for maneuvering target tracking, and a comprehensive review on this topic is provided by Li et al. in [8]. Recently, Yuan et al. in [9] presented a multiple IMM approach to estimate the state of thrusting projectiles, where an unbiased mixing process was introduced to obtain more accurate estimates of the drag coefficient and thrust. Glass et al. proposed IMM estimators with unbiased mixing for tracking targets performing coordinated turns [10]. Mao et al. studied the performance evaluation problem of IMM using the dynamic error spectrum in [11]. In addition, IMM has been wildly used in other fields such as industrial fault diagnosis and prognosis [12], intelligent transportation systems [13], and even automatic speech recognition systems [14].

Although IMM technique is well researched in the past, most of existing works are limited to single-sensor systems. However, multiple sensors are often used in many systems, such as sensor networks and target surveillance systems, in order to improve accuracy, reliability and robustness. Compared with the rich literature on IMM for single-sensor systems, only a limited number of results can be available on the IMM fusion filtering and estimation problem for multiple sensor systems. Ding et al. in [15] proposed a distributed IMM fusion algorithm for multi-platform tracking. A standard IMM was carried out at each platform based on its own independent mode set. The combined tracks at local platforms were transmitted to the fusion center and further fused there with a constructed global model. Based on the work in [15], Hong et al. in [16] developed a distributed multirate IMM fusion algorithm, where out-of-sequence measurements were considered.

The works mentioned above are still restricted to systems with multiple synchronous sensors. However, in practice, sensors may have different sampling rates, different initial sampling instants, or even different communication delays to the fusion center, which leads to the result that data from different sensors are not coincident in time. In other words, sensors in practical applications are more likely to be asynchronous, rather than synchronous. Watson et al. in [17] tried to derive an IMM track fusion algorithm for multiple asynchronous measurements, but only a numeric solution was provided. Moreover, to simplify the problem, it was assumed that in each fusion interval model transition took place at most once at the beginning of this time interval. Since the real model transition of the system concerned is unknown and may take place many times at any time instants during a fusion time interval, such an unreasonable assumption will inevitably lead to the degradation of estimation performance. Ruan et al. in [18] fused two asynchronous sensors by updating the IMM filtering sequentially once an asynchronous measurement was obtained. However, the illustrative examples showed that the performance of IMM was degraded, rather than improved if the additional sensor was relatively less accurate than the original one. To sum up, very little attention has been paid to the IMM fusion estimation problem for systems with multiple asynchronous sensors despite its important engineering significance due to the widespread presence of asynchronous sensors in practical systems. The asynchronous IMM fusion estimation problem is still an open yet challenging issue. There still lacks effective asynchronous IMM algorithms, which are applicable to arbitrary number of asynchronous sensors, take into account all the possible model switchings, have explicit analytical solutions, and most important, can achieve better fusion performance than single sensor alone.

Motivated by above considerations, the purpose of this paper is to develop a novel IMM fusion estimation algorithm for asynchronous multi-sensor multi-model hybrid systems. We consider the most general cases, where the number of sensors is arbitrary as well as their sampling rates and initial sampling instants. The asynchronous measurements collected in each filtering interval are transformed to the fusion time instant as an equivalent measurement and then fused together. By doing this, the asynchronous IMM fusion filtering algorithm we proposed avoids the counter-intuitive performance degradation phenomenon in [18]. However, due to the common process noise, the equivalent measurement noises after direct measurement transformation are correlated with each other. The process noise and the equivalent measurement noises are also correlated. These correlations are carefully calculated and taken into account in the proposed algorithm. Moreover, unlike the work in [17] where model transition was assumed to take place at most only once at the beginning of the fusion interval, model transitions at all asynchronous sampling time instants during the fusion interval are considered in this paper, which means elemental filters are both re-initialized and updated conditioned on the model transition sequence during the fusion filtering interval. Finally, to reduce the computational complexity of the proposed algorithm, an equivalent recursive form is also derived.

The rest of this paper is organized as follows. In Section 2, the fusion filtering problem for Markov jump linear system with multiple asynchronous sensors is formulated. Then the asynchronous IMM fusion algorithm is developed in Section 3. Section 4 gives an alternative solution to reduce the computational complexity of the proposed algorithm. Simulation results are shown in Section 5 and finally conclusions are drawn in Section 6.

Section snippets

Problem formulation

In this paper, we consider a jump linear system with altogether M candidate models in the model set. For a given model m, the continuous-time dynamic equation is given by $x (t_{j}) = Φ^{m} (t_{j}, t_{i}) x (t_{i}) + ω^{m} (t_{j}, t_{i})$ where $x \in R^{d_{x}}$ is the continuous-valued system state, d_x is the dimension of x, $Φ^{m} (t_{j}, t_{i})$ is the corresponding state transition matrix, $ω^{m} (t_{j}, t_{i})$ is zero mean white Gaussian noise with covariance $Q^{m} (t_{j}, t_{i})$ . The mode evolution is described by a continuous-time homogeneous Markov chain ${(t), t \geq 0}$ with

Asynchronous IMM fusion filtering algorithm

Using the total probability formula, we have ${\hat{x}}_{k} = E {x (t_{k}) | Z^{k}} = \sum_{m = 1}^{M} {\hat{x}}_{k}^{m} μ_{k}^{m}$ where ${\hat{x}}_{k}^{m} = E {x (t_{k}) | (t_{k}) = m, Z^{k}}$ is the state estimate conditioned on model m, $μ_{k}^{m} = Prob ((t_{k}) = m | Z^{k})$ is the posterior probability that model m is in effect at time t_k. Consequently, it follows [6], [8] $P_{k} = E {({\hat{x}}_{k} - x (t_{k})) {({\hat{x}}_{k} - x (t_{k}))}^{T}} = \sum_{m = 1}^{M} [P_{k}^{m} + ({\hat{x}}_{k} - {\hat{x}}_{k}^{m}) {({\hat{x}}_{k} - {\hat{x}}_{k}^{m})}^{T}] μ_{k}^{m}$

Similarly, we have ${\hat{x}}_{k}^{m} = \sum_{l = 1}^{L} {\hat{x}}_{k}^{M_{k, l}^{m}} μ_{k}^{M_{k, l} | m}$ and $P_{k}^{m} = E {({\hat{x}}_{k}^{m} - x (t_{k})) {({\hat{x}}_{k}^{m} - x (t_{k}))}^{T}} = \sum_{l = 1}^{L} [P_{k}^{M_{k, l}^{m}} + ({\hat{x}}_{k}^{m} - {\hat{x}}_{k}^{M_{k, l}^{m}}) {({\hat{x}}_{k}^{m} - {\hat{x}}_{k}^{M_{k, l}^{m}})}^{T}] μ_{k}^{M_{k, l} | m}$ where ${\hat{x}}_{k}^{M_{k, l}^{m}} = E {x (t_{k}) | M_{k, l}, (t_{k}) = m$

Reduction of computational complexity for large N_k

The computational complexity of the asynchronous IMM fusion algorithm presented in Section 3 is $O ({(\sum_{i = 1}^{N_{k}} d_{z}^{i})}^{3})$ , determined by $\sum_{i = 1}^{N_{k}} d_{z}^{i}$ dimensional matrix inversions of ${[R_{k}^{⁎} - {(S_{k}^{⁎})}^{T} {(P_{k | k - 1}^{⁎})}^{- 1} S_{k}^{⁎}]}^{- 1}$ and ${[H_{k}^{⁎} P_{k | k - 1}^{⁎} {(H_{k}^{⁎})}^{T} + R_{k}^{⁎}]}^{- 1}$ in (30), (31), (36). The computational complexity increases sharply with N_k, the number of asynchronous measurements collected in time interval $(t_{k - 1}, t_{k}]$ . For this reason, the proposed asynchronous IMM fusion algorithm is computationally inefficient, or even infeasible

Simulation results

We consider a single target tracking scenario in a 2D space. The target is located initially at the coordinate origin in $\vec{x} - \vec{y}$ plane. It starts to move according to the constant velocity (CV) model with initial velocity (10 m/s, 10 m/s), and then switches to the coordinated turn (CT) model at t=30 s, and then back to the CV model again at t=60 s. The simulation ends up at t=90 s. The state vector consists of the position and velocity along $\vec{x}$ and $\vec{y}$ , respectively. The CV and CT model are described

Conclusions

In this paper, we present a novel solution to IMM fusion estimation problem with multiple asynchronous sensors. The asynchronous measurements in each fusion interval are all transformed to the fusion time instant, and then fused together. The correlations between any two of the transformed measurement noises are carefully calculated and taken in to account in the proposed algorithm as well as the correlation between the equivalent measurement noise and the process noise. The structure of the

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61304105 and 61125306, and China Postdoctoral Science Foundation under Grant 2013M540864, and the Open Foundation of Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis.

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IMM fusion estimation with multiple asynchronous sensors

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Asynchronous IMM fusion filtering algorithm

Reduction of computational complexity for large Nk

Simulation results

Conclusions

Acknowledgments

Signal Process.

Math. Comput. Model.

Signal Process.

Maximum likelihood state estimation of semi-Markovian switching system in non-Gaussian measurement noise

IEEE Trans. Aerosp. Electron. Syst.

Algorithm for performance analysis of the IMM algorithm

IEEE Trans. Aerosp. Electron. Syst.

Multitarget-Multisensor Tracking: Principles and Techniques

IMMPDAF for radar management and tracking benchmark with ECM

IEEE Trans. Aerosp. Electron. Syst.

Survey of maneuvering target tracking. Part V. Multiple-model methods

IEEE Trans. Aerosp. Electron. Syst.

Reduction of computational complexity for large N_k