Multi-sensor federated unscented Kalman filtering algorithm in intermittent observations①

Hu Zhentao (胡振濤)

(*Institute of Image Processing and Pattern Recognition, Henan University, Kaifeng 475004, P.R.China)(**College of Computer and Information Engineering, Henan University, Kaifeng 475004, P.R.China )

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Multi-sensor federated unscented Kalman filtering algorithm in intermittent observations①

Hu Zhentao (胡振濤)②

(*Institute of Image Processing and Pattern Recognition, Henan University, Kaifeng 475004, P.R.China)(**College of Computer and Information Engineering, Henan University, Kaifeng 475004, P.R.China )

Aiming at the adverse effect caused by limited detecting probability of sensors on filtering precision of a nonlinear system state, a novel multi-sensor federated unscented Kalman filtering algorithm is proposed. Firstly, combined with the residual detection strategy, effective observations are correctly identified. Secondly, according to the missing characteristic of observations and the structural feature of unscented Kalman filter, the iterative process of the single-sensor unscented Kalman filter in intermittent observations is given. The key idea is that the state estimation and its error covariance matrix are replaced by the state one-step prediction and its error covariance matrix, when the phenomenon of observations missing occurs. Finally, based on the realization mechanism of federated filter, a new fusion framework of state estimation from each local node is designed. And the filtering precision of system state is improved further by the effective management of observations missing and the rational utilization of redundancy and complementary information among multi-sensor observations. The theory analysis and simulation results show the feasibility and effectiveness of the proposed algorithm.

nonlinear estimation, intermittent observations, unscented Kalman filter, federated filter

0 Introduction

The estimation problem of a nonlinear dynamic system exists widely in target tracking, navigation, fault diagnosis, signal processing and automatic control, etc. The optimal solution for this problem is to obtain a complete description of state posterior distribution, but it is usually difficult to be implemented in actual engineering. Since the accurate description generally needs endless parameters, optimal solution can be obtained only for some specific conditions[1]. Combined with analytical approximation, numerical approximation and Gaussian-sum approximation, some suboptimal filters are proposed, such as extended Kalman filter(EKF)[2], grid filter(GF)[3], and Gaussian-sum filter (GSF)[4], et al. However, with growing complexity and increasing perception requirement to estimated object, the filtering precision and real-time of above algorithms are difficult to meet the needs in actual engineering[5]. In recent years, along with the improvement of computer performance, sampling methods used to approximate posterior probability distribution have attracted much attention of scholars in related fields. According to the difference of sampling strategies, the existing nonlinear filters can be classified into two types: random sampling and deterministic sampling. The typical implementation of random sampling is particle filter (PF). It adopts a new sequential Bayesian estimation idea by using importance sampling and re-sampling technique in Monte Carlo simulation framework. In addition, PF is suitable for nonlinear system with free-form noise distribution[6]. However, the realization mechanism of sequential importance sampling and re-sampling will unavoidably lead to two drawbacks: the particles degeneracy and the depletion of particles diversity after re-sampling. Moreover, the filtering precision of PF is closely related to the dimension of the estimated system and the number of sampling particles, which make parameters lack of universality in applications[7]. The typical implementation of deterministic sampling is unscented Kalman filter (UKF). Its idea is not to approximate the nonlinear function but the probability density distribution of nonlinear function by UT transformation strategy[8]. And for nonlinear system with Gauss noise, it is superior to the existing nonlinear filters in filtering precision, computational burden and operability.

Recently designing work of nonlinear filter in theory and practical application has made great progress, while studies are usually subject to the hypothesis that no observation is missed. Namely, the detection probability of sensor is 1. However, in actual engineering, due to obstacles, bad weather, sensor faults and uncertain factors in surrounding, observations missing, out-of-sequence and communication delay are inevitable. These situations are also known as the intermittent observations[9,10]. The concrete manifestation is that some abnormal data caused by the random missing of observations lead the adverse effect on filtering performance. Therefore, the implementation structure needs to be improved in the design of nonlinear filter. For this problem, some scholars have carried out relevant studies and made some preliminary results. Sinopoli, et al, propose the discrete Kalman filter (KF) in single- sensor intermittent observations, and the statistical convergence property of estimation error covariance matrix is discussed[11]. On this basis, Kar, et al, studied further the asymptotic property of stochastic Riccati equations[12]. Chen, et al, studied the selection of parameters boundary in filter model by means of linear matrix inequality (LMI) in the constraint of estimation error covariance[13]. Aiming at the state estimation of nonlinear system in intermittent observations, Kluge, et al, proves the convergence boundary of error covariance matrix in EKF for different observation models[14]. Combined with the fuzzy theory, Gao, et al, discussed the design of nonlinear filter in intermittent observations[15]. For the problem when the nonlinear of an estimated system and the multi-sensor intermittent observations exist at the same time, Hu, et al, proposed a new realization method of EKF to solve state estimation in typical nonlinear system[16]. Through the introduction of cost function and risk function, Djuric, et al, constructed a cost reference particle filter which effectively improved the excessive dependence of filtering precision on the statistical information of observation noise[17].

In conclusion, the existing achievements for intermittent observations mainly focus on single- sensor observation system, and the estimation results rely heavily on sensor performance. When the detection probability of used sensor is too low, it will result in a sharp drop of filtering precision. It is considered that multi-sensor detection system can play the superiority of coordinated and complementary among multi-sensor observations, overcome the uncertainty and limitation of single sensor, and improve the reliability and robustness of the whole observation system. Therefore, it can improve necessarily the system estimation precision by reasonable utilization of multi-sensor observations information, especially when the estimated system is with the characteristic of intermittent observations.

In this work, multi-sensor intermittent observations model is firstly constructed according to the parameter characteristic of detection probability. And then utilizing the residual detection strategy, the iterative process of single-sensor unscented Kalman filtering algorithm (S-UKF) is designed in intermittent observations. Combining the multi-source information fusion idea with the federated filtering mechanism, a novel multi-sensor federated unscented Kalman filtering algorithm (MF-UKF) is proposed. And the influence of detection probability on state estimation is analyzed and validated. The simulation results show the feasibility and effectiveness of the proposed algorithm.

1 Multi-sensor intermittent observations model

Considering intermittent observations in practical applications, λmis defines as the detection probability of sensor m. And the nonlinear system with multi-sensor intermittent observations can be modeled as follows:

zk,m=dk,mhk(xk)+vk,m

(1)

where zk,mdenotes the observation of sensor m at time k, hk(·) denotes the mapping function from state space to observation space, xkdenotes the state vector of estimated system, and vk,m～N (0,σvk,m) denotes the observation noise, drawn from a Gaussian distribution of which mean and variance are 0 and σvk,m, respectively. dk,mdenotes the discrete random variable and the value is selected as 0 or 1. the probability distribution of dk,mis subject to Prob{dk,m=1}=λm. dm,k=1 indicates that the observation of sensor m is existing. On the contrary, observation is missing. Moreover, dk,mand vk,mshould be subject to the following relationship.

(2)

where δ adopts the limit form, namely, δ→∞. I denotes the unit matrix with appropriate dimension. The system state transfer equation is expressed as

xk=f(xk-1)+uk-1

(3)

where f(·) denotes the transfer function of the system state and uk-1～N (0, σuk-1,m) denotes process noise.

2 Single-sensor unscented Kalman filter in intermittent observations

Considering the influence caused by intermittent observations on the state estimation of nonlinear system, the structure of some existing nonlinear filters need to be improved. In this section, combined with the characteristic of intermittent observations, a single-sensor unscented Kalman filter (S-UKF) is constructed.

In the UKF algorithm, the UT transform strategy is adopted to approximate the probability density distribution of nonlinear function. And UKF employs a set of discrete deterministic sigma points to calculate the mean and variance of Gaussian random state variables. And then the mean and variance of posteriori probability are spread by the state transfer equation. Thereby, the iterative prediction and update process of state estimation are realized in the framework of Bayesian filter[18]. In filtering precision, UKF is superior to EKF using the local linearization technique. Meanwhile, in computational complexity, UKF is better than PF which uses the sequential Monte Carlo simulation method. Allowing the realization of S-UKF is still for single-sensor observation system, for simplicity, label m of sensor m in Eqs(1) and (2) can be omitted. Thus the estimated system model can be degenerated to be a single sensor observation model. The concrete realization of S-UKF can be expressed as follows:

1: Initializing state estimation and its error covariance matrix

(4)

(5)

(6)

(7)

(8)

(9)

(10)

where ε denotes the state distribution parameter. When estimated state is subject to Gaussian distribution, the value is 2[19].

(11)

(12)

(13)

(14)

6: Next, calculating residual vector to realize the identification of observation missing according to the residual detection strategy.

(15)

where j, j denotes jth row and jth column in matrix, C denotes the detection threshold value of residual vector, and its value is usually selected as 3 or 4. If C is too large, observation missing occurs, otherwise, observation false-alarm occurs[21]. In actual engineering, generally it needs to consider the overall performance from each component, and judges the credibility of sensor observation at current time. For example, observations information including radial distance, azimuth and elevation, are needed to take into account simultaneously in the target tracking system. So the constraint condition in Eq.(15) can be modified as

(16)

Kk=Pk,xz(Pk,zz)-1

(17)

when dkis equal to 1, and

(18)

(19)

when dkis equal to 0, and

(20)

Pk/k=Pk/k-1

(21)

(22)

(23)

In accordance with the above construction process of S-UKF, it is known that the object of S-UKF is to identify the random missing of observation by introducing residual detection strategy.

3 Multi-sensor federated unscented Kalman filter in intermittent observations

Although the averse effect on state estimation result can be decreased by the above treatment to some extent. However, it is difficult for the estimation precision and robustness to meet some requirements that higher filtering precision is required. According to the specific situation of intermittent observation, combined the mechanism of federal filter[19]with the utilization of redundancy and complementary information among multi-sensor observations, a novel multi-sensor federated unscented Kalman filter is constructed in this section. The structure frame of MF-UKF is shown in Fig.1.

Fig.1 The structure frame of MF-UKF

(24)

Pk/k,m=Pk/k-1,m-dk,mKk,mPk,zz,m(Kk,m)Τ

(25)

(26)

(27)

(28)

4 Simulation results and analysis

To verify the feasibility and validity of algorithm proposed in this paper, the typical target tracking problem in two-dimensional plane is set as simulation scenario. The target is observed by three radar sensors with different detection probability. According to physical properties of radar, radial distance and azimuth can be given in observation system. zk,mis defined as the observation of radar m, and zk,m=[rk,mθk,m]Τ. rk,mand θk,mdenote radial distance and azimuth at time k, respectively. Combined with the dynamic characteristic of uniform motion target, the state transition equation and the observation equation are as follows:

xk=Fxk-1+Γk-1wk-1

To illustrate the influence caused by the random missing of sensor observation on filtering precision, observation data from radar 1 are respectively adopted to UKF and S-UKF. For the above two algorithms, Fig.2 and Fig.3 show that the root mean square errors (RMSE) in horizontal direction and vertical direction in 1000 times Monte Carlo simulation. The results clearly show that RMSE of UKF is inferior for lack of processing mechanism of observation random missing. And due to error accumulation, the filter divergence even occurs. On the contrary, since the detecting and processing mechanism for the random missing of observation is introduced into S-UKF, the filtering precision is improved effectively relative to UKF.

Fig.2 Horizontal direction

Fig.3 Vertical direction

According to the realization framework of S-UKF, variable dk,mused to indicate whether observation from sensor m exists or not, can be obtained according to inequality constraint expressed in Eq.(16). A simulation experiment with one time is used to verify the estimation validity of dk,min the inequality constraint. Assuming the residual detection threshold C is equal to 3, comparisons between a true value and estimation, according to dk,mforming three radar observations, are given by Fig.4, Fig.5, and Fig.6, respectively. The result shows that the estimation of dk,mis consistent with the true value. On this basis, the misjudgment rate of three sensors (miscalculation times/observation times×100%) is obtained in 1000 times Monte Carlo simulations, and their results are 0.28%, 0.24% and 0.19%, respectively. Therefore, the validity and feasibility of residual detection strategy used to estimate dk,mis verified.

It is known that the misjudgment rates of dk,1, dk,2

Fig.4 λ1=0.95

Fig.5 λ2=0.85

Fig.6 λ3=0.65

and dk,3show the decline trend in figures. The key reason is that C is small, and the misjudgment of observation mainly belongs to false alarm. Namely, normal observation is mistaken as the observation missing. At this point, the less is the cardinality of normal observation, namely, the detection probability is lower, and relatively the less is the number of observation misjudgment. So the reason leads to the decrease of estimated error rate. If “misjudgment times/no missing observation times” is used to be expressed the misjudgment rate, the misjudgment rate are 0.29%, 0.28% and 0.29% respectively. Above situations well validate the un-correlate relationship between misjudgment rate obtained by the residual error detection strategy and the detection probability of sensor. In addition, if the value of detection threshold C is 4, the misjudgment rates will be 0.008%, 0.003% and 0.005% at the same simulation condition, respectively. It shows that C directly effects on observation misjudgment rate. Namely, the larger C is, the lower false alarm is.

Based on the above experiment conclusions, in the same simulation condition including hardware and software, RMSE of S-UKF for position estimation in horizontal direction and vertical direction is given by each sensor observation, respectively. Meanwhile, RMSE of MF-UKF is given by synthesizing the output information of each sub filter. For simplicity, assuming that the coefficients including β1, β2and β3of information allocation in local nodes, are all equal to 1/3, and simulation results are shown in Fig.7 and Fig.8. In order to more intuitively and quantitatively compare with the performance of algorithm, the mean of RMSE is given in Table1.

Fig.7 and Fig.8 show that the bigger is the detection probability of the radar sensor, the higher is the filtering precision of S-UKF. The results also indicate that S-UKF can improve the filtering precision to some extent. While a few affect on estimation precision still exist intermittent observations if in only directly taking state prediction and its error covariance matrix as state estimation at current time. And RMSE of S-UKF is in the inverse proportional relationship to detection probability of sensor. Moreover, the results in figures show that the estimation precision of MF-UKF is superior to S-UKF. Namely, In intermittent observations, the adverse influence caused by detection probability of sensor on estimation precision of S-UKF can be improved by utilizing effectively redundancy and complementary information among multi-sensor observations. The quantitative analysis of RMSE in Table 1 also verifies the above conclusion.

Fig.7 Horizontal direction

Fig.8 Vertical direction

AlgorithmHorizontaldirectionVerticaldirectionS-UKF-Sensor338.531927.9265S-UKF-Sensor232.882123.8102S-UKF-Sensor130.492822.1580MF-UKF20.653115.0018

5 Conclusions

Aiming at state estimation of nonlinear system in intermittent observations, a single-sensor unscented Kalman filter in intermittent observations is firstly constructed by the introduction of residual detection strategy. On this basis, combined with the federal filter algorithm, a novel multi-sensor federated unscented Kalman filter in intermittent observations is proposed. Relative to the existing nonlinear filter, the new algorithm effectively improves adverse effect caused by observation missing on filtering precision of state estimation. Simulation results show that MF-UKF has a higher target tracking precision than UKF and S-UKF. Meanwhile, the structure of MF-UKF applies to multi-sensor observation situation, so its application scope is wider. In addition, the algorithm has advantages of clear physical meaning and easy realization.

[ 1] Ronald P S Mahler. Statistical Multisource-multitarget information fusion. Boston, London: Artech House Publishers, 2007. 23-27

[ 2] Gustafsson F, Hendeby G. Some Relations between extended and unscented Kalman filters. IEEE Trans on Signal Processing, 2012,60(2): 545-555

[ 3] Laneuville D, Vignal H. Grid based target motion analysis. In: Proceedings of the IEEE Aerospace Conference, Big Sky, USA, 2007. 1-7

[ 4] Terejanu G, Singla P, Singh T, Scott P D. Adaptive Gaussian sum filter for nonlinear bayesian estimation. IEEE Trans on Automatic Control, 2011, 56(9):2151- 2156

[ 5] Cappe O, Godsill S J, Moulines E. An overview of existing methods and recent advances in sequential Monte Carlo. Proceedings of the IEEE, 2007,95(5):899-924

[ 6] Saha S, Gustafsson F. Particle filtering with dependent noise processes. IEEE Trans on Signal Processing, 2012 , 60(9): 4497-4508

[ 7] Derek Yee, Reilly J P, Kirubarajan T, Punithakumar K. Approximate conditional mean particle filtering for linear/nonlinear dynamic state space models. IEEE Trans on Signal Processing, 2008, 56(12): 5790 -5803

[ 8] Dunik J, Simandl M, Straka O. Unscented Kalman filter: aspects and adaptive setting of scaling parameter. IEEE Trans on Automatic Control, 2012,57(9): 2411- 2416

[ 9] Yilin Mo, Sinopoli B. Kalman filtering with intermittent observations: tail distribution and critical value. IEEE Transactions on Automatic Control, 2012, 57(3): 677- 689

[10] Plarre K, Bullo F. On Kalman filtering for detectable systems with intermittent observations. IEEE Trans on Automatic Control, 2009 , 54(2): 386 -390

[11] Sinopoli B, Schenato L, Fransceschetti L M, et al. Kalman filtering with intermittent observations. IEEE Trans on Automatic Control, 2004, 49(9):1453-1464

[12] Kar S, Sinopoli B, Moura J M. Kalman filtering with intermittent observations: Weak convergence to a stationary distribution. IEEE Trans on Automatic Control, 2012, 57(2): 405-420

[13] Chen S J, Qi G Q, Sheng A D. Admissible sampling frequency in measurements with variance-constrained and missing data. Control Theory & Applications, 2012, 29(5):629-634

[14] Kluge S, Reif K, Brokate M. Stochastic stability of the extended Kalman filter with intermittent observations. IEEE Trans on Automatic Control, 2012, 55(2): 514-518

[15] Gao H J, Zhao Y, Lam J, Chen K. H∞ fuzzy filtering of nonlinear systems with intermittent measurements. IEEE Trans on Fuzzy System, 2009, 17(2): 291-300

[16] Hu J, Wang Z D, Gao H J, Stergioulas L K. Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements. Automatica, 2012,48 (9): 2007- 2015

[17] Djuric P M, Zhang Z J, Bugallo M F. Target tracking by a new class of cost-reference particle filters. In: Proceedings of the IEEE Aerospace Conference, Big Sky, USA, 2008,1-9

[18] Julier S J, Uhlmann J K. Unscented filtering and nonlinear estimation. Proceedings of the IEEE, 2004 , 92(3): 401-422

[19] Julier S J. The spherical simplex unscented transformation. Proceedings of the American Control Conference, 2003, 2430 -2434

[20] Zhou H R, Jing Z L, Wang P D. Maneuvering Target Tracking. Beijing. National Defence Industry Press, 2011 (In Chinese)

[21] Liu J, Ma J, Tian J W. Pulsar/CNS integrated navigation based on federated UKF. Journal of Systems Engineering and Electronics, 2010 , 21(4): 675 - 681

[22] Qin Y Y. Kalman Filter and Integrated Navigation Principle. Xian: Northwestern Polytechnical University Press, 2012

Hu Zhentao, born in 1979. He received his Ph.D degrees in Control Science and Engineering from Northwestern Polytechnical University in 2010. He also received his B.S. and M.S. degrees from Henan University in 2003 and 2006 respectively. Now, he is an assistant professor of college of computer and information engineering, Henan University. His research interests include complex system modeling and estimation, target tracking and particle filter, etc.

10.3772/j.issn.1006-6748.2015.02.003

①Supported by the National Natural Science Foundation (NNSF) of China under Grant (No. 61300214), the Science and Technology Innovation Team Support Plan of Education Department of Henan Province (No.13IRTSTHN021), the Science and Technology Research Key Project of Education Department of Henan Province (No.13A413066), the Basic and Frontier Technology Research Plan of Henan Province (No. 132300410148), the Funding Scheme of Young Key Teacher of Henan Province Universities (No. 2013GGJS-026), the Key Project of Teaching Reform Research of Henan University (No. HDXJJG2013-07), the Postdoctoral Science Fund of Henan Province (No. 2013029) and the Postdoctoral Science Fund of China (No. 2014M551999).

②To whom correspondence should be addressed. E-mail: hzt@henu.edu.cn Received on Apr. 18， 2014***, Hu Yumei**, Li Song**