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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 716915, 14 pages
http://dx.doi.org/10.1155/2013/716915
Research Article

Adaptive Kalman Estimation in Target Tracking Mixed with Random One-Step Delays, Stochastic-Bias Measurements, and Missing Measurements

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

Received 5 November 2013; Accepted 21 November 2013

Academic Editor: Lifeng Ma

Copyright © 2013 Sujuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this paper is concerned with the estimation problem for linear discrete-time stochastic systems with mixed uncertainties involving random one-step sensor delay, stochastic-bias measurements, and missing measurements. Three Bernoulli distributed random variables are employed to describe the uncertainties. All the three uncertainties in the measurement have certain probability of occurrence in the target tracking system. And then, an adaptive Kalman estimation is proposed to deal with this problem. The adaptive filter gains can be obtained in terms of solutions to a set of recursive discrete-time Riccati equations. Examples in three scenarios of target tracking are exploited to show the effectiveness of the proposed design approach.

1. Introduction

In practical target tracking, the missing phenomenon of the sensor measurement often occurs. For example, the measurement value may contain noise only due to the shelter of the obstacles, a high noise environment, a failure in the measurement, intermittent sensor failures, high maneuverability of tracked target, and so forth. Thus, the estimation for systems with missing measurements has received much attention during the few years. In general, there are two ways to describe the missing phenomenon. One way is to model the uncertainty by using a stochastic Bernoulli binary switching sequence taking on values of 0 and 1 (see, e.g., [118] and the references therein). The suboptimal filtering algorithm [2] in the minimum variance sense with only missing measurements has been proposed and the robust filter [14] is designed. In [3] Sinopoli et al. studied the statistical behavior of the Kalman filter error covariance with varying Kalman gain with missing measurements and the existence of a critical value has been shown for the arrival rate of the observations. The stochastic stability of the extended kalman filter with missing measurements is analyzed in [8], while stochastic stability of the unscented kalman filter with missing measurements is studied in [13]. For benchmarking the performance of any estimation algorithm with missing measurements in advance, the modified Cramer-Rao bound (CRLB) and modified Riccati equation have been studied in [1, 1518]. Another way is to model the uncertainty as a Markovian jumping sequence (see [1921] and the references therein). The convergence for semi-Markov chains has been studied in [20], where the exponential filter is designed in [21].

For the case of random delayed measurements, the delays will cause performance degradation or instability with traditional Kalman filters. Therefore, a large volume of literature has been published on the topic of filtering problem with delayed measurements; see, for example, [6, 7, 1012, 2227]. Most of the publications assume that the delayed measurements are always deterministic. However, it is not often the case in actual target tracking, where the time delays in the measurement may occur in a random way. A stochastic Bernoulli binary switching sequence taking on values of 0 and 1 has been employed to describe the randomly delayed measurements. In such case, the minimum variance suboptimal filter [22, 26] and the robust filter with only delayed measurements [25] are obtained.

For the case of stochastic-bias measurements, a two-stage kalman estimator for state estimation in the presence of random bias and for tracking maneuvering targets is given in [28], while an optimal two-stage kalman filter in the presence of random bias is obtained in [29]. Moreover, optimal and suboptimal separate-bias kalman estimators for a stochastic bias are studied in [30]. The biased bearing-only parameter estimation for bistatic system is studied in [31] and the Cramer-Rao lower bound for biased bearings-only maneuvering target tracking is obtained in [32], where the stochastic-bias phenomenon is described by a stochastic Bernoulli binary switching sequence taking on values of 0 and 1.

Based on the stochastic parameterized description, recently, some suboptimal and optimal filtering algorithms have been proposed for systems with both sensor delays and missing measurements [1012, 21, 24, 33, 34]. Adaptive kalman filtering [10] is designed mixed with random sensor delays, multiple packet dropouts, and missing measurements. The problem of robust state estimation for two-dimensional stochastic time-delay systems [12] is studied with missing measurements and sensor saturation. An optimal linear filter for discrete-time systems with randomly delayed and lost measurements with/without time stamps is studied in [34]. Up to now, the state estimation problem for systems in target tracking mixed with random one-step delays, stochastic-bias measurements, and missing measurements is seldom reported, which almost exist simultaneously in target tracking. Motivated by the results above, a new mathematical model is developed to describe the random one-step delays, stochastic-bias measurements, and missing measurements using three Bernoulli distributed random variables taking on values of 0 and 1. By state augmentation, it is converted to a stochastic parameterized system. Based on the model, An adaptive kalman estimation is obtained in target tracking via projection theory. The rest of the paper is organized as follows. Section 2 establishes the new system model and introduces the augment state Kalman estimation to deal with the proposed problem. Section 3 presents an adaptive estimation algorithm. In Section 4, simulation results are analyzed and are used to evaluate the proposed algorithm. A summary of our conclusions is given in Section 5.

2. System Model and Problem Formulation

Consider the following stochastic-biased discrete time linear time-varying state-space system with missing measurements: where is the state vector, is the measured output, and and are Gaussian random vectors with zero mean and covariance matrices and , respectively. The bias vector is treated as a zero mean Gaussian random vector with a priori known covariance . and are constant matrices with compatible dimensions. The is a scalar binary Bernoulli distributed random variable taking values which are defined as and . If , the measurement at time is available; otherwise, the measurement is missing; that is, the received observation is only the measurement noise at time . The is a scalar binary Bernoulli distributed random variable taking values which are defined as and . If and , the measurement at time is available and unbiased; else if and , the measurement at time is available but biased.

Assume that the initial state satisfies the mean and covariance conditions: where is the mathematical expectation operator and the superscript is the transpose. The initial state is uncorrelated with , , and .

Next, we will establish a unified model to describe the mixed uncertainties of missing measurements, one-step delay, and stochastic biased measurements by Bernoulli distributed random variables.

Define

A compact representation of the system (1) with mixed uncertainties of missing measurements, one-step delay, and stochastic-biased measurements can be given in an augmented form as follows: where

Further, we have the following statistical information about and :

Let denote the three models corresponding, respectively, to systems with no missing measurement, one-step sensor delay, and missing measurement. They are defined by the following system matrices:(a)current measurement case: (b)one-step random sensor delay case: (c)missing measurement case:

The systems can be represented as follows: where satisfies

Assume that we are able to model the real system at time by analyzing the system. The probability of the system at time is . Obviously, we have .

Remark 1. In this paper, the uncertain model subject to missing measurements, one-step delay, and stochastic-biased measurements is considered; that is, the probabilities of the stochastic uncertainties are known. Here the missing measurements are replaced by zeros; that is, the measurement containing only noise is not a valid value. We can examine the “error detection” bits before sending them to the estimator. The delay may be much longer and the receiver (estimator side) cannot know the values without time stamps. Therefore, we cannot have an exact model ; that is, the value of is unknown when the measurement arrives in an estimator. In this case, we need to know the probability of each model .

Define That is, is an estimator vector of .

We can construct an estimator of the following structure:

Define

From (12), we know that is an estimate vector for . Therefore, it is obvious that we can choose and as follows:

The state estimation error covariance is defined by where

In order to obtain the optimal , we can minimize .

3. Estimation Algorithm Mixed with One-Step Sensor Delays, Missing Measurements, and Stochastic-Biased Measurements

In this section, based on the stochastic system models , we will derive the optimal before giving one-step predictions.

From (3) and (14), we can readily have that

Define Substituting (4) and (13) into (19) yields

Observing (7)–(10) and (14), we have the following results:

Furthermore, substituting (21) into (20), we can obtain the following equation:

Noting that , (22) can be rewritten as

Using the statistics of and in (6), the covariance matrix of can be computed by

From (17) and (19), we can easily have

Therefore, the problem of minimizing is equivalent to

Let and left- and right-multiply (24) by and , respectively. The estimation error covariance matrix can be computed by

Differentiate with respect to and set the derivative to zero. We can obtain the optimal as follows:

Therefore the covariance matrices of the estimation error for any (i.e., not necessarily optimal) can be computed by a set of recursive discrete-time Riccati-like equations. The corresponding optimal can be computed by

Let , , and . We may obtain the relevant formula for measurement update as follows: We may propose the following adaptive filtering scheme.

Consider the conceptual algorithm.

Step 1. The required input parameters are

Step 2. Initialization is as follows: where we have assumed to derive the above initialization.
Perform the following equations:

Step 3. Consider the state prediction
and the prediction covariance computation

Step 4. Innovation update is as follows.
If , then . Else if , then

Step 5. Update :
Considering error covariance matrix update, perform the following equations:

Remark 2. We consider that the model is unknown when the measurement arrives in an estimator. In this case, we need to know the probability of each model. However, we can get an exact model when the measurement with time stamp arrives in an estimator, in which case, we may have an improved adaptive algorithm with time stamp.

4. Simulation Results

In this section, Monte Carlo simulations were conducted to verify the performance of the algorithms in three scenarios of target tracking: uniform rectilinear motion scenario, uniform circular motion scenario, and uniformly accelerated motion scenario. A total of 100 runs were executed in each case.

(A) Uniform Rectilinear Motion Scenario. Consider the system (1) with parameters where is the sampling period, is the Kronecker product, and denotes the unity matrix of 3 orders.

Assume that the initial state of target is , the sampling period is set to be , the probability of stochastic-bias measurement is . The simulation results are obtained by the proposed adaptive estimation algorithm in Section 4. The root mean square error (RMSE) for the states of position and velocity with different is given in Figures 14. Seen from Figures 14, the proposed adaptive estimation algorithm is effective. Besides, it can be seen from Figure 1 that the tracking performance is better with higher probability when the delay probability is constant, where the black curves denote position and velocity RMSEs, respectively, with detection probability , delay probability , and the probability of missing probability , the green curve with , , and , and the red curve with , , and . Figure 2 gives comparison curves of position RMSE and velocity RMSE, respectively, where the black curves denote position RMSE with , , and , the green curve with , , and , and the red curve with , , and . From Figure 3, it is obvious that the estimation performance is better with lower missing measurement when the detection probability is constant. From Figure 4, we can see that the estimation algorithm has worse performance with a higher delay probability when the probability of missing measurement is constant.

fig1
Figure 1: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig2
Figure 2: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig3
Figure 3: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig4
Figure 4: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

Next, we consider the tracking performances with stochastic-bias probability . Figure 4 gives comparison curves of position RMSE and velocity RMSE with , , and , respectively, where the black curves denote position RMSE and velocity RMSE with , the green curve with , and the red curve with . From Figure 4, we can obtain that the estimation performance is better with lower probability of stochastic-bias measurement.

To further demonstrate the effectiveness of our proposed algorithm, simulation scenarios of uniform rectilinear motion and uniform accelerated motion are given in the following.

(B) Uniform Rectilinear Motion Scenario. Consider the system (1) with parameters. where is the sampling period, is the Kronecker product, and denotes the unity matrix of 3 orders.

Assume that the initial state of target is , the sampling period is set to be , the angular velocity is set to be , and the probability of stochastic-bias measurement is . The simulation results can be obtained as in Figures 5, 6, 7, and 8 in this scenario.

fig5
Figure 5: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig6
Figure 6: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig7
Figure 7: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig8
Figure 8: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

(C) Uniform Accelerated Motion Scenario. Consider the system (1) with parameters where is the sampling period, is the Kronecker product, and denotes the unity matrix of 3 orders. Assume that the initial state of target is , the sampling period is set to be , and the probability of stochastic-bias measurement is . The simulation results can be obtained as in Figures 9, 10, 11, and 12 in this scenario.

fig9
Figure 9: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig10
Figure 10: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig11
Figure 11: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.
fig12
Figure 12: Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

From Figures 58 in scenario B and Figures 912 in scenario C, it is clear that the proposed algorithm is effective. On the other hand, the estimation performance can be influenced by detection probability, one-step delay probability, missing probability, and stochastic-bias probability. Moreover, we have the corresponding results in scenarios B and C as well as in scenario A.

5. Conclusions

This paper presented an adaptive estimation algorithm in target tracking mixed with one-step delayed measurements, stochastic-bias measurements, and missing measurements. The contributions of this paper are as follows. First, the stochastic-bias measurement was incorporated into the proposed adaptive estimation algorithm. Second, the adaptive estimation algorithm mixed with one-step delayed measurements, stochastic-bias measurements, and missing measurements was implemented by augmenting the state vector. Examples and simulations are given to demonstrate the performances of the designed estimator. Through Monte Carlo simulations, the proposed algorithm was shown to be effective. Besides that, all the probabilities of missing measurements, stochastic-bias measurements, and one-step randomly delayed measurements have an influence on the estimation performance. Moreover, the estimation performance can be influenced by stochastic-bias probability. In this paper, we have only considered one-step sensor delays; furthermore, we can obtain the corresponding approach with multiple-step sensor delays by enlarging the state vectors.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (nos. 61104186, and 61273076) and the Natural Science Foundation of Jiangsu Province (no. BK2012801) and sponsored by Research and Innovation Project for College Graduates of Jiangsu Province (2011) (no. CXLX11-0261).

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