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

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., [1–18] 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, 15–18]. Another way is to model the uncertainty as a Markovian jumping sequence (see [19–21] 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, 10–12, 22–27]. 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 [10–12, 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: 𝑋_𝑘+1=𝐹_𝑘𝑋_𝑘+𝑤_𝑘,𝑋_𝑘(𝑘≤0)=𝑋₀,𝑦_𝑘=𝑑_𝑘𝐻_𝑘𝑋_𝑘+𝑑_𝑘1−𝑙_𝑘𝜏_𝑘+𝑣_𝑘, (1) where 𝑋_𝑘∈ℝ^𝑛 is the state vector, 𝑦_𝑘∈ℝ^𝑚 is the measured output, and 𝑤_𝑘∈ℝ^𝑝 and 𝑣_𝑘∈ℝ^𝑚 are Gaussian random vectors with zero mean and covariance matrices 𝜎²_𝑤≥0 and 𝜎²_𝑣≥0, respectively. The bias vector 𝜏_𝑘∈ℝ^𝑚 is treated as a zero mean Gaussian random vector with a priori known covariance 𝜎²_𝜏. 𝐹_𝑘 and 𝐻_𝑘 are constant matrices with compatible dimensions. The 𝑑_𝑘 is a scalar binary Bernoulli distributed random variable taking values {1,0} which are defined as Prob{𝑑_𝑘=1}=𝜆 and Prob{𝑑_𝑘=0}=1−𝜆. If 𝑑_𝑘=1, the measurement at time 𝑘 is available; otherwise, the measurement is missing; that is, the received observation is only the measurement noise at time 𝑘. The 𝑙_𝑘 is a scalar binary Bernoulli distributed random variable taking values {1,0} which are defined as Prob{𝑙_𝑘=1}=𝜂 and Prob{𝑙_𝑘=0}=1−𝜂. If 𝑑_𝑘=1 and 𝑙_𝑘=0, the measurement at time is available and unbiased; else if 𝑑_𝑘=1 and 𝑙_𝑘=1, the measurement at time 𝑘 is available but biased.

Assume that the initial state 𝑋₀ satisfies the mean and covariance conditions: 𝐸𝑋₀=𝜇₀,𝐸𝑋₀−𝜇₀𝑋₀−𝜇₀^T=𝑃₀, (2) where 𝐸{⋅} is the mathematical expectation operator and the superscript T is the transpose. The initial state 𝜇₀ is uncorrelated with 𝑤_𝑘, 𝑣_𝑘, and 𝜏_𝑘.

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 𝑋_𝑘+1=𝑋^T_𝑘+1,𝑋^T_𝑘,𝑦^T_𝑘^T. (3)

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: 𝑋_𝑘+1=𝐴_𝑘𝑋_𝑘+𝐵_𝑘𝑊_𝑘+𝑀_𝑘Λ_𝑘,𝑦_𝑘=𝐶_𝑘𝑋_𝑘+𝐷_𝑘𝑊_𝑘+𝑁_𝑘Λ_𝑘, (4) where 𝑊_𝑘=𝑤^T_𝑘,𝑣^T_𝑘,𝑣^T_𝑘−1^T,Λ_𝑘=𝜏^T_𝑘,𝜏^T_𝑘−1^T. (5)

Further, we have the following statistical information about 𝑊_𝑘 and Λ_𝑘: 𝐸𝑊_𝑘𝐸Λ=0,_𝑘𝑃=0,_𝑊𝑘=𝐸𝑊_𝑘𝑊^T_𝑘=diag𝜎²_𝑤,𝜎²_𝑣,𝜎²_𝑣,𝐸𝑣_𝑘𝑣^T_𝑟𝑃=0,𝑘≠𝑟,_Λ𝑘=𝐸Λ_𝑘Λ^T_𝑘=diag𝜎²_𝜏,𝜎²_𝜏,𝐸𝜏_𝑘𝜏^T_𝑟=0,𝑘≠𝑟. (6)

Let {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3} 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: 𝐴¹_𝑘=𝐹_𝑘𝐻00𝐼00_𝑘00,𝐵¹_𝑘=,𝑀𝐼000000𝐼0_1𝑘=1−𝑙_𝑘=0000𝐼01−𝑙_𝑘𝑀¹_𝑘,𝐶¹_𝑘=𝐻_𝑘,𝐷00_1𝑘=0𝐼0,𝑁¹_𝑘=1−𝑙_𝑘𝐼0=1−𝑙_𝑘𝑁¹_𝑘, (7)(b)one-step random sensor delay case: 𝐴²_𝑘=𝐹_𝑘00𝐼000𝐻_𝑘0,𝐵²_𝑘=,𝑀𝐼0000000𝐼_2𝑘=1−𝑙_𝑘=00000𝐼1−𝑙_𝑘𝑀²_𝑘,𝐶²_𝑘=0𝐻_𝑘−10,𝐷²_𝑘=00𝐼,𝑁²_𝑘=1−𝑙_𝑘0𝐼=1−𝑙_𝑘𝑁²_𝑘, (8)(c)missing measurement case: 𝐴³_𝑘=𝐹_𝑘00𝐼00000,𝐵³_𝑘=,𝑀𝐼000000𝐼0_3𝑘=1−𝑙_𝑘=0000001−𝑙_𝑘𝑀³_𝑘,𝐶³_𝑘=000,𝐷³_𝑘=,𝑁000_3𝑘=1−𝑙_𝑘00=1−𝑙_𝑘𝑁³_𝑘. (9)

The systems {𝐴_𝑘,𝐵_𝑘,𝐶_𝑘,𝐷_𝑘,𝑀_𝑘,𝑁_𝑘} can be represented as follows: 𝐴_𝑘,𝐵_𝑘,𝐶_𝑘,𝐷_𝑘,𝑀_𝑘,𝑁_𝑘∶=³_𝑞=1𝛽_𝑞𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘, (10) where 𝛽_𝑞 satisfies ³_𝑞=1𝛽_𝑞=1,𝛽_𝑞=0or1. (11)

Assume that we are able to model the real system at time 𝑘 by analyzing the system. The probability of the system {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3} at time 𝑘 is 𝜌_𝑞. Obviously, we have ∑³_𝑞=1𝜌_𝑞=1.

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 {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3}; that is, the value of 𝑞 is unknown when the measurement arrives in an estimator. In this case, we need to know the probability 𝜌_𝑞(𝑞=1,2,3) of each model {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3}.

Define 𝑋_𝑘+1=𝐸𝑋^T_𝑘+1,𝑋^T_𝑘^T. (12) That is, 𝑋_𝑘+1 is an estimator vector of [𝑋^T_𝑘+1,𝑋^T_𝑘]^T.

We can construct an estimator of the following structure: 𝑋_𝑘+1=𝐴_𝑘𝑋_𝑘+𝐺_𝑘𝑦_𝑘−𝐶^𝑞_𝑘𝑋_𝑘. (13)

Define 𝑆=𝐼0. (14)

From (12), we know that 𝑋_𝑘+1 is an estimate vector for [𝑋^T_𝑘+1,𝑋^T_𝑘]^T. Therefore, it is obvious that we can choose 𝐴_𝑘 and 𝐶_𝑘^𝑞 as follows: 𝐴_𝑘=𝐹_𝑘0,𝐶𝐼0^𝑞𝑘=𝐶^𝑞_𝑘𝑆^T. (15)

The state estimation error covariance is defined by 𝑃_𝑒𝑘=𝐸𝑒_𝑘𝑒^T_𝑘, (16) where 𝑒_𝑘=𝑋_𝑘−𝑋_𝑘. (17)

In order to obtain the optimal 𝐺_𝑘, we can minimize 𝑃_𝑒𝑘.

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

In this section, based on the stochastic system models {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3}, we will derive the optimal 𝐺_𝑘 before giving one-step predictions.

From (3) and (14), we can readily have that 𝑆𝑋_𝑘=𝑋^T_𝑘,𝑋^T_𝑘−1^T. (18)

Define 𝜉_𝑘+1=𝑆𝑋_𝑘+1−𝑋_𝑘+1. (19) Substituting (4) and (13) into (19) yields 𝜉_𝑘+1=³_𝑞=1𝛽_𝑞𝑆𝐴^𝑞_𝑘𝑋_𝑘+𝐴_𝑘−𝐺_𝑘𝐶^𝑞_𝑘𝑋_𝑘+𝑆𝐵^𝑞_𝑘−𝐺_𝑘𝐷^𝑞_𝑘𝑊_𝑘+𝑆𝑀^𝑞_𝑘−𝐺_𝑘𝑁^𝑞_𝑘Λ_𝑘−𝐺_𝑘𝐶^𝑞_𝑘𝑋_𝑘. (20)

Observing (7)–(10) and (14), we have the following results: 𝑆𝐴^𝑞_𝑘𝑋_𝑘=𝐴_𝑘𝑆𝑋_𝑘,𝐶^𝑞_𝑘𝑋_𝑘=𝐶^𝑞_𝑘𝑆^T𝑆𝑋_𝑘,𝐶^𝑞_𝑘𝑋_𝑘=𝐶^𝑞_𝑘𝑆𝑋_𝑘. (21)

Furthermore, substituting (21) into (20), we can obtain the following equation: 𝜉_𝑘+1=³_𝑞=1𝛽_𝑞𝐴_𝑘−𝐺_𝑘𝐶^𝑞_𝑘𝜉_𝑘+𝑆𝐵^𝑞_𝑘−𝐺_𝑘𝐷^𝑞_𝑘𝑊_𝑘+𝑆𝑀^𝑞_𝑘−𝐺_𝑘𝑁^𝑞_𝑘Λ_𝑘+𝐺_𝑘𝐶^𝑞_𝑘𝑆^T−𝐶^𝑞_𝑘𝑆𝑋_𝑘. (22)

Noting that 𝐶^𝑞_𝑘=𝐶^𝑞_𝑘𝑆^T, (22) can be rewritten as 𝜉_𝑘+1=³_𝑞=1𝛽_𝑞𝐴_𝑘−𝐺_𝑘𝐶^𝑞_𝑘𝜉_𝑘+𝑆𝐵^𝑞_𝑘−𝐺_𝑘𝐷^𝑞_𝑘𝑊_𝑘+𝑆𝑀^𝑞_𝑘−𝐺_𝑘𝑁^𝑞_𝑘Λ_𝑘. (23)

Using the statistics of 𝑊_𝑘 and Λ_𝑘 in (6), the covariance matrix of 𝜉_𝑘 can be computed by 𝑃_𝜉𝑘+1=³_𝑞=1𝛽_𝑞𝐴_𝑘−𝐺_𝑘𝐶^𝑞_𝑘𝑃_𝜉𝑘𝐴_𝑘−𝐺_𝑘𝐶^𝑞_𝑘^T+𝑆𝐵^𝑞_𝑘−𝐺_𝑘𝐷^𝑞_𝑘𝑃_𝑊𝑘𝑆𝐵^𝑞_𝑘−𝐺_𝑘𝐷^𝑞_𝑘^T+1−𝜂𝑆𝑀^𝑞_𝑘−𝐺_𝑘𝑁^𝑞_𝑘×𝑃_Λ𝑘𝑆𝑀^𝑞_𝑘−𝐺_𝑘𝑁^𝑞_𝑘^T. (24)

From (17) and (19), we can easily have 𝑒_𝑘+1=Φ𝜉_𝑘+1,Φ=𝐼0. (25)

Therefore, the problem of minimizing 𝑃_𝑒𝑘 is equivalent to min_𝐺𝑘𝐸TrΦ𝑃_𝜉𝑘+1Φ^Tsubjectto(24). (26)

Let 𝐾_𝑘=Φ𝐺_𝑘 and left- and right-multiply (24) by Φ and Φ^T, respectively. The estimation error covariance matrix can be computed by 𝑃_𝑒𝑘+1=³_𝑞=1𝛽_𝑞Φ𝐴_𝑘−𝐾_𝑘𝐶^𝑞_𝑘𝑃_𝜉𝑘+1Φ𝐴_𝑘−𝐾_𝑘𝐶^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞Φ𝑆𝐵^𝑞_𝑘−𝐾_𝑘𝐷^𝑞_𝑘𝑃_𝑊𝑘Φ𝑆𝐵^𝑞_𝑘−𝐾_𝑘𝐷^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞1−𝜂Φ𝑆𝑀^𝑞_𝑘−𝐾_𝑘𝑁^𝑞_𝑘×𝑃_Λ𝑘Φ𝑆𝑀^𝑞_𝑘−𝐾_𝑘𝑁^𝑞_𝑘^T. (27)

Differentiate 𝐸{Tr𝑃_𝑒𝑘+1} with respect to 𝐾_𝑘 and set the derivative to zero. We can obtain the optimal 𝐾_𝑘 as follows: 𝐾_𝑘=³_𝑞=1𝛽_𝑞Φ𝐴_𝑘𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞Φ𝑆𝐵^𝑞_𝑘𝑃_𝑊¹_𝑘𝐷^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞1−𝜂Φ𝑆𝑀^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T׳_𝑞=1𝛽_𝑞𝐶^𝑞_𝑘𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞𝐷^𝑞_𝑘𝑃_𝑊𝑘𝐷^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞1−𝜂𝑁^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T−1. (28)

Therefore the covariance matrices of the estimation error for any (i.e., not necessarily optimal) 𝐾_𝑘 can be computed by a set of recursive discrete-time Riccati-like equations. The corresponding optimal 𝐺_𝑘 can be computed by 𝐺_𝑘=𝐾^T_𝑘0^T. (29)

Let 𝐴_𝑘=𝐼, 𝐵^𝑞_𝑘=0, and 𝐿_𝑘=𝐾_𝑘. We may obtain the relevant formula for measurement update as follows: 𝑃_{𝑒𝑘+1∣𝑘+1}=³_𝑞=1𝛽_𝑞Φ−𝑀_𝑘𝐶^𝑞_𝑘𝑃_𝜉𝑘+1Φ−𝑀_𝑘𝐶^𝑞_𝑘^T+𝐿_𝑘𝐷^𝑞_𝑘𝑃_𝑊𝑘𝐷^𝑞_𝑘^T𝑀^T_𝑘+1−𝜂³_𝑞=1𝛽_𝑞𝐿_𝑘𝑁^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T,𝐿_𝑘=³_𝑞=1𝛽_𝑞Φ𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T+1−𝜂Φ𝑆𝑀^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T׳_𝑞=1𝛽_𝑞𝐶^𝑞_𝑘𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞𝐷^𝑞_𝑘𝑃_𝑊𝑘𝐷^𝑞_𝑘^T+³_𝑞=1𝛽_𝑞1−𝜂𝑁^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T−1. (30) We may propose the following adaptive filtering scheme.

Consider the conceptual algorithm.

Step 1. The required input parameters are 𝐹_𝑘,𝐻_𝑘,𝜎²_𝑤,𝜎²_𝑣,𝜎²_𝜏,𝑃₀,𝜌_𝑞𝑞=1,2,3,𝐸𝑋₀=𝜇₀. (31)

Step 2. Initialization is as follows: 𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘,𝑃∣𝑞=1,2,3_𝑊𝑘=diag𝜎²_𝑤,𝜎²_𝑣,𝜎²_𝑣,𝑃_Λ𝑘=diag𝜎²_𝜏,𝜎²_𝜏,𝑃_𝜉0=𝑃₀𝑃₀𝑃₀𝑃₀, (32) where we have assumed 𝑋₋₁=𝑋₀ to derive the above initialization.
Perform the following equations: 𝑋₀=𝐸𝑋^T₀𝑋^T₀^T=𝜇^T₀𝜇^T₀^T,𝐶₀=³_𝑞=1𝜌^𝑞₀𝐶^𝑞₀𝑆^T,𝐾₀=³_𝑞=1𝛽^𝑞₀Φ𝑃_𝜉0𝐶^𝑞₀^T׳_𝑞=1𝛽^𝑞₀𝐶^𝑞₀𝑃_𝜉0𝐶^𝑞₀^T+𝐷^𝑞₀𝑃_𝑊0𝐷^𝑞₀^T+1−𝜂𝑁^𝑞₀𝑃_Λ0𝑁^𝑞₀^T−1,𝑋_0∣0=𝑋₀+𝐾₀𝑦₀−𝐶₀𝑋₀,𝑃_𝑒0∣0=³_𝑞=1𝜌^𝑞₀𝐼−𝐾₀𝐶^𝑞₀𝑃_𝜉0𝐼−𝐾₀𝐶^𝑞₀^T׳_𝑞=1𝜌^𝑞_𝑘𝐾₀𝐷^𝑞₀𝑃_𝑊0𝐷^𝑞₀^T𝐾^T₀+³_𝑞=1𝜌^𝑞₀1−𝜂𝐾₀𝑁^𝑞₀𝑃_Λ0𝑁^𝑞₀^T𝐾^T₀. (33)

Step 3. Consider the state prediction 𝑋_𝑘+1=𝐹𝑋_𝑘∣𝑘,𝑋_𝑘+1=𝑋^T_𝑘+1,𝑋^T_𝑘^T (34)
and the prediction covariance computation 𝑃_𝜉𝑘+1=𝐴_𝑘𝑃_{𝜉𝑘|𝑘}𝐴^T_𝑘+³_𝑞=1𝛽^𝑞_𝑘𝑆𝐵^𝑞_𝑘𝑃_𝑊𝑘𝐵^𝑞_𝑘^T𝑆^T+1−𝜂𝑆𝑀^𝑞_𝑘𝑃_Λ𝑘𝑀^𝑞_𝑘^T𝑆^T. (35)

Step 4. Innovation update is as follows.
If 𝑑_𝑘=0, then 𝐾_𝑘=0. Else if 𝑑_𝑘=1, then 𝐾_𝑘=³_𝑞=1𝛽^𝑞_𝑘Φ𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T׳_𝑞=1𝛽^𝑞_𝑘𝐶^𝑞_𝑘𝑃_𝜉𝑘+1𝐶^𝑞_𝑘^T+𝐷^𝑞_𝑘𝑃_𝑊𝑘𝐷^𝑞_𝑘^T+1−𝜂𝑁^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T−1. (36)

Step 5. Update 𝑘=𝑘+1: 𝑋_𝑘∣𝑘=𝑋_𝑘+𝐾_𝑘𝑦_𝑘−𝐶_𝑘𝑋_𝑘. (37)
Considering error covariance matrix update, perform the following equations: 𝐺_𝑘=𝐾^T_𝑘0^T,𝑃_{𝜉𝑘+1|𝑘+1}=³_𝑞=1𝛽^𝑞_𝑘𝐼−𝐺_𝑘𝐶^𝑞_𝑘𝑃_𝜉𝑘𝐼−𝐺_𝑘𝐶^𝑞_𝑘^T+𝐺_𝑘𝐷^𝑞_𝑘𝑃_𝑊𝑘𝐷^𝑞_𝑘^T𝐺^T_𝑘+1−𝜂𝐺_𝑘𝑁^𝑞_𝑘𝑃_Λ𝑘𝑁^𝑞_𝑘^T𝐺^T_𝑘. (38)

Remark 2. We consider that the model {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3} is unknown when the measurement arrives in an estimator. In this case, we need to know the probability 𝜌_𝑞(𝑞=1,2,3) of each model. However, we can get an exact model {𝐴^𝑞_𝑘,𝐵^𝑞_𝑘,𝐶^𝑞_𝑘,𝐷^𝑞_𝑘,𝑀^𝑞_𝑘,𝑁^𝑞_𝑘∣𝑞=1,2,3} 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 𝐹_𝑘=𝐼_3×3⊗,𝜎1𝑇01_2𝑤=𝐼_3×3⊗𝑇³3𝑇²2𝑇²2𝑇,𝜎²_𝑣=25m²,𝜎²_𝜏=25m², (39) where 𝑇 is the sampling period, ⊗ is the Kronecker product, and 𝐼_3×3 denotes the unity matrix of 3 orders.

Assume that the initial state of target is 𝑋₀=[−8000,200,0,500,0,0,1000,0,0]^T, the sampling period is set to be 𝑇=0.2s, the probability of stochastic-bias measurement is 1−𝜂=0.1. 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 𝜌_𝑞(𝑞=1,2,3) is given in Figures 1–4. Seen from Figures 1–4, 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 𝜌₁=0.8, delay probability 𝜌₂=0.1, and the probability of missing probability 𝜌₃=0.1, the green curve with 𝜌₁=0.7, 𝜌₂=0.1, and 𝜌₃=0.2, and the red curve with 𝜌₁=0.6, 𝜌₂=0.1, and 𝜌₃=0.3. Figure 2 gives comparison curves of position RMSE and velocity RMSE, respectively, where the black curves denote position RMSE with 𝜌₁=0.8, 𝜌₂=0.1, and 𝜌₃=0.1, the green curve with 𝜌₁=0.8, 𝜌₂=0.05, and 𝜌₃=0.15, and the red curve with 𝜌₁=0.8, 𝜌₂=0.15, and 𝜌₃=0.05. 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.

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(b)

Figure 1 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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(b)

Figure 2 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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(b)

Figure 3 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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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 1−𝜂. Figure 4 gives comparison curves of position RMSE and velocity RMSE with 𝜌₁=0.8, 𝜌₂=0.1, and 𝜌₃=0.1, respectively, where the black curves denote position RMSE and velocity RMSE with 1−𝜂=0.1, the green curve with 1−𝜂=0.5, and the red curve with 1−𝜂=0.8. 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. 𝐹_𝑘=𝐼_3×3⊗1sin(𝜔𝑇)𝜔[1−cos(𝜔𝑇)]𝜔²()()0cos𝜔𝑇sin𝜔𝑇𝜔()(),𝜎0−𝜔sin𝜔𝑇cos𝜔𝑇_2𝑤=𝐼_3×3⊗𝑆,𝜎²_𝑣=25m²,𝜎²_𝜏=25m²,𝐴𝑆=¹¹𝐴₁₂𝐴₁₃𝐴₂₁𝐴₂₂𝐴₂₃𝐴₃₁𝐴₃₂𝐴₃₃,𝐴₁₁()+()=6𝜔𝑇−8sin𝜔𝑇sin2𝜔𝑇4𝜔⁵,𝐴₁₂=[2sin(𝜔𝑇/2)]⁴𝜔⁴,𝐴₁₃=[−2𝜔𝑇+4sin(𝜔𝑇)−sin(2𝜔𝑇)]4𝜔³,𝐴₂₁=2[sin(𝜔𝑇/2)]⁴𝜔⁴,𝐴₂₂=[2𝜔𝑇−sin(2𝜔𝑇)]4𝜔³,𝐴₂₃=[sin(𝜔𝑇)]²2𝜔²,𝐴₃₁=[−2𝜔𝑇+4sin(𝜔𝑇)−sin(2𝜔𝑇)]4𝜔³,𝐴₃₂=[sin(𝜔𝑇)]²2𝜔²,𝐴₃₃=[2𝜔𝑇+sin(2𝜔𝑇)],4𝜔 (40) where 𝑇 is the sampling period, ⊗ is the Kronecker product, and 𝐼_3×3 denotes the unity matrix of 3 orders.

Assume that the initial state of target is 𝑋₀=[−5000,200,−1,500,0,0,1000,0,0]^T, the sampling period is set to be 𝑇=0.2s, the angular velocity is set to be 𝜔=0.05rad/s, and the probability of stochastic-bias measurement is 1−𝜂=0.1. The simulation results can be obtained as in Figures 5, 6, 7, and 8 in this scenario.

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(b)

Figure 5 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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(b)

Figure 6 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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Figure 7 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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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 𝐹_𝑘=𝐼_3×3⊗𝑇1𝑇²2,𝜎01𝑇001_2𝑤=𝐼_3×3⊗𝑇⁵𝑇208⁴𝑇³6𝑇⁴8𝑇³3𝑇²2𝑇²2𝑇²2𝑇,𝜎²_𝑣=25m²,𝜎²_𝜏=25m², (41) where 𝑇 is the sampling period, ⊗ is the Kronecker product, and 𝐼_3×3 denotes the unity matrix of 3 orders. Assume that the initial state of target is 𝑋₀=[−5000,100,−5,500,0,0,1000,0,0]^T, the sampling period is set to be 𝑇=0.2s, and the probability of stochastic-bias measurement is 1−𝜂=0.1. The simulation results can be obtained as in Figures 9, 10, 11, and 12 in this scenario.

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Figure 9 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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Figure 10 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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Figure 11 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

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Figure 12 

Comparison of the RMSEs for the proposed algorithm with different : (a) position RMSE; (b) velocity RMSE.

From Figures 5–8 in scenario B and Figures 9–12 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).