Zhansheng Duan

Comments on "Unbiased converted measurements for tracking"

We show that there exists a compatibility problem in the derivation of the mean and covariance of the converted measurement errors in L. Mo et al. (ibid., vol.34, no.3, p.1023-7, 1998), and then present a modification to the computation of them, in which both the mean and the covariance are computed strictly conditioned on the measurements.

Lossless Linear Transformation of Sensor Data for Distributed Estimation Fusion

In distributed estimation fusion, processed data from each sensor is sent to the fusion center. By taking linear transformation of the raw measurements of each sensor, two optimal distributed fusion algorithms are proposed in this paper. Compared with existing fusion algorithms, they have three nice properties. First, they are optimal in the sense that they are equivalent to the optimal centralized fusion. Second, their communication requirements from each sensor to the fusion center are equal to or less than those of the centralized and most existing distributed fusion algorithms. Third, they do not need the inverses of estimation error covariance matrices, which are assumed to exist in most existing algorithms but can not be guaranteed to exist. So the proposed algorithms can be applied in more cases. Pros and cons of these two new algorithms are analyzed. A possible way to reduce the computational complexity of the new algorithms, an extension to the case of a singular covariance matrix of measurement noise, and an extension to the reduced-rate communication case for some simple systems are also discussed.

Estimation Fusion with General Asynchronous Multi-Rate Sensors

The asynchronous estimation fusion problem is investigated for an arbitrary number of sensors with arbitrary sampling rates. By constructing an augmented measurement equation at the fusion time instant, a centralized asynchronous fusion algorithm is developed based on the Kalman filter first without ignoring the correlation between the process noise and the augmented measurement noise. It is optimal in the minimum mean-squared error (MMSE) sense. A distributed asynchronous fusion algorithm is then proposed by reconstructing the optimal centralized fusion result with asynchronous local estimates and their error covariance matrices. It is equivalent to the centralized fusion algorithm under the full-rate communication assumption and outperforms the latter when at least one sensor communicates with the fusion center at a lower rate than its sampling rate. Compared with the existing distributed fusion algorithms for asynchronous sensors, the proposed distributed fusion algorithm avoids the complicated calculation of cross-covariance matrices between each pair of asynchronous local estimates. The communication burden can also be reduced since neither sensor measurement matrices nor local filtering gains need to be transmitted to the fusion center. Moreover all available local estimates are utilized as well as the fused one-step prediction. Some practical considerations of the proposed distributed fusion algorithm are also discussed. Performance of the proposed centralized and distributed fusion algorithms are illustrated through numerical simulations.

Modeling and State Estimation for Dynamic Systems With Linear Equality Constraints

The problem of modeling and estimation for linear equality constrained (LEC) systems is considered. The exact constrained dynamic model usually is not readily available or is too complicated, and hence in many studies an auxiliary dynamic model is employed in which the state does not necessarily obey the constraint strictly. Based on the understanding that the constraints, as prior information about the state, should be incorporated into the dynamics modeling, an LEC dynamic model (LECDM) is constructed first. The model optimally fuses the linear equality constraint (LEC) and the auxiliary dynamics. Some of its superior properties are presented. Next, the linear minimum mean squared error (LMMSE) estimate of the LEC state is proved to satisfy the constraint. The LMMSE estimator for linear systems, called the LEC Kalman filter (LECKF), and two approximate LMMSE estimators for nonlinear systems are presented. The LECKF is compared with other constrained estimators, and a sufficient condition is also provided under which the estimate projection method mathematically equals the LECKF. Furthermore, extensions of the LECDM for the LEC systems with uncertain or unknown constraint parameters are discussed. Finally, illustrative examples are provided to show the effectiveness and efficiency of the LECKF and to verify the theoretical results given in the paper.

Adaptive turn rate estimation using range rate measurements

The coordinated turn (CT) model is often used to track maneuvering target which performs CT motion. The key point to the successful use of this model is to determine the turn rate parameter. A new method to estimate the turn rate by using radar range-rate measurements is presented. First, four possible turn rates can be achieved with the range-rate measurements. Second, the minimum turn rate and its opposite value are chosen to be the possible turn rates. Finally, an interacting multiple model (IMM) algorithm with one constant velocity (CV) model and two CT models is designed to track the maneuvering target which performs uniform and ct motions. Monte-Carlo simulation results show that this algorithm is not only better than equivalent noise approach in tracking performance, but also better than the conventional IMM algorithm with the adaptive turn rate model. Further simulations show that the algorithm is robust with the accuracy of the range-rate measurements

Evaluation of Probability Transformations of Belief Functions for Decision Making

The transformation of belief function into probability is one of the most important and common ways for decision making under the framework of evidence theory. In this paper, we focus on the evaluation of such probability transformations (PTs), which are crucial for their proper applications and the design of new ones. Shannon entropy or probabilistic information content (PIC) measure is traditionally used in evaluating PTs. The transformation having the lowest entropy or highest PIC is considered as the best one. This standpoint is questioned in this paper by comparing a PT based on uncertainty minimization with other available PTs. It shows experimentally that entropy or PIC is not comprehensive to evaluate a PT. To make a comprehensive evaluation, some new approaches are proposed by the joint use of PIC and the distance of evidence according to the value- and rank-based fusion. A pattern classification application oriented evaluation approach for PTs is also proposed. Some desired properties for PTs are also discussed. Experimental results and related analysis are provided to show the rationality of the new evaluation approaches.

The Role of Pseudo Measurements in Equality-Constrained State Estimation

The pseudo measurement method is a main approach to equality-constrained state estimation due to its simplicity. It is, however, not popular due to possible numerical problems and increased computational complexity. The work presented here further develops the pseudo measurement method. To avoid numerical problems resulting from singular measurement noise when a matrix inverse is used, the Moore-Penrose (MP) inverse is used instead. Also, to reduce the computational load without performance loss and to simplify the analysis of this type of estimation problem, two sequential forms are obtained. They differ only in the processing order of the physical measurement and the pseudo measurement (i.e., the equality constraint). Although form 1 is the same as some existing results, form 2 is new. This motivates the discussion of processing order for this type of estimation problem, especially in the extension to the nonlinear case. It is found that under certain conditions, the use of the pseudo measurement for filtering is redundant. This differs in effect from update by the physically error-free measurement. However, if there exists model mismatch, update by the pseudo measurement is necessary and helpful. Supporting numerical examples are provided.

Performance Analysis of the Kalman Filter With Mismatched Noise Covariances

The Kalman filter is a powerful state estimator and has been successfully applied in many fields. To guarantee the optimality of the Kalman filter, the noise covariances need to be exactly known. However, this is not necessarily true in many practical applications. Usually, they are either completely unknown or at most partially known. In this technical note, we study performance of the Kalman filter with mismatched process and measurement noise covariances. For this purpose, three mean squared errors (MSEs) are used, namely the ideal MSE (IMSE), the filter calculated MSE (FMSE), and the true MSE (TMSE). The main contribution of this work is that the relationships between the three MSEs are disclosed from two points of views. The first view is about their ordering and the second view is about the relative closeness from the FMSE and TMSE to the IMSE. Using the first view, it is found that for the case with positive (definite) deviation from the truth, the FMSE is the worst and the IMSE is the best. And for the case with negative (definite) deviation, the TMSE is the worst and the best is the FMSE. Using the second view, it is found that the TMSE is relatively closer to the IMSE than the FMSE if the deviation is larger than certain threshold, and the TMSE will be farther away otherwise. Numerical examples further verify these conclusions.

Analysis, design, and estimation of linear equality-constrained dynamic systems

The state of many dynamic systems evolves subject to some equality constraints. Most existing work focuses on developing state estimation algorithms for an equality-constrained dynamic system assuming the system is given. How to design and analyze such a system is rarely addressed, even though it is critically important for application and performance evaluation. In this paper, we first analyze the underlying structural information of linear equality-constrained (LEC) dynamic systems through direct elimination. It is found that the process noise of such a system is state dependent in general. This means that existing formulations of such a system are quite limited. Based on the analysis, we then propose a new systematic way to design LEC systems. The key idea is to design only the unconstrained part of the system. Unlike existing system conversion-based design techniques, in our work the desired model class is given and only the distributions of the initial state and process noise need to be determined, which is comparatively easier. To handle state dependency of the process noise, a new constrained state estimation algorithm based on direct elimination is also proposed. Three examples are provided to illustrate the effectiveness of the proposed methods of analysis, design, and estimation, respectively.

Hybrid grid multiple-model estimation with application to maneuvering target tracking

Estimation for discrete-time stochastic systems with parameters varying in a continuous space is considered in this paper. Justified by an analysis of model approximation, a novel approach, called hybrid grid multiple model (HGMM), is proposed for state estimation. The model set used by HGMM is a combination of a fixed coarse grid and an adaptive fine grid to cover the mode space with a relatively small number of models. Next, two fundamental problems of the HGMM approach-model-set sequence-conditioned estimation and design of adaptive fine models-are addressed. Then, based on two model-set designs by moment matching, HGMM estimation algorithms are presented. Finally, performance of the developed HGMM estimation algorithms is evaluated on benchmark tracking scenarios, and simulation results demonstrate their superiority to the state-of-the-art MM estimation algorithms in terms of accuracy and computational complexity.