In a Bayesian framework, estimation is to infer a random quantity x based on prior infor- mation and measurement z. It is well known that the posterior mean E[x|z] is the opti- mal estimator that minimizes mean-square error (MSE). This minimum mean-square error (MMSE) estimation in general requires knowledge of the entire distribution ofx, which is not achievable for most applications other than the linear Gaussian case. However, it provides a theoretical basis for analysis and approximation techniques.
Instead of looking for the best estimator among all estimators, the best one in the set of all linear (in the measurement z) estimators may be a good choice, resulting in the linear minimum mean-square error (LMMSE) estimation. It makes a good compromise between technical simplicity and performance for problems with moderate nonlinearity. The well- known Kalman filter (KF) [110, 111] is a special case of the recursive LMMSE estimator, which is optimal (i.e., MMSE estimator) for a linear Gaussian system. LMMSE estimation
for example, the first-order extended Kalman filter (EKF) [22, 96, 178, 179, 231, 248], the unscented filter (UF) [100–107, 251], the divided difference filter [196] and the Gaussian filter [94], are either approximation of or based on LMMSE estimation. Further, LMMSE estimation was successfully applied directly to radar tracking [58, 252] with linear target dynamics model and nonlinear measurement model. It was also applied to state estimation for Markovian jump linear systems [43, 76], which, however, should be better handled by multiple-model estimation [152].
Intuitively, LMMSE estimation should work well for problems with low degree of nonlin- earity. However, if x and z are related highly nonlinearly, a linear estimator, even the best one, may not be adequate to provide acceptable accuracy. Instead of searching for the best in the set of all linear (in z) estimators, as LMMSE estimation does, we can look for the best estimator in a larger or different set, and hence improve estimation performance. We propose generalized LMMSE (GLMMSE) estimation by employing this idea. The candidate set in GLMMSE estimation can be determined by a vector-valued measurement transform function (MTF) g(z), and we may search for the best estimator within the set L(g) of all estimators that are linear in g(z), rather than in z. Clearly, this includes LMMSE estima- tion as a special case with g being an identity function. Theoretically speaking, GLMMSE estimation with a proper MTF g(z) performs at least as well as LMMSE estimation if the moments involved can be evaluated exactly. However, similar to LMMSE estimation, ana- lytical evaluation of the moments needed is difficult to achieve in general. Fortunately, many approximation techniques, e.g., unscented transform and numerical integration, developed for LMMSE estimation are also applicable to GLMMSE estimation.
The idea of applying LMMSE estimation to converted measurements exists in the target tracking literature. In [58, 252], the radar measurements in polar coordinates were converted to Cartesian coordinates, resulting in the pseudo-linear measurements, and then LMMSE estimation was applied to the converted measurement. The converted-measurement Kalman filters proposed in [134, 182, 239] also followed this idea. However, they are all special cases
of our proposed GLMMSE. First, they all converted the measurements to be pseudo-linear in the target (partial) state, which may be difficult to achieve for some cases (e.g., the range rate). Besides, the pseudo-linear conversion may not always be beneficial. Further, most often the converted measurements have the same dimension as the original measurements, which is not a necessity in our GLMMSE estimation framework. Actually, we may argu- ment the original measurements by the converted measurements in MTF g(z) to enhance performance, provided they are not linearly dependent.
Clearly, “what g(z) to use?” is the key to GLMMSE estimation. Design of g(z) is evidently problem dependent and up to the preference of users. It is difficult to come up with rigorous rules for constructing MTF in general. However, design guidelines are provided and discussed based on a numerical example in this chapter. A general guideline can be made that x should be less nonlinear in the converted data y = g(z) than in z. A measure of nonlinearity [145] can be employed to quantify the degree of nonlinearity and thus provides criterion for designing functiong. Also, an MTF can be selected from a family of functions by solving an optimization problem, i.e., minimizing the estimated mean-square errors computed in the estimator. Cautions are needed to consider the problems of the numerical instability and algorithm’s sensitivity to the prior assumption.
Our method is illustrated by two nonlinear problems in target tracking: a) single target tracking with nonlinear dynamics model and measurement model; b) multi-target tracking (MTT) with a linear dynamics model and a nonlinear measurement model. Note that MTT is a nonlinear estimation problem due to the uncertainty of measurement origin, i.e., data association problem, no matter if the target dynamics model and measurement model are linear or not.
This chapter is organized as follows. LMMSE estimation is briefly reviewed in Sec. 3.2. GLMMSE estimation is proposed in Sec. 3.3 and the design guidelines for MTF are discussed in Sec. 3.4. Computation of GLMMSE estimator based on Gaussian density approximation
target tracking problems in Sec. 3.6 and 3.7, respectively. Its performance is compared with the conventional LMMSE estimation based on the results of Monte Carlo simulation. A summary is made in Sec. 3.8.