A performance bound for maneuvering target tracking using best-fitting Gaussian distributions

In this paper, we consider the problem of calculating the posterior Cramer-Rao lower bound (PCRLB) in the case of tracking a maneuvering target. In a recent article (A. Bessel, et al., 2003) the authors calculated the PCRLB conditional on the maneuver sequence and then determined the bound as a weighted average, giving an unconditional PCRLB (referred to herein as the Enumer-PCRLB). However, we argue that this approach can produce an optimistic lower bound because the sequence of maneuvers is implicitly assumed known. Indeed, in simulations we show that in tracking a target that can switch between a nearly constant-velocity (NCV) model and a coordinated turn (CT) model, the Enumer-PCRLB can be lower than the PCRLB in the case of tracking a target whose motion is governed purely by the NCV model. Motivated by this, in this paper we develop a general approach to calculating the maneuvering target PCRLB based on utilizing best-fitting Gaussian distributions. The basis of the technique is, at each stage, to approximate the multi-modal prior target probability density function using a best-fitting Gaussian distribution. We present a recursive formula for calculating the mean and covariance of this Gaussian distribution, and demonstrate how the covariance increases as a result of the potential maneuvers. We are then able to calculate the PCRLB using a standard Riccati-like recursion. Returning to our previous example, we show that this best-fitting Gaussian approach gives a bound that shows the correct qualitative behavior, namely that the bound is greater when the target can maneuver. Moreover, for simulated scenarios taken from (A. Bessel, et al., 2003), we show that the best-fitting Gaussian PCRLB is both greater than the existing bound (the Enumer-PCRLB) and more consistent with the performance of the variable structure interacting multiple model (VS-IMM) tracker utilized therein.