MLE in presence of equality and inequality nonlinear constraints for the ballistic target problem

Abstract

This paper focuses on the estimation of the impact point of a ballistic target by means of a batch processing approach which can be applied more specifically when the number of radar measurements is poor. In particular, the Maximum Likelihood (ML) estimator is proposed for the estimation of the ballistic target trajectory approximated to a pure parabolic curve; also the Cramer-Rao Lower Bound (CRLB), which gives the minimum theoretically achievable variance of the estimate, is calculated. The estimator accuracy is improved by the application of equality and inequality constraints on the trajectory characteristics such as maximum range, maximum height and trajectory plane. In the case of application of equality constraints also the CRLB can be computed and the constrained estimator improves strongly its accuracy. In real operational scenarios, a smoothed information can be expressed by means of inequality constraints. The performance of the corresponding estimator has been analyzed by means of Monte Carlo simulation, because of the unavailability of computation of the CRLB for this case.

Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements

“…As demonstrated in [33] (see also [34]), assuming θ ∈ R n , when the MLE problem has an equality constraint in the form f (θ) = 0 q , f : R n → R q , the CRLB can be computed starting from the unconstrained case, according to the following equation…”

“…This additional information is exploited by developing a constrained MLE problem and an approach for the selection of the ownship's trajectory mimicking the Artificial Potential Fields technique [31,32], which is typically used by mobile robots to aim at a goal (in this case, the region where the target is assumed to be) while avoiding obstacles (i.e., the region that is assumed not to intersect the target's trajectory). Moreover, from a theoretical standpoint, the CRLB on the estimation covariance matrix is characterized in the case of MLE problems with inequality constraints; this is performed by extending the approach in [33,34], where equality constraints where discussed, via the cast of inequality constraints into nonsmooth equality ones and by the adoption of generalized Jacobian matrices [35], which are set-valued on a zero-measure set where the derivative of the resulting nonsmooth function is not defined.…”