Recursive estimators and hybrid Cramér–Rao bounds for discrete-time Markovian dynamic systems

Abstract

This article addresses the joint estimation of random states and deterministic parameters within a wide class of linear discrete state-space (DSS) models when a large set of dependent observations is available. Two models are considered, depending on whether the initial state is assumed to be deterministic or random: the generalized conditional signal model and the generalized unconditional signal model. A unified recursive method for estimating the initial state, current state, and unknown deterministic parameters, is obtained by resorting to the Kalman Filter (KF), then generalized to the class of nonlinear DSS models compatible with the extended KF. Hence, to assess the efficiency of these estimators, we also derive recursive hybrid Cramér–Rao bounds for Markovian dynamic systems. A unified framework is established for bounds on complex-valued initial and current states in the presence of unknown deterministic parameters. Finally, the article includes numerical simulations of the empirical mean squared error of the proposed estimators, provides closed-form expressions of the bounds, and demonstrates their practical relevance through two radar application examples.