Asymptotically Optimal Nonlinear Filtering: Theory and Examples with Application to Target State Estimation

Abstract

The State-Dependent Riccati Equation (SDRE) filter, which is derived by constructing the dual of the well-known SDRE nonlinear regulator control design technique, has been studied in various papers, with mainly practical investigations of the filter. Until recently, theoretical aspects of the filter had not been fully investigated, leaving many unanswered questions, such as stability and convergence of the filter. The authors conducted an investigation of the conditions under which the state estimate given by this algorithm converges asymptotically to the first order minimum variance estimate given by the extended Kalman filter (EKF). Conditions for determining a region of stability for the SDRE filter were also investigated. The analysis was based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. In this paper, the motivation for introducing HJB equations is justified with mathematical rigor, which is given by reference to the maximum likelihood approach to deriving the EKF. The application of the SDRE filter is then demonstrated on challenging examples to illustrate the theoretical aspects and design flexibility (additional degrees of freedom) of the algorithm when loss of observability is encountered. In particular, a realistic and detailed evaluation of the filter is carried out for the problem of target state estimation in an advanced tactical missile guidance application for analysis in the optimal guidance problem for air-air engagements using only passive sensor (angle-only) information. Simulation results are presented which show dramatic tracking improvement using the SDRE target tracker.