Abstract
In this paper, a theoretical investigation of the state-dependent Riccati equation (SDRE) filter is carried out, which is derived by constructing the dual of the well-known SDRE nonlinear regulator control design technique. The SDRE filter has been studied in various papers, with mainly practical investigations of the filter. However, the theoretical aspects of the filter have not been fully investigated and there remain many unanswered questions, such as stability and convergence of the filter. This paper investigates 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 are also investigated. The analysis is based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. The motivation for introducing HJB equations is given by reference to the maximum likelihood approach to deriving the EKF. The application of the SDRE filter will be demonstrated on a simple pendulum problem to illustrate the theory. The behavioral differences and similarities between the SDRE filter, the linearized Kalman filter (LKF) and the EKF are also discussed using this example.
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