Real-time Recursive State Estimation for Nonlinear Discrete Dynamic Systems with Gaussian or non-Gaussian Noise

Abstract

Many systems in the real world are more accurately described by nonlinear models. Since the original work of Kalman (Kalman, 1960; Kalman & Busy, 1961), which introduces the Kalman filter for linear models, extensive research has been going on state estimation of nonlinear models; but there do not yet exist any optimum estimation approaches for all nonlinear models, except for certain classes of nonlinear models; on the other hand, different suboptimum nonlinear estimation approaches have been proposed in the literature (Daum, 2005). These suboptimum approaches produce estimates by using some sorts of approximations for nonlinear models. The performances and implementation complexities of these suboptimum approaches surely depend upon the types of approximations which are used for nonlinear models. Model approximation errors are an important parameter which affects the performances of suboptimum estimation approaches. The performance of a nonlinear suboptimum estimation approach is better than the other estimation approaches for specific models considered, that is, the performance of a suboptimum estimation approach is model-dependent. The most commonly used recursive nonlinear estimation approaches are the extended Kalman filter (EKF) and particle filters. The EKF linearizes nonlinear models by Taylor series expansion (Sage & Melsa, 1971) and the unscented Kalman filter (UKF) approximates a posteriori densities by a set of weighted and deterministically chosen points (Julier, 2004). Particle filters approximates a posterior densities by a large set of weighted and randomly selected points (called particles) in the state space (Arulampalam et al., 2002; Doucet et al., 2001; Ristic et al., 2004). In the nonlinear estimation approaches proposed in (Demirbas, 1982; 1984; Demirbas & Leondes, 1985; 1986; Demirbas, 1988; 1989; 1990; 2007; 2010): the disturbance noise and initial state are first approximated by a discrete noise and a discrete initial state whose distribution functions the best approximate the distribution functions of the disturbance noise and initial state, states are quantized, and then multiple hypothesis testing is used for state estimation; whereas Grid-based approaches approximate a posteriori densities by discrete densities, which are determined by predefined gates (cells) in the predefined state space; if the state space is not finite in extent, then the state space necessitates some truncation of the state space; and grid-based estimation approaches assume the availability of the state Real-time Recursive State Estimation for Nonlinear Discrete Dynamic Systems with Gaussian or non-Gaussian Noise 1