Biases and accuracy of, and an alternative to discrete nonlinear filters

Abstract

Dynamical systems encountered in reality are essentially nonlinear. A number of approaches were propped for nonlinear filter with different accuracy. They are mainly investigated from the Bayesian point of view, and may be classified into three kinds of methods: linearization, statistical approximation and Monte-Carlo simulation (Jazwinski 1970; Tanizaki 1993; Gelb 1994). Very often, linearization of the nonlinear system is done using either a precomputed nominal trajectory or the estimate of the state vector. This second linearization approach is better known as the extended Kalman filter (Jazwinski 1970). Since the solution based on one-step linearization may be poor for a highly nonlinear system, iteration in this case is expected in order to obtain a more accurate estimate. Higher order approaches should probably be taken into account. It should be noted, however, that if the system presents a significant nonlineaaity, even the mean and covariance matrix of the nonlinear filter can be misleading, since the means of the estimated state variables may deviate appreciably from their true parameter values. The basic idea of statistical approximation is to replace a nonlinear function of random variables by a series expansion (Gelb 1994) or to approximate thea posteriori conditional probability density function of the state vector (Sorenson & Stubberud 1968; Kramer & Sorenson 1988). The Monte Carlo simulation technique may be used to determine the mean and covariance matrix of the nonlinear filter (if properly designed), which requires a large sample to obtain statistically meaningful results (see e.g. Brown & Mariano 1989; Carlin et al. 1992; Gelb 1994; Tanizaki 1993)