This paper considers the problem of applying the Kalman filters to nonlinear systems. The Kalman filter (KF) is an optimal linear estimator when the process noise and the measurement noise can be modeled by white Gaussian noise. The KF only utilizes the first two moments of the state (mean and covariance) in its update rule. In situations when the problems are nonlinear or the noise that distorts the signals is non-Gaussian, the Kalman filters provide a solution that may be far from optimal. Nonlinear problems can be solved with the extended Kalman filter (EKF). This filter is based upon the principle of linearization of the state transition matrix and the observation matrix with Taylor series expansions. Exploiting the assumption that all transformations are quasi-linear, the EKF simply makes linear all nonlinear transformations and substitutes Jacobian matrices for the linear transformations in the KF equations. The linearization can lead to poor performance and divergence of the filter for highly non-linear problems. An improvement to the extended Kalman filter is the unscented Kalman filter (UKF). The UKF approximates the probability density resulting from the nonlinear transformation of a random variable. It is done by evaluating the nonlinear function with a minimal set of carefully chosen sample points. The posterior mean and covariance estimated from the sample points are accurate to the second order for any nonlinearity. The paper presents a comparison of the estimation quality for two nonlinear measurement models of the following Kalman filters: covariance filter (KF), extended filter (EKF) and unscented filter (UKF).
