دانلود کتاب The Unscented Kalman Filter for Nonlinear Estimation

The Extended Kalman Filter (EKF) has become a standardtechnique used in a number of nonlinear estimation and machinelearning applications. These include estimating thestate of a nonlinear dynamic system, estimating parametersfor nonlinear system identification (e.g., learning theweights of a neural network), and dual estimation (e.g., theExpectation Maximization (EM) algorithm) where both statesand parameters are estimated simultaneously.This paper points out the flaws in using the EKF, andintroduces an improvement, the Unscented Kalman Filter(UKF), proposed by Julier and Uhlman [5]. A central andvital operation performed in the Kalman Filter is the propagationof a Gaussian random variable (GRV) through thesystem dynamics. In the EKF, the state distribution is approximatedby a GRV, which is then propagated analyticallythrough the first-order linearization of the nonlinearsystem. This can introduce large errors in the true posteriormean and covariance of the transformed GRV, which maylead to sub-optimal performance and sometimes divergenceof the filter. The UKF addresses this problem by using adeterministic sampling approach. The state distribution isagain approximated by a GRV, but is now represented usinga minimalset of carefully chosen sample points. These samplepoints completely capture the true mean and covarianceof the GRV, and when propagated through the true nonlinearsystem, captures the posterior mean and covarianceaccurately to the 3rd order (Taylor series expansion) for anynonlinearity. The EKF, in contrast, only achieves first-orderaccuracy. Remarkably, the computational complexity of theUKF is the same order as that of the EKF.Julier and Uhlman demonstrated the substantial performancegains of the UKF in the context of state-estimationfor nonlinear control. Machine learning problems were notconsidered. We extend the use of the UKF to a broader classof nonlinear estimation problems, including nonlinear systemidentification, training of neural networks, and dual estimationproblems. Our preliminary results were presentedin [13]. In this paper, the algorithms are further developedand illustrated with a number of additional examples.