Nonlinear filtering using the complex -step derivative approximation
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This dissertation presents a new approach to nonlinear filtering using a divided difference with a complex-step derivative approximation to compute numerical derivatives. In this dissertation we show using a complex-step on polynomial approximations obtained with an interpolation formula has considerable advantages compared to standard Divided Difference (DD2) filter and also other well known filters such as the Extended Kalman Filter (EKF) and Unscented Filter (UF). The generalized complex-step first- and second-order derivatives are applied to the DD2 filter to maximize its potential. Formal analysis of the approximations of mean estimates and covariance estimates is conducted to prove that the new approach with complex-step introduces errors in higher order terms than other filter, which means that it will give a better approximation in general by introducing less error in the mean and covariance estimates. Simulation results are provided to show the performance of the new filter. For simulation, the classic falling body example is invoked to evaluate the performance of the new filter and compared with various well known filters including the standard DD2 filter, EKF and UF.