Non-linear Filtering Based On Monte Carlo Method

In order to obtain a posterior density function(PDF) of the state for dynamic systems, we usually use Bayesian estimation principle. For linear dynamic systems,Kalman filter(KF) is the optimal solution in the Bayesian framework. But in the major application areas, KF can only be applied for linear systems. For nonlinear systems, the most common algorithm is the extended Kalman filter(EKF). It has been proved that the effect of the state estimation for first-order Taylor expansion is not better than the effect based on second-order expansion. It is very complicated to calculate for the second or higher order expansion, but the estimation effect does not behave significantly better than for the first-order. So, in practical applications, people often use the first order extended Kalman filter.In the Bayesian framework, for nonlinear systems, it requires two procedures consisting of propagation and updating to construct the probability density function of the state based on all the available information. In the two process, we need to calculate the inverse function of nonlinear transformations, Jacobian matrix and high dimensional integral. The analytic solution expressions can not be directly calculated.Therefore, this discussion will develop such methods that do not need complicated calculation. The algorithm of nonlinear filtering based on Monte Carlo method is called Particle filter(PF). The required density of the state vector is represented as a set of random samples, and this method is not restricted by assumptions of linearity or Gaussian noise. But in the sampling process, it will produce a degeneracy of weight problem which needs to use re-sampling. Although the re-sampling process solves the degradation problem of the sample weight, it increases another problem of the lack of sample diversity. Therefore, two methods are introduced in discussion. They are roughening method and prior editing method. Through these two methods, we hope to improve the estimation performance of the system and to overcome sample diversity issue.Lastly, two examples are presented. For these two simulation examples, the performance of the PF algorithm is greatly superior to the first order EKF method. At the same time, it also verifies that roughening method and prior editing method are effective to overcome the problem of sample diversity in the second model on target tracking.