Bayesian Filtering Based On Sequential Monte Carlo Method And Rao-Blackwellisation

For the filtering problem of dynamical systems,the Bayesian filtering provides a theoretical recursive formula.But in the major application areas,this formula involves the high-dimensional integral,which cannot be computed analytically.In this paper,we introduce the particle filtering which uses the sequential Monte Carlo(SMC)method to approximate the integral.The key idea of the SMC method is to obtain the estimate of the state recursively by updating the discrete samples and their weights constantly.However,the variance of weights can increase continually with iteration.This gives rise to the fact that all but one of weights are very close to zero and the particle degeneracy phenomenon will occur inevitably.Two methods are proposed to reduce the variance.One is to select the appropriate importance density function.From the viewpoint of degenerate distribution,we give the selection criteria of the importance density function in the sense of the minimum conditional variance.Then we obtain the expressions of the optimal importance density function(OPDF)or the suboptimal importance density function(SOPDF)for the specific models.The other method to reduce the variance is the Rao-Blackwellisation technology.This technique constructs a conditional expectation with smaller variance by exploiting any possible analytical substructure in the model,and then obtains a variance-less state estimation via the SMC method.The particle filter based on this technique is usually known as the Rao-Blackwellised particle filter(RBPF).We present the general framework of the RBPF algorithm,and for two important classes of model,the jump Markov linear system(JMLS)and the mixed linear/nonlinear Gaussian model(ML/NLGM),we develop their RBPF algorithms.In addition,this paper also discusses how to calculate the OPDF of the JMLS,the OPDF of the special ML/NLGM,and the SOPDF of the general ML/NLGM.Lastly,we employ four simulation examples to compare the performance of the standard particle filter algorithm(PF)with the bootstrap RBPF algorithm.Obviously it can be found that the estimation accuracy of the RBPF is higher than that of the PF.For the JMLS,we compare the RBPF filtering algorithm based on the OPDF with that based on the bootstrap RBPF;for the ML/NLGM,we compare the RBPF filtering algorithm based on the SOPDF with that based on bootstrap RBPF.We conclude that the filtering accuracy can be improved by using the OPDF or the SOPDF.