Forward modeling and inverse estimation for nonlinear filtering

The highly accurate state estimation of nonlinear systems is needed in many different application fields. In many existing nonlinear filters, the uncompensated nonlinear approximation error and widely used numerical sampling methods both seriously affect the estimation accuracy and numerical stability. In this paper, based on the variational Bayesian framework, a novel iterative nonlinear filter consisting of forward modeling and inverse estimation is proposed. In the forward modeling stage, a linear Gaussian regression process with distributed variational parameters (VPs) is designed to fit the whole nonlinear measurement likelihood. The distributed VPs consider both first and second moments information, so as to compensate for approximation error from the whole likelihood perspective, instead of only the nonlinear function approximation, which can further improve the accuracy of Bayesian posterior update. In the linear Gaussian regression process, the original nonlinear measurement function (NMF) is just viewed as a prior and no longer directly participates in the modeling of the measurement likelihood. Hence, in the inverse estimation stage, when maximizing the evidence lower bound to update the posteriors of the state and VPs, we can obtain closed-form solutions without numerical sampling methods to approximate the integral of NMF required in many existing methods, which can guarantee numerical stability. Moreover, to avoid the random setting of hyperparameters and to block the propagation of the accumulated error, we propose two rules, estimation and lower bound consistencies, to conduct the hyperparameters' initialization at the beginning of iteration at each sampling time and also derive an optimization method to update hyperparameters during iteration. Finally, the performance of our proposed method is demonstrated in the nonlinear orbital estimation problems.