Research On Nonlinear Filtering Algorithm Based On Variational Bayesian Inference

Recently,nonlinear filtering algorithms had attracted much attention due to the importance.Although lots of important theoretical results had been achieved,researches still need to be studied.For nonlinear stochastic systems,it was difficult or impossible to obtain the posterior distribution of the state by using traditional Kalman filter,so it was impossible to obtain accurate optimal filtering estimation.Therefore,many classical sub-optimal nonlinear filtering methods had been proposed.With the improvement of technology,the theory of nonlinear filtering had also obtained remarkable progress.Many significant theoretical research results had been obtained in the past four decades.Some classical nonlinear filtering methods had emerged,especially Sigma point Kalman filter and particle filter which flourished nonlinear filtering.However,researches on non-linear filtering were still based on the improvement of existing methods in past 10 years.With the development of machine learning theory and the great improvement of hardware performance,the application of machine learning theory to nonlinear filtering had attracted much attention.Variational Bayesian learning and maximum expectation algorithm were the representative learning algorithms.Variational Bayesian learning was proposed by using variational approximation theory on the basis of traditional Bayesian inference and EM iterative estimation algorithm.Because it was faster than Monte Carlo Markov chain method,its application field had extended from image processing,blind source separation,speech enhancement,channel estimation and other parameter inference fields to a larger signal processing field.It became an important topic in this field.The variational Bayesian method was not good enough in model adaptability and needed further development.How to estimate the state of a nonlinear system when there existed uncertain parameters? How to identify nonlinear uncertain systems? How to robustly estimate the state with non-Gaussian random uncertainties? In this paper,the estimation of uncertain parameters,the identification of uncertain systems and the suppression of non-Gaussian random uncertain disturbances were studied by using the variational Bayesian reasoning method for nonlinear uncertain systems.Firstly,the model of uncertain parameters was established,and then the state and uncertain parameters were estimated iteratively based on variational inference.Then,the convergence of the proposed algorithm was analyzed based on the theory of functional analysis to ensure the stability of these presented algorithm.Finally,simulation experiments in typical application scenarios were done to verify the effectiveness of the presented algorithms,which made the algorithm adapt in engineering applications.The main contribution of the paper was organized as follows:1.Variational Bayesian inference was used to study the adaptive filtering algorithm for nonlinear systems with uncertain parameters.The variable Bayesian inference and Monte Carlo sampling technology were combined together to solve this problem.A set of nonlinear recursive adaptive filtering algorithm was proposed to solve unknown parameters and probability density functions.The proposed algorithm approximated the joint posterior distribution of real system parameters and uncertain parameters and states by generating separable variational distributions.The convergence analysis of variational reasoning was also given to ensure the convergence and robustness of the estimation.2.A class of nonlinear filtering algorithm based on variational Bayesian theory was proposed to solve the unknown measurement noise problem in target tracking system.When the unknown measurement noise was conditionally independent of the state,estimation of the state probability density function was transformed into approximating two probability density functions of the unknown noise and the nonlinear state based on the variational idea.Then,an iterative algorithm for joint estimation of state and unknown measurement noise was established by using variational Bayesian inference.Therefore,unknown measurement noise could be estimated as a hidden state.The convergence results of the approximation algorithm for the nonlinear probability density function were given.3.An estimation algorithm for parameters with non-Gaussian random uncertainties in nonlinear systems was proposed.Bayesian learning had two main objectives: the first was approximate marginal likelihood(non-gaussian random parameters)for model comparison;the second was the posterior distribution of approximate state(also known as system model),which could be used for prediction.These two objectives constituted an iterative algorithm.The rule to determine the cycle was the Kullback-Leibler divergence between the real distribution of the state and the selected fixed and easy-to-handle distribution,which was used to approximate the real distribution,and an iterative algorithm was derived.The algorithm was initialized based on the idea of sampling.4.A variational algorithm was proposed for parameter estimation of nonlinear systems,which combined expectation maximization algorithm,particle smoother and particle filter to directly estimate uncertain parameters in general nonlinear discrete-time state space model.Maximum likelihood criterion was used to ensure the general statistical efficiency of the algorithm.From the perspective of probability density function,uncertain parameters were estimated,in which the uncertain parameters considered in nonlinear systems were distributed arbitrarily.For uncertain parameters and states,the estimation was transformed into two probability density functions,and the sequential importance sampling method was used to initialize the algorithm.Then,an iterative algorithm was established to estimate the state and uncertain parameters jointly.5.A Bayesian learning method based on state transition and expansion of observation function was proposed for identification of nonlinear state-space model.It was learned by basis functions from data,used by Gauss process to connect,and developed by a priori knowledge of coefficients to adjust the flexibility of the model and prevent over-fitting of data,similar to the Gauss process state space model.The prior information could also be seen as a regularization and helped the model to generalize data without sacrificing the richness provided by base function expansion.In order to effectively learn coefficients and other unknown parameters,an identification algorithm was formed by using the latest sequential Monte Carlo method,which provided a theoretical guarantee for Bayesian learning.