EM-based adaptive divided difference filter for nonlinear system with multiplicative parameter

This paper is concerned with application of expectation maximization (EM) algorithm for deriving an adaptive version of divided difference filter for joint state estimation and multiplicative parameter identification of nonlinear system with the colored measurement noise. Owing to the fact that there exist a mutual coupling and interaction of state and parameter on each other, it requires a joint or simultaneous estimation of both state and parameter by a mutual iteration, and justly, EM iterates Expectation (E-)step and Maximization (M-)step to meet such requirement. Firstly, E-step involves state filtering and smoothing issues under knowing the previous parameter identification results, which is well solved by resorting to the Gaussian approximation with a trade-off between accuracy and complexity. Further, such Gaussian approximation estimators are applied for evaluating the condition expectation of complete-data likelihood function, nonlinearly characterized by the multiplicative parameter needed to be optimized. Secondly, M-step deals with the maximization of the condition expectation by directly making its derivative as zero to obtain the current general parameter identification equation as the nonlinear integral. Thirdly, by iteratively operating E-step and M-step, an adaptive divided difference filter is proposed for joint state estimation and parameter identification by using the second-order Stirling interpolation to compute the associated nonlinear integral. Finally, the robust performance of the EM-based adaptive version of divided difference filter to the unknown or time-varying multiplicative parameter, as compared with the standard augmentation method, is demonstrated by a maneuvering target tracking example.