A Comprehensive Expectation Identification Framework for Multirate Time-Delayed Systems

Abstract

The expectation maximization (EM) algorithm has been extensively used to solve system identification problems with hidden variables. It needs to calculate a derivative equation and perform a matrix inversion in the EM-M step. The equations related to the EM algorithm may be unsolvable for some complex nonlinear systems, and the matrix inversion has heavy computational costs for large-scale systems. This paper provides two expectation based algorithms with the aim of constructing a comprehensive expectation framework concerning different kinds of time-delayed systems: (1) for a small-scale linear system, the classical EM algorithm can quickly obtain the parameter and time-delay estimates; (2) for a complex nonlinear system with low order, the proposed expectation gradient descent (EGD) algorithm can avoid derivative function calculation; (3) for a large-scale system, the proposed expectation multi-direction (EMD) algorithm does not require eigenvalue calculation and has less computational costs. These two algorithms are developed based on the gradient descent and multi-direction methods. Under such an expectation framework, different kinds of models are identified on a case-by-case basis. The convergence analysis and simulation examples show effectiveness of the algorithms.