A Comprehensive Expectation Identification Framework for Multirate Time-Delayed Systems

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 article 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 algorithm can avoid derivative function calculation; 3) for a large-scale system, the proposed expectation multidirection algorithm does not require eigenvalue calculation and has less computational costs. These two algorithms are developed based on the gradient descent and multidirection 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 the effectiveness of the algorithms.