Optimal joint estimation and identification theorem to linear Gaussian system with unknown inputs

Abstract

Abstract Joint estimation and identification to the linear system with unknown input(s) (UI, UIs) is critical in the control community as well as signal processing. In this paper we present the solution to the problem based on the expectation-maximization (EM) method to alternately estimate system states and identify the UIs. The dominant advantage of the proposed method is that we could handle the UI(s) in not only the system dynamics model but also the measurement model. Specifically we make the following contributions: (1) providing the rigorous mathematical definitions of the problem, (2) theoretically proving the existence and uniqueness of the solution to the joint estimation and identification problem, (3) presenting the theoretical proof of convergence and effectiveness of the EM-based algorithm, and (4) supplying with sufficiently insightful explanations for the mathematical derivation.