  In this work, we propose a subsystem decomposition approach and a distributed
estimation scheme for a class of implicit two-time-scale nonlinear systems.
Taking the advantage of the two-time-scale separation, these processes are
decomposed into smaller subsystems such as fast subsystem and slow subsystem.
In the proposed method, an approach, composite solution, combines the
approximate solutions obtained from both fast and slow subsystems. Based on the
fast and slow subsystems, a distributed state estimation scheme is proposed to
handle the implicit time-scale multiplicity. In the proposed design, an
extended Kalman filter (EKF) is designed for the fast subsystem and a moving
horizon estimator (MHE) is designed for the slow subsystem. There is a
communication between the estimators: the slow subsystem is only required to
send information to the fast subsystem one-directionally. The fast subsystem
estimator does not send out any information. The estimators use different
sampling of the state measurements, i.e., fast sampling of the fast state
variables is considered in the fast EKF and slow sampling of the slow state
variables is considered in the slow MHE. Extensive simulations based on a
chemical process are performed to illustrate the effectiveness and
applicability of the proposed subsystem decomposition and composite state
estimation architecture.
