${\cal H}_{2}$ Filtering for Discrete-Time Nonlinear Singularly Perturbed Systems

In this paper, we consider the
H
₂
(or Kalman)-filtering problem for discrete-time singularly-perturbed (two time-scale) nonlinear systems. Two types of filters, namely, i) decomposition and ii) aggregate, are discussed, and sufficient conditions for the solvability of the problem in terms of new discrete-time Hamilton-Jacobi-Bellman equations (DHJBEs) are presented. For each type of filter above, first-order approximate filters are derived, and reduced-order filters are also derived in each case. The results are also specialized to linear systems, in which case the DHJBEs reduce to a system of linear-matrix-inequalities (LMIs) which are efficient to solve. Examples are also presented to illustrate the results.