L2 optimal filter reduction: a closed-loop approach

This paper is concerned with the problem of order reduction of a full-order Kalman filter for a stable linear signal model so that the steady-state filtering error variance associated with the reduced order filter is minimized. By an orthogonal parameterization, the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices. Both continuous and iterative algorithms are derived to compute an optimal reduced-order filter. The algorithms are shown to possess good properties, including the desirable convergence property. The proposed algorithms are simple and effective. Numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open-loop methods such as the well-known balanced truncation method