Array Algorithms for H2 and H1 Estimation

Currently, the preferred method for implementing H 2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms which are typically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically ooer an order of magnitude reduction in the computational eeort. Using our recent observation that H 1 ltering coincides with Kalman ltering in Krein space, in this paper we develop array algorithms for H 1 ltering. These can be regarded as natural generalizations of their H 2 counterparts, and involve propagating the indeenite square-roots of the quantities of interest. The H 1 square-root and fast array algorithms both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H 1 lters. These conditions are built into the algorithms themselves so that an H 1 esti-mator of the desired level exists if, and only if, the algorithms can be executed. However, since H 1 square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H 2 case, further investigation is needed to determine the numerical behaviour of such algorithms.