Array Algorithms for H 2 and H ∞ Estimation

Currently, the preferred method for implementing H2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms, that are typically more stable than conventional ones, and fast array algorithms, which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Using our recent observation that H ∞ filtering coincides with Kalman filtering in Krein space, in this chapter we develop array algorithms for H∞ filtering. These can be regarded as natural generalizations of their H2 counterparts, and involve propagating the indefinite square roots of the quantities of interest. The H ∞ 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 ∞ filters. These conditions are built into the algorithms themselves so that an H ∞ estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H ∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H2 case, further investigation is needed to determine the numerical behavior of such algorithms.