Square-root arrays and Chandrasekhar recursions for H∞ problems

Using their previous observation that H^∞ filtering coincides with Kalman filtering in Krein space the authors develop square-root arrays and Chandrasekhar recursions for H^∞ filtering problems. The H^∞ square-root algorithms involve propagating the indefinite square-root of the quantities of interest and have the property that the appropriate inertia of these quantities is preserved. For systems that are constant, or whose time-variation is structured in a certain way, the Chandrasekhar recursions allow a reduction in the computational effort per iteration from O(n³) to O(n²), where n is the number of states. The H^∞ square-root and Chandrasekhar recursions both have the interesting feature that one does not need to explicitly check for the positivity conditions required of the 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.