Finite horizon H ∞ filter and its 2N algorithm

Abstract

A 2^N algorithm will double a time (or space) step in each evaluation for initial value problem. The 2^N algorithm for the integration of filtering differential equation of the finite horizon H ∞ filter is presented in this paper. Since it is a boundary value problem within a time range, a new 2^N algorithm is introduced by merging two intervals each time so that the time interval is doubled in each evaluation. If one divides the original time range into one million intervals, 20 evaluations will complete the whole process. Owing to the extremely small initial time interval, the first few terms of the Taylor expansion of the interval matrices are sufficient for very accurate results. Since the filter gain matrices are the solution of the Riccati differential equation and the existence of the solution depends on the induced norm γ, the computation of critical value y^CT_-2 is reviewed first. Then, according to the result and the prespecified performance index, the suitable parameter γ⁻² can be selected and the precise numerical solution of the Riccati differential equation and the filtering differential equation can be obtained by using the 2^N algorithm, although the filtering equation is time varying. The 2^N algorithm for interval merging is given explicitly.