Finite horizon H filter and its 2N algorithm

A 2N algorithm will double a time (or space) step in each evaluation for initial value problem. The 2N algorithm for the integration of (ltering di:erential equation of the (nite horizon H∞ (lter is presented in this paper. Since it is a boundary value problem within a time range, a new 2N 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 (rst few terms of the Taylor expansion of the interval matrices are su>cient for very accurate results. Since the (lter gain matrices are the solution of the Riccati di:erential equation and the existence of the solution depends on the induced norm , the computation of critical value −2 cr is reviewed (rst. Then, according to the result and the prespeci(ed performance index, the suitable parameter −2 can be selected and the precise numerical solution of the Riccati di:erential equation and the (ltering di:erential equation can be obtained by using the 2N algorithm, although the (ltering equation is time varying. The 2N algorithm for interval merging is given explicitly