H∞-optimality of H2 predictors

Given past observations of a process, {y_j,j<i}, suppose we are interested in constructing one-step-ahead predictors of y _i, denoted by yˆ_i|i-1. We show that, irrespective of whether the process {y_j} is stationary or non-stationary, or whether it is scalar- or vector-valued, the H² -optimal one-step-ahead predictor is also H^∞-optimal. Since the H² and H^∞ paradigms represent fundamentally different approaches to estimation and control, the estimators and controllers obtained from each formalism have often drastically different performances with respect to the other criterion. Our result, however, provides a nontrivial example of when the two formalisms lead to the same optimal design