The robust $H_2$ control problem : a worst case design

The worst-case effect of a disturbance system on the H 2 norm of the system is analyzed. An explicit expression is given for the worst-case H2 norm when the disturbance system is allowed to vary over all nonlinear, time-varying and possibly noncausal systems with bounded L2-induced operator norm. An upper bound for this measure, which is equal to the worst-case H2 norm if the exogeneous input is scalar, is defined. Some further analysis of this upper bound is done, and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given. The latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature.