On upper bounds on worst-case H 2 performance obtained with dynamic multipliers

In this note, an iterative procedure is presented for obtaining a decreasing sequence of upper bounds on the worst-case H ₂ performance of a given stabilizing controller in the presence of normalized coprime-factor perturbations. To obtain such bounds, a descent procedure is introduced for a dual Lagrangean functional which gives upper bounds on the worst-case H ₂ performance index and is defined on the set of real-rational and non-negative functions (dynamic multipliers). Specific ways are presented for selecting feasible and descent directions in this set and lower bounds are derived for the corresponding decreases on the dual functional. At any step, the dynamics of the current multiplier gives rise to a linear class over which the optimization of the dual functional is shown to be equivalent to linear optimization subject to linear matrix inequalities (LMI). This allows for a combination of function space and LMI techniques in the process of obtaining increasingly tighter upper bounds on worst-case H ₂ performance, as illustrated in a numerical example.