  We consider measurement-feedback control in linear dynamical systems from the
perspective of regret minimization. Unlike most prior work in this area, we
focus on the problem of designing an online controller which competes with the
optimal dynamic sequence of control actions selected in hindsight, instead of
the best controller in some specific class of controllers. This formulation of
regret is attractive when the environment changes over time and no single
controller achieves good performance over the entire time horizon. We show that
in the measurement-feedback setting, unlike in the full-information setting,
there is no single offline controller which outperforms every other offline
controller on every disturbance, and propose a new $H_2$-optimal offline
controller as a benchmark for the online controller to compete against. We show
that the corresponding regret-optimal online controller can be found via a
novel reduction to the classical Nehari problem from robust control and present
a tight data-dependent bound on its regret.
