  Inspired by online learning, data-dependent regret has recently been proposed
as a criterion for controller design. In the regret-optimal control paradigm,
causal controllers are designed to minimize regret against a hypothetical
optimal noncausal controller, which selects the globally cost-minimizing
sequence of control actions given noncausal access to the disturbance sequence.
  We extend regret-optimal control to the more challenging measurement-feedback
setting, where the online controller must compete against the optimal noncausal
controller without directly observing the state or the driving disturbance. We
show that no measurement-feedback controller can have bounded competitive ratio
or regret which is bounded by the pathlength of the measurement disturbance. We
do derive, however, a controller whose regret has optimal dependence on the
joint energy of the driving and measurement disturbances, and another
controller whose regret has optimal dependence on the pathlength of the driving
disturbance and the energy of the measurement disturbance. The key technique we
introduce is a reduction from regret-optimal measurement-feedback control to
$H_{\infty}$-optimal measurement-feedback control in a synthetic system. We
present numerical simulations which illustrate the efficacy of our proposed
control algorithms.
