  We study the online robust control problem for linear dynamical systems with
disturbances and uncertainties in the cost functions, with limited preview of
the future disturbances and the cost functions, $N$. Our goal is to find an
online control policy that can minimize the disturbance gain, defined as the
ratio of the cumulative cost and the cumulative energy in the disturbances over
a period of time, in the face of the uncertainties, and characterize its
achievable gain in terms of the system relevant parameters. Our goals contrast
with prior online control works for the same problem, which either focus on
minimizing the static regret, a weaker performance metric, or assume a very
large preview of the future uncertainties. Specifically, we consider a class of
cost functions characterized by $\beta$ ($\beta < 1$), a number whose inverse
bounds the variation of the cost functions. We propose a novel variation of the
Receding Horizon Control as the online control policy. We show that, under
standard system assumptions, when $N > 4/\beta^3$, the proposed algorithm can
achieve a disturbance gain $(2/\beta+\rho(N)) \overline{\gamma}^2$, where
$\overline{\gamma}^2$ is the best (minimum) possible disturbance gain for an
oracle policy with full knowledge of the cost functions and disturbances, with
$\rho(N) = O(1/N)$. We also demonstrate through simulations that the proposed
policy satisfies the derived bounds and is consistently better than the
standard RHC approach.
