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We study the online robust control problem for linear dynamical systems withdisturbances and uncertainties in the cost functions, with limited preview ofthe future disturbances and the cost functions, $N$. Our goal is to find anonline control policy that can minimize the disturbance gain, defined as theratio of the cumulative cost and the cumulative energy in the disturbances overa period of time, in the face of the uncertainties, and characterize itsachievable gain in terms of the system relevant parameters. Our goals contrastwith prior online control works for the same problem, which either focus onminimizing the static regret, a weaker performance metric, or assume a verylarge preview of the future uncertainties. Specifically, we consider a class ofcost functions characterized by $\beta$ ($\beta < 1$), a number whose inversebounds the variation of the cost functions. We propose a novel variation of theReceding Horizon Control as the online control policy. We show that, understandard system assumptions, when $N > 4/\beta^3$, the proposed algorithm canachieve a disturbance gain $(2/\beta+\rho(N)) \overline{\gamma}^2$, where$\overline{\gamma}^2$ is the best (minimum) possible disturbance gain for anoracle policy with full knowledge of the cost functions and disturbances, with$\rho(N) = O(1/N)$. We also demonstrate through simulations that the proposedpolicy satisfies the derived bounds and is consistently better than thestandard RHC approach.

Online Robust Control of Linear Dynamical Systems with Limited Prediction