  We propose a robust model predictive control (MPC) method for discrete-time
linear time-invariant systems with norm-bounded additive disturbances and model
uncertainty. In our method, at each time step we solve a finite time robust
optimal control problem (OCP) which jointly searches over robust linear state
feedback controllers and bounds the deviation of the system states from the
nominal predicted trajectory. By leveraging the System Level Synthesis (SLS)
framework, the proposed robust OCP is formulated as a convex quadratic program
in the space of closed-loop system responses. When an adaptive horizon strategy
is used, we prove the recursive feasibility of the proposed MPC controller and
input-to-state stability of the origin for the closed-loop system. We
demonstrate through numerical examples that the proposed method considerably
reduces conservatism when compared with existing SLS-based and tube-based
robust MPC methods, while also enjoying low computational complexity.
