A novel set-based reachability method for optimal robust control of constrained linear systems

This paper addresses finite horizon optimal control of linear dynamical systems affected by an additive bounded disturbance and subject to polyhedral state and input constraints. The goal is to design a static state feedback control law that minimizes a quadratic nominal cost while robustly satisfying the state/input constraints. We propose a novel set-based reachability approach and compare it against two alternative set-based approaches that were proposed for robust model predictive control (MPC) a few years back. All the three approaches offer a computational procedure to the design of a control policy, which is expressed as the sum of a state feedback term and an open loop term. While in the robust MPC methods the feedback term is a-priori fixed and only the open loop term is optimized, in the proposed method both of them are design parameters that are jointly optimally tuned. This is achieved by adopting a zonotopic parametrization of the control law that makes constraints and cost function respectively linear and quadratic in the parameters. As a result, the set-based reachability method provides a feasible solution for tighter constraints than the two alternative set-based methods.