Soft-constrained ℓasso-MPC for robust LTI tracking: Enlarged feasible region and an ISS gain estimate

This paper investigates the robustness of a soft constrained LTI MPC for set-point tracking, using an ℓ₁-regularised cost. The MPC using this type of cost (informally dubbed ℓ_asso-MPC) is suitable, for instance, for redundantly-actuated systems. This is because of its ability to select a set of preferred actuators, leaving the other ones at rest for most of the time. The proposed approach aims to recover from state constraint violation and to track a changing set-point. Nominal stability is guaranteed for all feasible references and robustness to additive uncertainties is formally characterised, under certain assumptions. In particular, sufficient conditions are given for the feasible region to be robustly invariant, this region being larger than the MPC for regulation. The closed-loop system is input-to-state stable (ISS), and a local ISS gain is computed. All results apply to stabilisable LTI systems. Results hold as well for the more common quadratic MPC, a special case of the proposed controller.