Measured-State Conditioned Recursive Feasibility forStochastic Model Predictive Control

In this paper, we address the problem of designing stochastic model predictive control (SMPC) schemes for linear systems affected by unbounded disturbances. The contribution of the paper is rooted in a measured-state initialization strategy. First, due to the nonzero probability of violating chance-constraints in the case of unbounded noise, we introduce ellipsoidal-based probabilistic reachable sets, and we include constraint relaxations to recover recursive feasibility conditioned on the measured state. Second, we prove that the solution of this novel SMPC scheme guarantees closed-loop chance constraints satisfaction under minimum relaxation. Last, we demonstrate that, in expectation, the need to relax the constraints vanishes over time, which leads the closed-loop trajectories steered toward the unconstrained LQR invariant region. This novel SMPC scheme is proven to satisfy the recursive feasibility conditioned on the state realization, and its superiority with respect to open-loop initialization schemes is shown through numerical examples.