Recursive feasibility for stochastic MPC and the rationale behind fixing flat tires

TL;DR

This work tackles recursive feasibility for stochastic model predictive control (SMPC) of linear systems under unbounded disturbances, introducing a measurement-dependent feasibility concept. It proposes ellipsoidal probabilistic reachable sets ${\mathcal{E}}_{W_x}$ and ${\mathcal{E}}_{W_u}$ with minimal relaxation variables to enforce chance constraints conditioned on the measured state $x_k$. Theoretical results prove chance-constraint satisfaction in expectation and show that relaxations vanish over time, guiding trajectories toward the unconstrained LQR invariant region, while numerical results compare MS-SMPC to open-loop initializations and demonstrate its advantages near feasibility boundaries. The framework lays groundwork for mission-wide constraints and polytopic extensions, highlighting practical impact for robust, data-informed SMPC under uncertainty.

Abstract

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 to 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 of relaxing the constraints vanishes over time, which leads the closed-loop trajectories steered towards the unconstrained LQR invariant region. This novel SMPC scheme is proven to satisfy the recursive feasibility conditioned to the state realization, and its superiority with respect to open-loop initialization schemes is shown through numerical examples.

Figures (5)

• Figure 2: State and input trajectories for SMPC enforcing no bounds (solid lines), hard bounds (dotted lines), and soft bounds (dashed lines) on the first input $v_{0|k}$.
• Figure 3: Comparison among the realized state trajectories starting from $x_0=(-40,\,40)$ for $k \in [0,10]$ enforcing on $v_{0|k}$: a) no bounds (solid line); b) hard bounds (dotted line); and, c) soft bounds (dashed lined).
• Figure 4: Cost comparison over 1000 simulations with $N=10$ for $x_0 = (-40,\, 40)$ of the three input bounds strategies. Mean cost: no bounds $J_{MPC} = 9999$; hard bounds $J_{MPC}=15460$; soft bounds $J_{MPC}=11552$.
• Figure 5: Cost comparison over 1000 simulations with $N=10$ for $x(0) = (-30, 0)$. Mean cost: for the new MS-SMPC (with no initial bound) $J_{MPC}=2951$, for the IS-SMPC $J_{MPC}=2956$, with an almost unitary ratio.
• Figure 6: Cost comparison over 1000 simulations of 10 steps for $x(0) = (-40, 37)$, close to the IS-SMPC feasibility bounds. Mean cost: for the MS-SMPC (with no initial bound) $8584$, for the IF-SMPC $11085$, with ratio of around $0.77$.