Collective Tube Model Predictive Control With Distribution-Free Joint Safety Certificates

Abstract: Data-calibrated stochastic MPC typically builds separate risk margins for many events along the horizon, such as times, facets, state/input components, or obstacles, and then combines them with a union bound. This approach is valid, but it does not match the key object used in the tube-MPC recursive-feasibility proof, which shifts a complete error tube. This paper develops collective tube MPC (CT-MPC), where the calibrated uncertainty object is the finite-horizon prediction-error trajectory. A reusable trajectory tube is calibrated offline, its cross-sections define deterministic Pontryagin tightenings online, and the certified violation event is that a fresh prediction-error trajectory leaves the tube. For linear systems with additive uncertainty and fixed ancillary feedback, we prove joint state-input safety over the prediction horizon, one-step recursive feasibility from an explicit shifted candidate, a finite-deployment risk bound, and a practical value-decrease inequality. The finite-sample certificate is distribution-free under split calibration and has beta-binomial form; its complexity is the certified number of residual trajectories that can define the tube, rather than the number of horizon-constraint blocks. We also give implementable shift-compatibility tests for polytopic tubes and a stable-compression fallback for irregular tube designers. Numerical experiments compare CT-MPC with Bonferroni tightening, a sample-envelope tube, and a joint-in-time conformal MPC baseline. The collective tube reduces deterministic tightening while preserving empirical safety and recursive feasibility.