Learning-Based Shrinking Disturbance-Invariant Tubes for State- and Input-Dependent Uncertainty

TL;DR

This paper introduces a learning-based method for constructing shrinking disturbance-invariant tubes in control systems, leveraging Gaussian Processes and fixed-point iterations to improve safety and reduce uncertainty as more data is collected.

Contribution

It presents a novel framework combining Gaussian Process posteriors with fixed-point methods to create adaptive, shrinking tubes for robust control under state- and input-dependent uncertainty.

Findings

Tubes become tighter with more data, improving safety margins.

The method maintains invariance and hard constraints during learning.

Demonstrated effectiveness on a double-integrator system.

Abstract

We develop a learning-based framework for constructing shrinking disturbance-invariant tubes under state- and input-dependent uncertainty, intended as a building block for tube Model Predictive Control (MPC), and certify safety via a lifted, isotone (order-preserving) fixed-point map. Gaussian Process (GP) posteriors become $(1-\alpha)$ credible ellipsoids, then polytopic outer sets for deterministic set operations. A two-time-scale scheme separates learning epochs, where these polytopes are frozen, from an inner, outside-in iteration that converges to a compact fixed point $Z^\star\!\subseteq\!\mathcal G$; its state projection is RPI for the plant. As data accumulate, disturbance polytopes tighten, and the associated tubes nest monotonically, resolving the circular dependence between the set to be verified and the disturbance model while preserving hard constraints. A double-integrator study illustrates shrinking tube cross-sections in data-rich regions while maintaining invariance.