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

TL;DR

This work tackles safety in tube Model Predictive Control under state- and input-dependent disturbances by learning a disturbance map with Gaussian Processes and embedding it into a lifted, fixed-graph space. A two-time-scale scheme freezes GP-based uncertainty per learning epoch and performs a monotone outside-in fixed-point iteration in the lifted space to compute a robust, RPI set $Z^ullet$, whose projection yields plant-level disturbance-invariant tubes. By converting GP ellipsoids to polytopic outer bounds and enforcing a fixed graph constraint, the method achieves epoch-nested, data-adaptive tubes with formal safety guarantees, while reducing conservatism as data accumulate. A two-dimensional double-integrator example demonstrates meaningful shrinkage of tube cross-sections in data-rich regions and explicit safety certification through the lifted fixed-point framework.

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 $(1-α)$ 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.

Figures (1)

• Figure 1: GP–learned disturbance structure and lifted–space RPI. (a,b) GP posteriors expose state- and input–dependent effects used to size local disturbance sets. (c,d) The lift–and–project iteration converges to a compact invariant set (yellow) contained in the graph constraint $\mathcal{G}$ (blue). The final projected RPI facet count is ${\operatorname{Proj}_{\mathbf{x}}(Z^{\star,(q)})}_{n_f} = 1466$, convergence metric: Hausdorff distance

Theorems & Definitions (10)

• Lemma 1: Monotonicity property
• proof
• Lemma 2: Cantor–Bolzano fixed point
• proof
• Lemma 3: RPI via measurable selector
• proof
• Lemma 4: Uniform safety of Anchors
• proof : Proof sketch
• Theorem 1: RPI existence, projection, and per-epoch uniform safety
• proof