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

2026-01-16

IEEE Control Systems Letters

AI Summary

This work addresses safe tube model predictive control under state- and input-dependent uncertainties by proposing a learning-based two-timescale framework. The approach employs Gaussian process regression to construct posterior credible ellipsoids, which are then outer-approximated by polyhedra and embedded into the inner-layer optimization to generate monotonically shrinking disturbance-invariant tubes. This is the first method to integrate Gaussian process learning with disturbance-invariant tube construction, effectively decoupling the circular dependency between the disturbance model and the verification set via a dimension-lifting order-preserving fixed-point mapping. Experimental results on a double integrator system demonstrate that, as data accumulates, the cross-sections of the disturbance tubes contract significantly in information-rich regions while rigorously preserving robust positive invariance and hard constraint satisfaction.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 <inline-formula> <tex-math notation="LaTeX">$\text {(}1-\alpha \text {)}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$Z^{\star }\subseteq {\mathcal {G}}$ </tex-math></inline-formula>; 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.

Problem

• disturbance-invariant tubes
• state- and input-dependent uncertainty
• tube Model Predictive Control
• safety certification
• set invariance

Innovation

• learning-based tube MPC
• disturbance-invariant tubes
• Gaussian Process uncertainty
• robust positive invariance
• two-time-scale iteration