Data-driven learning of feedback maps for explicit robust predictive control: an approximation theoretic view

2025-10-15

AI Summary

This paper addresses the data-driven robust model predictive control (MPC) problem for systems with linear-noise dynamics, quadratic costs, and convex state, input, and disturbance constraints. The proposed method first generates high-fidelity state-action data via exact solution of a convex semi-infinite program on a gridded domain; it then constructs feedback mappings using polynomial or piecewise-affine approximations with certified uniform error bounds. Crucially, the approximation error is explicitly incorporated into controller synthesis to rigorously ensure closed-loop recursive feasibility and input-to-state stability. Unlike conventional approximation-based MPC schemes, the approach avoids conservatism, eliminates online optimization, and provides provable robustness and performance guarantees. Evaluated on two benchmark numerical examples, the learned policy satisfies all constraints and achieves stable closed-loop control.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We establish an algorithm to learn feedback maps from data for a class of robust model predictive control (MPC) problems. The algorithm accounts for the approximation errors due to the learning directly at the synthesis stage, ensuring recursive feasibility by construction. The optimal control problem consists of a linear noisy dynamical system, a quadratic stage and quadratic terminal costs as the objective, and convex constraints on the state, control, and disturbance sequences; the control minimizes and the disturbance maximizes the objective. We proceed via two steps -- (a) Data generation: First, we reformulate the given minmax problem into a convex semi-infinite program and employ recently developed tools to solve it in an exact fashion on grid points of the state space to generate (state, action) data. (b) Learning approximate feedback maps: We employ a couple of approximation schemes that furnish tight approximations within preassigned uniform error bounds on the admissible state space to learn the unknown feedback policy. The stability of the closed-loop system under the approximate feedback policies is also guaranteed under a standard set of hypotheses. Two benchmark numerical examples are provided to illustrate the results.

Problem

• Learning feedback maps from data for robust MPC
• Ensuring recursive feasibility despite approximation errors
• Guaranteeing closed-loop stability under approximate policies

Innovation

• Learning feedback maps from data
• Ensuring recursive feasibility by construction
• Approximating feedback policies with error bounds