Polynomial Parametric Koopman Operators for Stochastic MPC

TL;DR

This paper introduces a polynomial chaos-based parametric Koopman operator framework for efficient stochastic model predictive control, enabling convex optimization for nonlinear systems with uncertainty.

Contribution

It develops a data-driven, convex reformulation of stochastic MPC using polynomial chaos and Koopman operators, scalable with control horizon and input dimension.

Findings

Efficient nonlinear SMPC with expectation and moment constraints.

Convex reformulation scales independently of lifted state dimension.

Numerical examples validate the framework's effectiveness.

Abstract

This paper develops a parametric Koopman operator framework for Stochastic Model Predictive Control (SMPC), where the Koopman operator is parametrized by Polynomial Chaos Expansions (PCEs). The model is learned from data using the Extended Dynamic Mode Decomposition -- Dictionary Learning (EDMD-DL) method, which preserves the convex least-squares structure for the PCE coefficients of the EDMD matrix. Unlike conventional stochastic Galerkin projection approaches, we derive a condensed deterministic reformulation of the SMPC problem whose dimension scales only with the control horizon and input dimension, and is independent of both the lifted state dimension and the number of retained PCE terms. Our framework, therefore, enables efficient nonlinear SMPC problems with expectation and second-order moment constraints with standard convex optimization solvers. Numerical examples demonstrate the efficacy of our framework for uncertainty-aware SMPC of nonlinear systems.