Polynomial Parametric Koopman Operators for Stochastic MPC
This work addresses uncertainty-aware control for nonlinear systems, offering an incremental improvement in computational efficiency for SMPC.
The paper tackled the challenge of efficient Stochastic Model Predictive Control (SMPC) for nonlinear systems by developing a parametric Koopman operator framework using Polynomial Chaos Expansions, which enables condensed deterministic reformulation with dimension scaling only with control horizon and input dimension, independent of lifted state dimension and PCE terms, allowing use of standard convex optimization solvers.
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.