A Koopman--Polynomial Chaos Framework for SMPC of Nonlinear Stochastic Processes

This paper develops a polynomial parametric Koopman operator (PPKO) framework for stochastic model predictive control (SMPC) of nonlinear systems with parametric uncertainty and time-varying stochastic disturbances. The Koopman operator is parameterized by polynomial chaos expansions (PCEs), enabling data-driven nonlinear dynamics to be integrated with PCE-based uncertainty propagation and control. We show that the polynomial parameterization introduces no irreducible approximation gap beyond PCE truncation. We then derive a condensed deterministic SMPC reformulation that handles both time-invariant parametric uncertainty and time-varying disturbances while retaining an online optimization dimension that depends only on the control horizon and input dimension, independent of the lifted-state and PCE dimensions. The resulting formulation yields convex programs with expectation and second-moment constraints that can be solved using standard optimization methods. Numerical studies on the Duffing oscillator and the electrothermal cooling system demonstrate the accuracy, scalability, and closed-loop performance of the proposed framework.