Polynomial Parametric Koopman Operators for Stochastic MPC

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.

Figures (6)

• Figure 3: Open-loop Duffing oscillator under parametric uncertainty for a representative set of initial conditions $x_0 = [1,1]^\top$: mean trajectories with $\pm1\sigma$ bands for the true system and the PPKO model. Trajectories are generated from 30,000 random parameter samples $[\delta, \alpha, \beta]$, input profiles, and initial conditions over a 40-step horizon ($\Delta t = 0.02$).
• Figure 4: Closed-loop dynamics and control input profiles for 3 representative parametric uncertainty realizations corresponding to three different open-loop dynamical regimes: damped double-well (left), damped single-well (middle), and undamped double-well (right). Uncontrolled open-loop trajectories (dotted) are shown for comparison. $x_0$ is the initial point, $x_f$ is the final point of the simulation, and the target is the origin (0,0). Equilibria correspond to either a stable point or saddle point.
• Figure 5: Series-parallel reaction network and CSTR configuration studied (from von2020stochastic).
• Figure 6: Open-loop CSTR dynamics under parametric uncertainty: mean trajectories with $\pm 1\sigma$ bands for the true system and the PPKO model. Trajectories are generated from random parameter samples $[k_1,k_2]$, input profiles, and initial conditions over a 10-step horizon ($\Delta t = 0.1$).
• Figure 7: Closed-loop CSTR regulation under parametric uncertainty. Top: mean trajectories with $\pm 1\sigma$ bands for the PPKO model and uncontrolled model. Bottom: Mean profile and $\pm 1\sigma$ bands for the control input $q_1$.