Contraction-Based Model Predictive Control using Bilinear Koopman Realizations with Proportional Error Bounds

Abstract

Data-driven model predictive control based on Koopman operator theory is a promising approach for constrained control of nonlinear systems with unknown dynamics. This paper proposes a robust model predictive control framework for such systems using bilinear Koopman realizations with state- and input-dependent proportional approximation-error bounds that vanish at the target equilibrium. Since prediction in lifted coordinates with a finite-dimensional Koopman realization need not remain on the manifold of valid lifted states, multi-step prediction may leave the region where one-step error certificates apply. We avoid this difficulty by reprojecting each predicted lifted state onto the original state coordinates and lifting it again, yielding an error-aware discrete-time control-affine predictor in the original state space without assuming invariance of the Koopman dictionary. For this predictor, we develop a homothetic tube construction based on a discrete-time robust control contraction metric whose radius explicitly accounts for the proportional approximation error bounds. The tube tightening handles arbitrary continuously differentiable nonlinear constraints, and the resulting model predictive control problem includes terminal ingredients and a tube-radius penalty that exploits the vanishing uncertainty near the target. We prove robust satisfaction of the original nonlinear constraints, recursive feasibility, and exponential stability of the sampled true closed-loop system with high probability over the training data without requiring a globally optimal solution to the model predictive control problem. Numerical examples, including nonlinear obstacle-avoidance constraints, demonstrate robust stabilization and higher performance of the proposed approach compared to existing Koopman-based model predictive control methods in terms of smaller closed-loop cost and flexibility.