发表机构
The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于双线性Koopman实现与比例误差界的鲁棒模型预测控制框架，通过重投影提升状态构建误差感知预测器，并利用同胚管收紧处理非线性约束，证明了约束满足、递归可行性与指数稳定性，数值实验显示其性能优于现有Koopman方法。
AI 中文摘要
基于Koopman算子理论的数据驱动模型预测控制是处理动力学未知的非线性系统约束控制的一种有前景的方法。本文针对此类系统提出了一种鲁棒模型预测控制框架，采用双线性Koopman实现，并带有状态和输入相关的比例逼近误差界，该误差界在目标平衡点处消失。由于在提升坐标下使用有限维Koopman实现的预测不必保持在有效提升状态流形上，多步预测可能离开单步误差证书适用的区域。我们通过将每个预测的提升状态重新投影到原始状态坐标并再次提升来避免这一困难，从而在原始状态空间中产生一个误差感知的离散时间控制仿射预测器，无需假设Koopman字典的不变性。针对该预测器，我们基于离散时间鲁棒控制收缩度量构建了一个同胚管，其半径明确考虑了比例逼近误差界。管的收紧处理任意连续可微的非线性约束，所得的模型预测控制问题包含终端项和管半径惩罚，利用目标附近消失的不确定性。我们证明了原始非线性约束的鲁棒满足性、递归可行性以及采样真实闭环系统在训练数据上的高概率指数稳定性，且不要求模型预测控制问题的全局最优解。数值示例，包括非线性避障约束，展示了所提方法相比现有基于Koopman的模型预测控制方法在更小闭环代价和灵活性方面的鲁棒稳定性和更高性能。
英文摘要
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.
