AI 中文总结
本文针对未知非线性系统，基于数据驱动的双线性库普曼实现提出鲁棒MPC框架。解决了有限维库普曼预测器的问题后，开发相关构造并制定问题。证明了闭环轨迹对约束的鲁棒满足等性质，数值实验显示该方法在性能上优于现有方法。
AI 中文摘要
基于库普曼算子理论的数据驱动模型预测控制是控制未知非线性系统的一种很有前景的方法。虽然线性库普曼实现因其简单性而常用，但双线性库普曼实现能为非线性控制系统提供更高的逼近精度。然而，考虑双线性库普曼实现中建模误差的鲁棒MPC公式仍然有限。本文基于数据驱动的双线性库普曼实现，为具有一般非线性约束的未知非线性系统提出了一个鲁棒MPC框架。一个核心困难是有限维库普曼预测器不一定能保持有效提升状态的流形，所以在提升坐标中的多步预测可能会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流形上来解决这个问题，从而在原始状态空间中获得一个误差感知的离散时间控制仿射预测器，而无需不切实际的假设。对于这个预测器，我们开发了一种基于离散时间鲁棒控制收缩度量的相似管构造，然后用终端成分制定了一个基于管的鲁棒MPC问题。在所提出的公式下，我们证明了真实闭环轨迹对原始非线性约束的鲁棒满足、递归可行性以及收敛到目标状态的邻域。数值实验证明了非线性系统的鲁棒镇定以及所提方法在性能上优于现有的基于库普曼的鲁棒MPC方法。
英文摘要
Data-driven model predictive control (MPC) using Koopman operator theory is a promising approach for constrained control of unknown nonlinear systems. While linear Koopman realizations are commonly used due to their simplicity, bilinear Koopman realizations can provide significantly higher approximation accuracy for nonlinear control systems. However, robust MPC (RMPC) formulations that account for modeling errors in bilinear Koopman realizations remain limited. This paper proposes a RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations. A central difficulty is that finite-dimensional Koopman predictors need not preserve the manifold of valid lifted states, so multi-step prediction in lifted coordinates may leave the region where one-step error certificates apply. We address this issue by reprojecting each predicted lifted state back onto the manifold, thereby obtaining an error-aware discrete-time control-affine predictor in the original state space without impractical assumptions. For this predictor, we develop a discrete-time robust control contraction metric based homothetic tube construction, and then formulate a tube-based RMPC problem with terminal ingredients. Under the proposed formulation, we prove robust satisfaction of the original nonlinear constraints by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state. Numerical experiments demonstrate robust stabilization of nonlinear systems and the advantages of the proposed method over existing Koopman-based RMPC approaches in terms of performance.
