The accurate modeling and control of nonlinear dynamical effects are crucial for numerous robotic systems. The Koopman formalism emerges as a valuable tool for linear control design in nonlinear systems within unknown environments. However, it still remains a challenging task to learn the Koopman operator with control from data, and in particular, the simultaneous identification of the Koopman linear dynamics and the mapping between the state and Koopman spaces. Conventional approaches, based on single-level unconstrained optimization, may lack model robustness, training efficiency, and long-term predictive accuracy. This paper presents a bi-level optimization framework that jointly learns the Koopman embedding mapping and Koopman dynamics with explicit multi-step dynamical constraints, eliminating the need for heuristically-tuned loss terms. Leveraging implicit differentiation, our formulation allows back-propagation in standard learning framework and the use of state-of-the-art optimizers, yielding more stable and robust system performance over various applications compared to conventional methods.

翻译：非线性动力学效应的精确建模与控制对于众多机器人系统至关重要。Koopman形式化方法为未知环境中非线性系统的线性控制设计提供了有效工具。然而，从数据中学习含有控制输入的Koopman算子仍是一项具有挑战性的任务，特别是需要同时识别Koopman线性动力学以及状态空间与Koopman空间之间的映射关系。基于单层无约束优化的传统方法可能缺乏模型鲁棒性、训练效率及长期预测精度。本文提出一种双层优化框架，通过显式的多步动力学约束联合学习Koopman嵌入映射与Koopman动力学，无需依赖启发式调参的损失项。借助隐式微分技术，该方法支持标准学习框架中的反向传播及先进优化器的使用，从而在多个应用场景中相比传统方法获得更稳定鲁棒的系统性能。