Nonlinear dynamical systems with input delays pose significant challenges for prediction, estimation, and control due to their inherent complexity and the impact of delays on system behavior. Traditional linear control techniques often fail in these contexts, necessitating innovative approaches. This paper introduces a novel approach to approximate the Koopman operator using an LSTM-enhanced Deep Koopman model, enabling linear representations of nonlinear systems with time delays. By incorporating Long Short-Term Memory (LSTM) layers, the proposed framework captures historical dependencies and efficiently encodes time-delayed system dynamics into a latent space. Unlike traditional extended Dynamic Mode Decomposition (eDMD) approaches that rely on predefined dictionaries, the LSTM-enhanced Deep Koopman model is dictionary-free, which mitigates the problems with the underlying dynamics being known and incorporated into the dictionary. Quantitative comparisons with extended eDMD on a simulated system demonstrate highly significant performance gains in prediction accuracy in cases where the true nonlinear dynamics are unknown and achieve comparable results to eDMD with known dynamics of a system.

翻译：带输入延迟的非线性动力系统因其固有的复杂性以及延迟对系统行为的影响，给预测、估计和控制带来了重大挑战。传统的线性控制技术在此类情境下往往失效，因此需要创新的方法。本文提出了一种新颖的方法，利用LSTM增强的深度Koopman模型来逼近Koopman算子，从而实现对带有时滞的非线性系统的线性表示。通过引入长短期记忆（LSTM）层，所提出的框架能够捕捉历史依赖性，并将时滞系统动力学高效编码到潜在空间中。与依赖预定义字典的传统扩展动态模式分解（eDMD）方法不同，LSTM增强的深度Koopman模型无需字典，这缓解了因需要已知并纳入字典的底层动力学所引发的问题。在模拟系统上与扩展eDMD的定量比较表明，在真实非线性动力学未知的情况下，该方法在预测精度上取得了极为显著的性能提升，并在系统动力学已知时达到了与eDMD相当的结果。