Deep Dictionary-Free Method for Identifying Linear Model of Nonlinear System with Input Delay

2025-06-03

International Conference on Process Control

AI Summary

Modeling nonlinear systems with input delays remains challenging due to the failure of conventional linear methods and the difficulty of constructing appropriate basis function dictionaries. To address this, we propose an LSTM-enhanced, dictionary-free deep Koopman framework. Our method eliminates reliance on predefined basis dictionaries by leveraging LSTM networks to automatically learn latent temporal dependencies between past inputs and states, thereby enabling linear approximation of nonlinear dynamics in a learned embedding space. Crucially, input delays are implicitly encoded within the Koopman operator learning process, obviating explicit delay feature engineering. Experimental results demonstrate that our approach achieves significantly higher prediction accuracy than extended dynamic mode decomposition (eDMD) on unknown nonlinear systems, while matching eDMD’s performance on systems with known dynamics—indicating strong generalization capability and robustness.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 com-parisons 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.

Problem

• Modeling nonlinear systems with input delays for prediction and control
• Overcoming limitations of traditional linear control techniques
• Eliminating dependency on predefined dictionaries in system identification

Innovation

• LSTM-enhanced Deep Koopman model for nonlinear systems
• Dictionary-free approach using LSTM layers
• Encodes time-delayed dynamics into latent space