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

TL;DR

Addresses the challenge of predicting nonlinear systems with input delays. Proposes a dictionary-free, LSTM-enhanced Deep Koopman framework that encodes history into a latent state $z$ and learns a linear update $z[k+1] = A_K z[k] + B_K u[k]$. Demonstrates superior prediction accuracy over extended Dynamic Mode Decomposition (eDMD) when true nonlinear terms are unknown and matches eDMD with known dynamics, while using a smaller lifted-state dimension. Shows empirical gains on a delayed two-tank system and yields more accurate Koopman eigenvalues. The approach enables robust data-driven identification and control design for delayed nonlinear systems without relying on predefined dictionaries.

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 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.

Figures (3)

• Figure 1: The LSTM-enhanced Deep Koopman algorithm architecture. The model is built on the autoencoder architecture of the Deep Koopman Operator. The history of state and input data is first extracted using the LSTM layer. The hidden states of the last LSTM layer are then concatenated with the last state as input to the autoencoder part. After lifting the states, there is a prediction layer represented with the matrices $A_\mathcal{K}$ and $B_\mathcal{K}$. After the prediction layer, the lifted states are projected back to states using the decoder part of the autoencoder.
• Figure 2: Predicted and actual water levels in tanks $1$ and $2$ for the two tank system. Blue line shows the simulation of \ref{['eq:two_tanks_system']}, while gray shows the noisy data, as used for training. The sequence shows a snippet of testing set which was unseen during system identification.
• Figure 3: Comparison of eigenvalues of the Koopman operator for the two tank system. Subplot "Original" shows the eigenvalues of the original system, lienarized around the tank levels of 1 m.