[Paper Review] Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians

Zhiyuan Liu, Soumya Kundu+2 more|arXiv (Cornell University)|Oct 4, 2017

Model Reduction and Neural Networks23 references3 citations

TL;DR

This paper proposes a data-driven decomposition method for nonlinear dynamical systems using Koopman Gramians, leveraging input-Koopman operators and deep dynamic mode decomposition to learn input-state separable Koopman representations even when mixed terms exist. The key contribution is a nonlinear decomposition algorithm that maximizes subsystem observability and disturbance rejection, validated on an IEEE 39-bus power system with sub-0.4% multi-step prediction error and balanced controllability/observability clustering.

ABSTRACT

In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians. We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which input and state observable functions are well separated. We then extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition to systems with controls or disturbances. We illustrate the accuracy of the method in learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. We next introduce a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise from other subsystems. We derive a relaxation based on Koopman Gramians and multi-way partitioning for the resulting NP-hard decomposition problem. We lastly illustrate the proposed algorithm with the swing dynamics for an IEEE 39-bus system.

Motivation & Objective

• To develop a principled, data-driven method for decomposing complex nonlinear dynamical systems into subsystems with improved distributed control performance.
• To address the lack of systematic decomposition criteria in large-scale systems where engineering intuition is insufficient, especially in ad-hoc or cyber-physical networks.
• To maximize internal subsystem observability and resilience to external disturbances from other subsystems through a Koopman Gramian-based optimization framework.
• To extend deep dynamic mode decomposition to systems with controls, enabling learning of input-state separable Koopman operators from data.
• To provide a scalable, relaxation-based approach to the NP-hard decomposition problem using multi-way partitioning and Koopman Gramians.

Proposed method

• Introduces the input-Koopman operator to model the evolution of observables in nonlinear systems with inputs, enabling a lifted linear representation.
• Uses deep dynamic mode decomposition with a 20-wide, 10-layer ResNet and ELU/Dropout in TensorFlow to learn Koopman operators from 100 random trajectories of the IEEE 39-bus system.
• Derives Koopman Gramians from the learned input-Koopman operator to quantify subsystem observability ($\kappa_o$) and controllability ($\kappa_c$).
• Develops a nonlinear decomposition algorithm that maximizes internal observability and disturbance rejection by balancing $\kappa_o$ and $\kappa_c$ across clusters.
• Applies a relaxation technique based on Koopman Gramians and multi-way partitioning to solve the NP-hard decomposition problem.
• Normalizes $\kappa_o$ and $\kappa_c$ to balance their disparate scales and enable equitable clustering.

Experimental results

Research questions

• RQ1 Under what conditions can a nonlinear system with mixed state-input terms be approximated by an input-state separable Koopman operator?
• RQ2 Can deep dynamic mode decomposition accurately learn Koopman operators for systems with controls, even when input and state terms are not separable?
• RQ3 How can Koopman Gramians be used to define a principled decomposition criterion that maximizes subsystem observability and disturbance rejection?
• RQ4 Can a relaxation-based multi-way partitioning algorithm effectively solve the NP-hard decomposition problem using Koopman-based metrics?
• RQ5 How does the proposed decomposition compare to spatial or connectivity-based clustering in terms of resilience and observability in power system dynamics?

Key findings

• The deep dynamic mode decomposition method achieved a one-step prediction error of less than 0.01% and a multi-step prediction error of less than 0.4% per time-step on unseen initial conditions in the IEEE 39-bus system.
• The algorithm successfully identified zonal clusterings of generators that were mostly spatially coherent, with generator 6 isolated due to low controllability and observability scores.
• The decomposition achieved a maximum variation in $\kappa_c$ of 0.33434 and in $\kappa_o$ of 0.09744 across clusters, indicating balanced subsystem performance.
• Generator 2 showed high observability ($\kappa_o$) but low controllability ($\kappa_c$), indicating vulnerability to external disturbances, which the algorithm addressed through strategic clustering.
• The proposed method outperformed spatial-based area assignments by focusing on input-output resilience and internal observability, not just physical proximity.
• The Koopman Gramian-based decomposition provided a scalable, data-driven alternative to heuristic or topology-based subsystem partitioning in large-scale systems.