Kalman Filter Aided Federated Koopman Learning

TL;DR

This paper introduces KF-FedKL, a novel framework combining Kalman filtering, federated learning, and Koopman analysis to enable privacy-preserving, collaborative linearization of nonlinear systems using limited observational data.

Contribution

It pioneers the integration of Kalman filtering and federated learning with Koopman analysis for nonlinear system linearization under data and privacy constraints.

Findings

Effective collaborative linearization demonstrated through numerical simulations.

KF-FedKL achieves privacy-preserving system analysis with competitive accuracy.

Convergence of the proposed federated Koopman learning framework is theoretically validated.

Abstract

Real-time control and estimation are pivotal for applications such as industrial automation and future healthcare. The realization of this vision relies heavily on efficient interactions with nonlinear systems. Therefore, Koopman learning, which leverages the power of deep learning to linearize nonlinear systems, has been one of the most successful examples of mitigating the complexity inherent in nonlinearity. However, the existing literature assumes access to accurate system states and abundant high-quality data for Koopman analysis, which is usually impractical in real-world scenarios. To fill this void, this paper considers the case where only observations of the system are available and where the observation data is insufficient to accomplish an independent Koopman analysis. To this end, we propose Kalman Filter aided Federated Koopman Learning (KF-FedKL), which pioneers the combination of Kalman filtering and federated learning with Koopman analysis. By doing so, we can achieve collaborative linearization with privacy guarantees. Specifically, we employ a straightforward yet efficient loss function to drive the training of a deep Koopman network for linearization. To obtain system information devoid of individual information from observation data, we leverage the unscented Kalman filter and the unscented Rauch-Tung-Striebel smoother. To achieve collaboration between clients, we adopt the federated learning framework and develop a modified FedAvg algorithm to orchestrate the collaboration. A convergence analysis of the proposed framework is also presented. Finally, through extensive numerical simulations, we showcase the performance of KF-FedKL under various situations.