KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

2025-01-20

AI Summary

To address insufficient robustness in state estimation for autonomous nonlinear systems—arising from model uncertainty and measurement noise—this paper proposes a physics-informed, learnable KKL observer. Methodologically, it reformulates KKL observer design as a differentiable neural approximation problem, employing invertible neural networks to embed system states into a linearly stable, high-dimensional latent space and reconstruct them faithfully. A stability-driven loss function is introduced, jointly incorporating physical constraints and numerical simulation data to provide theoretical bounds on estimation error. Furthermore, a systematic parameter-optimization framework is developed to enhance observer performance. Extensive evaluation across multiple benchmark nonlinear systems demonstrates superior accuracy and robustness in state estimation. The approach is successfully applied to sensor fault detection and isolation in Kuramoto oscillator networks, validating its practical efficacy.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Learning to Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

This paper proposes a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose to sequentially approximate these maps by neural networks that are trained using physics-informed learning. We generate synthetic data for training by numerically solving the system and observer dynamics. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided. Additionally, a systematic method for optimizing observer performance through parameter selection is presented. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples and its application to sensor fault detection and isolation in a network of Kuramoto oscillators using learned KKL observers.

Problem

• KKL Observer
• Complex Nonlinear Systems
• State Estimation

Innovation

• Physics-Informed Neural Networks
• KKL Observer Design
• Fault Detection and Isolation