Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks (2402.12360v1)

Abstract: We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series solution.

• The paper introduces a novel PINN-based observer framework that linearizes error dynamics for enhanced state estimation in nonlinear systems.
• It leverages a greedy-wise training strategy to boost approximation accuracy in regions with steep nonlinearity and singular behaviors.
• The approach outperforms conventional power-series methods, offering robust observer design without relying on traditional linearization.

Enhancing Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

Introduction

Recent advancements in the field of control systems have brought machine learning, particularly neural networks, to the forefront of design and optimization of state estimators or observers for nonlinear systems. Traditionally, the challenge of nonlinear discrete-time observer design has been approached through various linearization techniques, which, despite their theoretical rigor, come with practical limitations and challenges, especially in systems with considerable nonlinear dynamics.

In this paper, we delve into the application of Physics-Informed Neural Networks (PINNs) to address the nonlinear discrete-time observer design problem. By incorporating physical laws directly into the training process of neural networks, PINNs aim to discover nonlinear state transformation maps, enabling the design of discrete-time observers with linearizable error dynamics and assignable convergence rates under a set of relatively mild assumptions. This approach seeks to merge the theoretical richness of conventional observer design methods with the versatile, data-driven capabilities of machine learning.

Methodological Framework

The paper outlines an exact observer linearization framework for discrete-time nonlinear systems, leveraged by PINNs to learn the state transformation maps essential for observer design. Central to this framework is the reformulation of the observer design problem to eliminate the need for linearizing the system output map, thereby broadening the scope of applications to systems with more complex dynamics.

A set of illustrative benchmark problems is introduced to validate the effectiveness of the PINN scheme. These case studies highlight not only the potential of PINNs in approximating nonlinear maps amid steep gradients and singular points but also the advantages of adopting a greedy-wise training strategy. This strategy incrementally trains the network within nested domains, progressively approaching areas with high nonlinearity to improve approximation accuracy and convergence behavior.

Comparative Analysis and Findings

The performances of the proposed PINN method are meticulously benchmarked against conventional power-series numerical implementations. The analysis demonstrates the PINN method's superior numerical approximation accuracy, particularly when employing the greedy-wise training approach. It effectively addresses the challenges posed by singularities in the nonlinear transformation map, achieving remarkable precision in regions where traditional methods exhibit significant limitations.

Practical Implications and Theoretical Contributions

This investigation not only expands the toolkit available for nonlinear observer design but also underscores the potential of integrating machine learning with classic control theories. It opens avenues for the design of state estimators in complex systems where direct application of traditional methods is constrained by computational or model-based limitations.

Future Directions in AI and Control Systems

Looking forward, the fusion of physics-informed machine learning models with control systems design promises to revolutionize how we approach problems of nonlinear dynamics and observer design. It beckons a future where data-driven methodologies and traditional control theories converge to offer robust, adaptable, and efficient solutions for complex systems ranging from advanced manufacturing processes to autonomous vehicle navigation and beyond.

In essence, this paper lays foundational work that marries the precision of physics-based models with the adaptability of neural networks, setting the stage for new breakthroughs in control systems design and optimization.