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PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems

Ayoub Farkane, Mohamed Boutayeb, Mustapha Oudani, Mounir Ghogho

TL;DR

The paper tackles nonlinear state estimation from partial measurements by designing PINN-Obs, an adaptive physics-informed neural network observer that jointly learns the state estimate $\\hat{x}(t)$ and the observer gain $L(t)$. By embedding the observer equation into a physics-constrained loss $\\mathrm{MSE} = \\mathrm{MSE}_0 + \\mathrm{MSE}_g + \\mathrm{MSE}_y$, the framework achieves convergence guarantees under mild observability conditions without requiring system transformations. Theoretical results show uniform convergence of the estimated state as training data grows, and extensive simulations on an induction motor model, satellite dynamics, and academic examples demonstrate superior accuracy and robustness compared to existing observer designs. The approach bridges model-based observer design and data-driven learning, offering a flexible tool for reliable state estimation in complex nonlinear systems with partial observations.

Abstract

State estimation for nonlinear dynamical systems is a critical challenge in control and engineering applications, particularly when only partial and noisy measurements are available. This paper introduces a novel Adaptive Physics-Informed Neural Network-based Observer (PINN-Obs) for accurate state estimation in nonlinear systems. Unlike traditional model-based observers, which require explicit system transformations or linearization, the proposed framework directly integrates system dynamics and sensor data into a physics-informed learning process. The observer adaptively learns an optimal gain matrix, ensuring convergence of the estimated states to the true system states. A rigorous theoretical analysis establishes formal convergence guarantees, demonstrating that the proposed approach achieves uniform error minimization under mild observability conditions. The effectiveness of PINN-Obs is validated through extensive numerical simulations on diverse nonlinear systems, including an induction motor model, a satellite motion system, and benchmark academic examples. Comparative experimental studies against existing observer designs highlight its superior accuracy, robustness, and adaptability.

PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems

TL;DR

The paper tackles nonlinear state estimation from partial measurements by designing PINN-Obs, an adaptive physics-informed neural network observer that jointly learns the state estimate and the observer gain . By embedding the observer equation into a physics-constrained loss , the framework achieves convergence guarantees under mild observability conditions without requiring system transformations. Theoretical results show uniform convergence of the estimated state as training data grows, and extensive simulations on an induction motor model, satellite dynamics, and academic examples demonstrate superior accuracy and robustness compared to existing observer designs. The approach bridges model-based observer design and data-driven learning, offering a flexible tool for reliable state estimation in complex nonlinear systems with partial observations.

Abstract

State estimation for nonlinear dynamical systems is a critical challenge in control and engineering applications, particularly when only partial and noisy measurements are available. This paper introduces a novel Adaptive Physics-Informed Neural Network-based Observer (PINN-Obs) for accurate state estimation in nonlinear systems. Unlike traditional model-based observers, which require explicit system transformations or linearization, the proposed framework directly integrates system dynamics and sensor data into a physics-informed learning process. The observer adaptively learns an optimal gain matrix, ensuring convergence of the estimated states to the true system states. A rigorous theoretical analysis establishes formal convergence guarantees, demonstrating that the proposed approach achieves uniform error minimization under mild observability conditions. The effectiveness of PINN-Obs is validated through extensive numerical simulations on diverse nonlinear systems, including an induction motor model, a satellite motion system, and benchmark academic examples. Comparative experimental studies against existing observer designs highlight its superior accuracy, robustness, and adaptability.

Paper Structure

This paper contains 27 sections, 3 theorems, 49 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Governing System
  4. Dynamic System
  5. Nonlinear State Observer
  6. Proposed Adaptive PINN-Based Model
  7. Neural Network Training and Testing
  8. Discussion
  9. Convergence Analysis
  10. Formulation of the Optimization Problem
  11. Neural Network Configuration
  12. Discussion
  13. Applications
  14. Ablation studies
  15. Neural Network Depth
  16. ...and 12 more sections

Key Result

Lemma 1

Let $x \in \mathscr{D}\subset \mathbb{R}^n$ and $y \in \mathbb{R}^m$. Suppose $h:\Omega \rightarrow \mathbb{R}^{d_{\text{out}}}$ is a neural network in $\mathcal{NN}_{\vec{\ell}}$ with $h \in C^{k, \beta}(\bar{\Omega})$ for some $\beta \in (0,1]$. Assume further that $y(\cdot, t)$ is continuously Li satisfies

Figures (7)

  • Figure 1: Adaptive PINNS for nonlinear state observer design.
  • Figure 2: (a) Comparison of predicted and actual state trajectories, and (b) Comparison of our method with other approaches using various error metrics.
  • Figure 3: (a) Comparison of predicted and actual state trajectories, and (b) corresponding estimation errors for the induction motor system.
  • Figure 4: (a) Comparison of estimated and actual states, and (b) corresponding estimation errors for the harmonic oscillator system.
  • Figure 5: (a) Predicted and actual state vectors and (b) corresponding estimation errors for academic example \ref{['ex3']}.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Definition 3
  • Lemma 1
  • Proof 1
  • Lemma 2: Theorem 3.1 in shin2020convergence
  • Theorem 1
  • Proof 2
  • ...and 5 more