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Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis

Yasmine Marani, Israel Filho, Tareq Al-Naffouri, Taous-Meriem Laleg-Kirati

TL;DR

A novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer’s correction term by enforcing the partial differential inequality in the loss function is proposed.

Abstract

Contraction analysis offers, through elegant mathematical developments, a unified way of designing observers for a general class of nonlinear systems, where the observer correction term is obtained by solving an infinite dimensional inequality that guarantees global exponential convergence. However, solving the matrix partial differential inequality involved in contraction analysis design is both analytically and numerically challenging and represents a long-lasting challenge that prevented its wide use. Therefore, the present paper proposes a novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer's correction term by enforcing the partial differential inequality in the loss function. The performance of the proposed PINN-based nonlinear observer is assessed in numerical simulation as well as its robustness to measurement noise and neural network approximation error.

Unsupervised Physics-Informed Neural Network-based Nonlinear Observer design for autonomous systems using contraction analysis

TL;DR

A novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer’s correction term by enforcing the partial differential inequality in the loss function is proposed.

Abstract

Contraction analysis offers, through elegant mathematical developments, a unified way of designing observers for a general class of nonlinear systems, where the observer correction term is obtained by solving an infinite dimensional inequality that guarantees global exponential convergence. However, solving the matrix partial differential inequality involved in contraction analysis design is both analytically and numerically challenging and represents a long-lasting challenge that prevented its wide use. Therefore, the present paper proposes a novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer's correction term by enforcing the partial differential inequality in the loss function. The performance of the proposed PINN-based nonlinear observer is assessed in numerical simulation as well as its robustness to measurement noise and neural network approximation error.

Paper Structure

This paper contains 13 sections, 3 theorems, 19 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries on Contraction Theory
  3. Contraction analysis in the context of nonlinear observer design
  4. Problem Formulation
  5. Unsupervised learning of the observer's correction term
  6. Collocation point generation
  7. Construction of the loss function
  8. Contraction loss
  9. Boundary condition loss
  10. Training Pseudo-Code
  11. Robustness to neural network approximation error and measurement noise
  12. Numerical Simulations
  13. Conclusion

Key Result

Theorem 1

BERNARD2022 Consider the smooth nonlinear system model. If there exists a positive definite matrix $P\in \mathbb{R}^{n\times n}$, a $\mathcal{C}^1$ function $k: \mathbb{R}^n \times \mathbb{R}^p \rightarrow \mathbb{R}^n$ and a real positive number $\lambda$, such that Then the observer in observer is a globally exponentially stable observer for system model.

Figures (3)

  • Figure 1: Block diagram of the proposed unsupervised learning-based contraction nonlinear observer design.
  • Figure 2: State estimation of the Van der Pol Oscillator under 15$\%$ of measurement noise using the proposed PINN-based contraction observer.
  • Figure 3: State estimation of the Inverse Duffing Oscillator under 15$\%$ of measurement noise using the proposed PINN-based contraction observer

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • proof