Lyapunov-Based Physics-Informed Deep Neural Networks with Skew Symmetry Considerations

TL;DR

This paper develops a Lyapunov-based physics-informed deep neural network controller for Euler-Lagrange systems, leveraging skew symmetry properties to improve stability and function approximation in dynamic control tasks.

Contribution

It introduces the first physics-informed DNN controller using Lyapunov stability analysis that explicitly incorporates skew symmetry in Euler-Lagrange systems.

Findings

Improved function approximation over non-symmetry-aware methods

Guaranteed asymptotic convergence of tracking and prediction errors

Enhanced stability and robustness in control simulations

Abstract

Deep neural networks (DNNs) are powerful black-box function approximators which have been shown to yield improved performance compared to traditional neural network (NN) architectures. However, black-box algorithms do not incorporate known physics of the system and can yield results which are physically implausible. Physics-informed neural networks (PINNs) have grown in popularity due to their ability to leverage known physical principles in the learning process which has been empirically shown to improve performance compared to traditional black-box methods. This paper introduces the first physics-informed DNN controller for an Euler-Lagrange dynamic system where the adaptation laws are designed using a Lyapunov-based stability analysis to account for the skew-symmetry property of the inertia matrix and centripetal-Coriolis matrix. A Lyapunov-based stability analysis is provided to guarantee asymptotic convergence of the tracking error and the skew-symmetric prediction error. Simulations indicate that the developed update law demonstrates improvement in individual and overall function approximation capabilities when compared to a physics-informed adaptation law which does not incorporate knowledge of system symmetries.