A Generalized Metriplectic System via Free Energy and System~Identification via Bilevel Convex Optimization

TL;DR

This paper extends the classical metriplectic formalism to include nonconservative dissipation by introducing free energy, and proposes a bilevel convex optimization method for system identification based on measurements.

Contribution

It introduces a generalized metriplectic framework that relaxes Casimir invariance and incorporates free energy, along with a bilevel convex optimization approach for system identification.

Findings

Demonstrates the generalized formalism on 2D Hamiltonian systems.

Applies the approach to SO(3) systems.

Provides a systematic method for system identification from measurements.

Abstract

This work generalizes the classical metriplectic formalism to model Hamiltonian systems with nonconservative dissipation. Classical metriplectic representations allow for the description of energy conservation and production of entropy via a suitable selection of an entropy function and a bilinear symmetric metric. By relaxing the Casimir invariance requirement of the entropy function, this paper shows that the generalized formalism induces the free energy analogous to thermodynamics. The monotonic change of free energy can serve as a more precise criterion than mechanical energy or entropy alone. This paper provides examples of the generalized metriplectic system in a 2-dimensional Hamiltonian system and $\mathrm{SO}(3)$. This paper also provides a bilevel convex optimization approach for the identification of the metriplectic system given measurements of the system.