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

Sangli Teng, Kaito Iwasaki, William Clark, Xihang Yu, Anthony Bloch, Ram Vasudevan, Maani Ghaffari

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.

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

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 . This paper also provides a bilevel convex optimization approach for the identification of the metriplectic system given measurements of the system.
Paper Structure (15 sections, 1 theorem, 31 equations, 4 figures, 2 algorithms)

This paper contains 15 sections, 1 theorem, 31 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Poisson structure
  5. Positive Semi-Definite Inner Product
  6. Metriplectic System
  7. Generalized Metriplectic System
  8. System identification
  9. Problem formulation
  10. Systems identification via convex optimization
  11. Numerical Analysis
  12. Linear system
  13. $\mathrm{SO}(3)$ system
  14. Discussion and future work
  15. Conclusion

Key Result

Theorem 1

Assume that Problem prob:entropy-search and prob:metric-search returns a feasible solution with value no greater than previous iteration, the cost of Problem prob:bilevel-cvx is guaranteed to converge.

Figures (4)

  • Figure 1: Trajectory of the 2D metriplectic system in the phase space. The trajectories converge to the zero-level set of the free energy ($\infty$ shape) via a selected entropy function and metric. The red dots denote the randomly sampled initial conditions.
  • Figure 2: The convergence of the loss of Algorithm \ref{['alg:2']} for the 2D metriplectic system. The loss after solving each sub-problems are shown. Superlinear convergence rate is observed. As the loss converges to the trivial lower bound $0$, we conclude that the algorithm find the global optimizer in this case.
  • Figure 3: Trajectory of the metriplectic system on $\mathrm{SO}(3)$. The trajectories converges to the zero-level set of the free energy as a sphere.
  • Figure 4: The convergence of the loss of Algorithm \ref{['alg:2']} for the metriplectic system on $\mathrm{SO}(3)$. The loss after solving each sub-problems are shown. The chattering at the last few iterations are due to the numerical issue of SDP solvers.

Theorems & Definitions (2)

  • Theorem 1: Convergence of Algorithm I
  • proof