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Koopman Generator Decomposition for Port-Hamiltonian System

Victor M. Preciado

Abstract

We establish a canonical decomposition of the infinitesimal Koopman generator of any port-Hamiltonian (pH) system into skew-adjoint (energy-conserving), positive-semidefinite (dissipative), and input-port components, proving that the generator satisfies an energy-dissipation inequality on a dense subdomain of $L^2(μ)$ for any invariant measure $μ$ satisfying a mild joint-invariance condition stated in Theorem 1. This infinite-dimensional splitting carries over exactly to finite-dimensional Galerkin approximations, yielding structure-constrained surrogate models that provably inherit passivity with a quadratic storage function in the lifted observable space. Leveraging this structure, we design passivity-based controllers directly in the lifted space and establish asymptotic stability of the lifted closed-loop system via LaSalle's invariance principle under a mild detectability condition. For linear pH systems, the decomposition recovers the true pH matrices exactly, confirming that the structural constraints arise naturally from the operator theory rather than being imposed by hand. The framework unifies port-Hamiltonian systems theory and Koopman spectral methods, providing a rigorous operator-theoretic foundation for energy-consistent lifting of nonlinear pH dynamics.

Koopman Generator Decomposition for Port-Hamiltonian System

Abstract

We establish a canonical decomposition of the infinitesimal Koopman generator of any port-Hamiltonian (pH) system into skew-adjoint (energy-conserving), positive-semidefinite (dissipative), and input-port components, proving that the generator satisfies an energy-dissipation inequality on a dense subdomain of for any invariant measure satisfying a mild joint-invariance condition stated in Theorem 1. This infinite-dimensional splitting carries over exactly to finite-dimensional Galerkin approximations, yielding structure-constrained surrogate models that provably inherit passivity with a quadratic storage function in the lifted observable space. Leveraging this structure, we design passivity-based controllers directly in the lifted space and establish asymptotic stability of the lifted closed-loop system via LaSalle's invariance principle under a mild detectability condition. For linear pH systems, the decomposition recovers the true pH matrices exactly, confirming that the structural constraints arise naturally from the operator theory rather than being imposed by hand. The framework unifies port-Hamiltonian systems theory and Koopman spectral methods, providing a rigorous operator-theoretic foundation for energy-consistent lifting of nonlinear pH dynamics.
Paper Structure (15 sections, 7 theorems, 33 equations)

This paper contains 15 sections, 7 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Port-Hamiltonian Systems
  4. Structure matrices and dissipation.
  5. Koopman Operator and Generator
  6. Koopman Generator Decomposition
  7. Finite-Dimensional Structure-Preserving Approximations
  8. Galerkin Projection and Structure Inheritance
  9. Passivity of the Lifted Model
  10. Passivity-Based Control in the Lifted Space
  11. Koopman-MPC with Passivity Constraints
  12. Illustrative Analytical Verification
  13. Linear pH System: Exact Recovery
  14. Nonlinear Pendulum: Explicit Generator Computation
  15. Conclusion

Key Result

Theorem 1

Let the autonomous drift $\mathbf{f}_0$ of the control-affine system eq:control_affine be port-Hamiltonian as in eq:ph_nonlinear with $\mathbf{u}=\mathbf{0}$, so that $\mathbf{v}(\mathbf{x}) = (\mathbf{J}(\mathbf{x})-\mathbf{R}(\mathbf{x})) \nabla\mathcal{H}(\mathbf{x})$. Then for any measure $\mu$ where $(\mathcal{K}_J f)(\mathbf{x})\triangleq \nabla f^\intercal\mathbf{J}(\mathbf{x})\nabla\mathc

Theorems & Definitions (21)

  • Definition 1: Dissipative operator and contraction semigroup
  • Theorem 1: Koopman Generator Decomposition
  • proof
  • Remark 1
  • Remark 2: Joint invariance condition on $\mu$
  • Proposition 1: Finite-dimensional splitting
  • proof
  • Corollary 1: Passivity of the Lifted KpH Model
  • proof
  • Remark 3: Lifting gap and injectivity condition
  • ...and 11 more