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Passivity encoding representations of nonlinear systems

Attila Karsai, Tobias Breiten, Justus Ramme, Philipp Schulze

TL;DR

The paper addresses how passive nonlinear systems can be encoded as algebraic port-Hamiltonian structures by linking passivity to representations (A) and (B) and identifying conditions under which one can construct $J,R$ (and $j,r$) from the system dynamics and Hamiltonian. The main approach hinges on the gradient $\eta = \nabla \mathcal{H}$, with injectivity of $\eta$ ensuring that any passive system can be recast in the (B) form, and subsequently connected to the (A) form via constructive procedures (P-A and P-A-fix). It provides sufficient conditions and explicit constructions (e.g., $M(z) = \int_0^1 Df(sz) D\eta(sz)^{-1} ds$) to realize port-Hamiltonian realizations, including special treatment of conservative systems and a general framework for the nonconservative case. The results are demonstrated on finite-dimensional examples (and a formal infinite-dimensional example), illustrating energy-preserving interconnections and the potential for a systematic pathway to pH models in nonlinear passive settings.

Abstract

Passive systems are characterized by their inability to generate energy internally, providing a powerful tool for modeling physical phenomena. Additionally, algebraically encoding passivity in the system description can be advantageous. For this, port-Hamiltonian systems are a prominent approach. Another possibility is writing the system in suitable coordinates. In this paper, we investigate the equivalence between passivity and the feasibility of passivity encoding representations, thereby elaborating upon existing results for port-Hamiltonian systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known.

Passivity encoding representations of nonlinear systems

TL;DR

The paper addresses how passive nonlinear systems can be encoded as algebraic port-Hamiltonian structures by linking passivity to representations (A) and (B) and identifying conditions under which one can construct (and ) from the system dynamics and Hamiltonian. The main approach hinges on the gradient , with injectivity of ensuring that any passive system can be recast in the (B) form, and subsequently connected to the (A) form via constructive procedures (P-A and P-A-fix). It provides sufficient conditions and explicit constructions (e.g., ) to realize port-Hamiltonian realizations, including special treatment of conservative systems and a general framework for the nonconservative case. The results are demonstrated on finite-dimensional examples (and a formal infinite-dimensional example), illustrating energy-preserving interconnections and the potential for a systematic pathway to pH models in nonlinear passive settings.

Abstract

Passive systems are characterized by their inability to generate energy internally, providing a powerful tool for modeling physical phenomena. Additionally, algebraically encoding passivity in the system description can be advantageous. For this, port-Hamiltonian systems are a prominent approach. Another possibility is writing the system in suitable coordinates. In this paper, we investigate the equivalence between passivity and the feasibility of passivity encoding representations, thereby elaborating upon existing results for port-Hamiltonian systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known.
Paper Structure (10 sections, 9 theorems, 50 equations)

This paper contains 10 sections, 9 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Passivity encoding representations
  4. Towards (B)
  5. Towards (A)
  6. Constructing port-Ha-mil-to-ni-an representations
  7. Conservative systems
  8. The general case
  9. Examples
  10. Conclusion

Key Result

Proposition 4

Systems of the form eq:ph-a or eq:ph-b are passive, and trajectories $z$ of eq:ph-a or eq:ph-b satisfy the energy balance for all $t_1 \geq t_0$, where $d(z) = \eta(z)^{\mathsf{T}} R(z) \eta(z) \geq 0$ or $d(z) = \eta(z)^{\mathsf{T}} r(\eta(z)) \geq 0$, respectively. In particular, any of the conditions (A) and (B) implies (P).

Theorems & Definitions (30)

  • Remark 1: uniqueness
  • Remark 2
  • Definition 3
  • Proposition 4: vanderschaft14-portgiesselmann24-energy
  • Lemma 5
  • Proposition 6: (P) $\Rightarrow$ (B)
  • proof
  • Remark 7
  • Proposition 8: (A) $\Rightarrow$ (B)
  • proof
  • ...and 20 more