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Reciprocity of nonlinear systems

Arjan van der Schaft

Abstract

One of the key contributions of the 1972 seminal paper by Willems was the analysis of symmetry (also called reciprocity) of input-state-output systems, both from an external (input-output) and internal (state) point of view. The developed theory also included the combination of reciprocity with passivity, and the consideration of relaxation systems, which are passive reciprocal systems without any oscillatory behavior. The paper was motivated from a fundamental system-theoretic point of view (how is external structure reflected into internal structure), as well as by a wide range of application areas, including electrical network synthesis, thermodynamics, and viscoelastic materials. On the other hand, the obtained results are for linear systems, and the extension to the nonlinear case, even for subclasses of nonlinear systems, is far from trivial. The present paper aims at taking some steps into this direction.

Reciprocity of nonlinear systems

Abstract

One of the key contributions of the 1972 seminal paper by Willems was the analysis of symmetry (also called reciprocity) of input-state-output systems, both from an external (input-output) and internal (state) point of view. The developed theory also included the combination of reciprocity with passivity, and the consideration of relaxation systems, which are passive reciprocal systems without any oscillatory behavior. The paper was motivated from a fundamental system-theoretic point of view (how is external structure reflected into internal structure), as well as by a wide range of application areas, including electrical network synthesis, thermodynamics, and viscoelastic materials. On the other hand, the obtained results are for linear systems, and the extension to the nonlinear case, even for subclasses of nonlinear systems, is far from trivial. The present paper aims at taking some steps into this direction.
Paper Structure (18 sections, 12 theorems, 88 equations)

This paper contains 18 sections, 12 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The linear case; recall and geometric interpretation
  3. Reciprocity in the linear case
  4. Passivity of linear systems
  5. Combining reciprocity and passivity for linear systems
  6. Port-Hamiltonian formulation of linear passive pseudo-gradient systems
  7. Reciprocity of nonlinear systems
  8. Pseudo-gradient systems
  9. External characterization of pseudo-gradient systems
  10. Combining nonlinear reciprocity with passivity
  11. Nonlinear passivity
  12. Reciprocal port-Hamiltonian systems
  13. Passive Hessian pseudo-gradient systems
  14. Nonlinear relaxation systems
  15. Examples
  16. ...and 3 more sections

Key Result

Proposition 2.2

A subspace $\mathcal{L} \subset \mathcal{V} \times \mathcal{V}^*$ is Lagrangian if and only if $\mathcal{L}=\mathcal{L}^{\perp \!\!\!\perp}$, where ${}^{\perp \!\!\!\perp}$ denotes orthogonal companion with respect to $\langle \cdot, \cdot \rangle$. Any Lagrangian subspace satisfies $\dim \mathcal{L

Theorems & Definitions (37)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 27 more