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Towards Polynomial Immersion of Port-Hamiltonian Systems

Mohammad Itani, Manuel Schaller, Karl Worthmann, Timm Faulwasser

Abstract

Port-Hamiltonian (pH) systems offer a highly structured and energy-based modular framework for control systems. Many pH systems exhibit non-polynomial non-linearities. We consider the problem of immersing such systems into a higher-dimensional polynomial representation. We prove that, along system trajectories, important features of the non-polynomial pH system are preserved such as the internal interconnection geometry, the energy balance relation with passivity supply rate, as well as energy dissipation. We illustrate how the lifted system enables the design of stabilizing feedback laws by combining sum-of-squares optimization with concepts from passivity-based control. We draw upon several examples to illustrate our findings.

Towards Polynomial Immersion of Port-Hamiltonian Systems

Abstract

Port-Hamiltonian (pH) systems offer a highly structured and energy-based modular framework for control systems. Many pH systems exhibit non-polynomial non-linearities. We consider the problem of immersing such systems into a higher-dimensional polynomial representation. We prove that, along system trajectories, important features of the non-polynomial pH system are preserved such as the internal interconnection geometry, the energy balance relation with passivity supply rate, as well as energy dissipation. We illustrate how the lifted system enables the design of stabilizing feedback laws by combining sum-of-squares optimization with concepts from passivity-based control. We draw upon several examples to illustrate our findings.
Paper Structure (15 sections, 7 theorems, 88 equations, 4 figures)

This paper contains 15 sections, 7 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries and Problem Statement
  4. Problem Statement
  5. Immersions
  6. Structure-preserving Polynomial Immersion of Port-Hamiltonian Systems
  7. Part 1
  8. Part 2
  9. Part 3
  10. Part 4
  11. Illustrative Examples
  12. Example with Exponential Terms
  13. The rolling coin
  14. SOS-IDA-PBC via Polynomial Immersions
  15. Conclusion and Outlook

Key Result

Theorem 1

Given a system eqn:affine_control_sys whose functions are all differential algebraic, i.e., satisfy eqn:ADE. Then there exists a finitely-generated rational field $\mathbb{R}(\psi_1(x),\ldots,\psi_{\tilde{n}}(x))$ with $\psi_1(x), \ldots, \psi_{\tilde{n}}(x)$ satisfying eqn:ADE that is equivalent to the differential field $\mathbb{R}\langle g_0(x), \ldots, g_m(x),h(x) \rangle_\nabla$. More define

Figures (4)

  • Figure 1: Illustration of the difference between an immersion and an invariant immersion.
  • Figure 2: A schematic of the rolling coin.
  • Figure 3: State and output trajectories: Left = system \ref{['eqn:pH_euro']}, right = polynomial port-Hamiltonian system.
  • Figure 4: Time evolution of the Hamiltonian $H_d$ (top left), the input $u$ (top right), and the states.

Theorems & Definitions (20)

  • Definition 1: Immersibility fliess1983finitenessohtsuka2005model
  • Definition 2: Invariant immersibility ohtsuka2005model
  • Remark 1: Manifold immersions
  • Definition 3: Differential algebraic immersion ohtsuka2005modelohtsuka2002differentially
  • Definition 4: Differential algebraic function ohtsuka2005model
  • Remark 2: Properties of diff. algebraic functions
  • Theorem 1: Immersibility into rational systems ohtsuka2005model
  • Theorem 2: Immersibility into polynomial systems ohtsuka2005model
  • proof
  • Definition 5: Lifted immersion
  • ...and 10 more