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On Port-Hamiltonian Formulation of Hysteretic Energy Storage Elements: The Backlash Case

Jurrien Keulen, Bayu Jayawardhana, Arjan van der Schaft

Abstract

This paper presents a port-Hamiltonian formulation of hysteretic energy storage elements. First, we revisit the passivity property of backlash-driven storage elements by presenting a family of storage functions associated to the dissipativity property of such elements. We explicitly derive the corresponding available storage and required supply functions `a la Willems [1], and show the interlacing property of the aforementioned family of storage functions sandwiched between the available storage and required supply functions. Second, using the proposed family of storage functions, we present a port-Hamiltonian formulation of hysteretic inductors as prototypical storage elements in port-Hamiltonian systems. In particular, we show how a Hamiltonian function can be chosen from the family of storage functions and how the hysteretic elements can be expressed as port-Hamiltonian system with feedthrough term, where the feedthrough term represents energy dissipation. Correspondingly, we illustrate its applicability in describing an RLC circuit (in parallel and in series) containing a hysteretic inductor element.

On Port-Hamiltonian Formulation of Hysteretic Energy Storage Elements: The Backlash Case

Abstract

This paper presents a port-Hamiltonian formulation of hysteretic energy storage elements. First, we revisit the passivity property of backlash-driven storage elements by presenting a family of storage functions associated to the dissipativity property of such elements. We explicitly derive the corresponding available storage and required supply functions `a la Willems [1], and show the interlacing property of the aforementioned family of storage functions sandwiched between the available storage and required supply functions. Second, using the proposed family of storage functions, we present a port-Hamiltonian formulation of hysteretic inductors as prototypical storage elements in port-Hamiltonian systems. In particular, we show how a Hamiltonian function can be chosen from the family of storage functions and how the hysteretic elements can be expressed as port-Hamiltonian system with feedthrough term, where the feedthrough term represents energy dissipation. Correspondingly, we illustrate its applicability in describing an RLC circuit (in parallel and in series) containing a hysteretic inductor element.

Paper Structure

This paper contains 11 sections, 4 theorems, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dissipativity Theory
  4. Port-Hamiltonian systems
  5. Passivity of Backlash-driven Storage Elements
  6. Port-Hamiltonian formulation
  7. Backlash inductor element in pH
  8. RC network with backlash ferromagnet inductor
  9. Parallel interconnection
  10. Series interconnection
  11. Conclusion

Key Result

Proposition III.1

For any $\gamma\in\left[-h, h\right]$, the function defines an admissible storage function for the backlash inductor with minimum value of $0$.

Figures (4)

  • Figure 1: Phase plot of a simple 'ferromagnetic' backlash operator, with the variables $I$ and $\phi$. The adjective 'ferromagnetic' here stems from the physical law that we attach to this operator, where $\phi$ is coupled to a port variable $V$ by $\frac{\textrm{d} \phi}{\textrm{d} t}=V$. The slope of the diagonal line defines the inductance $L$.
  • Figure 2: Admissible storage functions $S_\gamma(\phi)$ for the backlash operator. The extreme cases $\gamma=\pm h$ bound the family, while the shaded region represents all admissible functions for $\gamma\in[-h,h]$. All functions attain a minimum value of zero. The dashed line indicates the admissible Hamiltonian of the backlash inductor, derived in \ref{['eq:Hbf']}.
  • Figure 3: Parallel interconnected resistor, ferromagnetic backlash and capacitor circuit.
  • Figure 4: Series interconnected resistor, ferromagnetic backlash and capacitor circuit.

Theorems & Definitions (7)

  • Definition II.1
  • Proposition III.1
  • Proposition III.2
  • Definition III.1
  • Proposition III.3
  • Definition III.2
  • Proposition III.4