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A discrete gradient scheme for preserving QSR-dissipativity

Attila Karsai, Philipp Schulze

Abstract

The notion of dissipative dynamical systems provides a formal description of processes that cannot generate energy internally. For these systems, changes in energy can only occur due to an external energy supply or dissipation effects. Unfortunately, dissipative properties tend to deteriorate in numerical computations, especially in nonlinear systems. Discrete gradient methods can help mitigate this problem. In this paper, we present a class of structure-preserving time discretization schemes based on discrete gradients for a special class of systems that are dissipative with respect to a quadratic supply rate.

A discrete gradient scheme for preserving QSR-dissipativity

Abstract

The notion of dissipative dynamical systems provides a formal description of processes that cannot generate energy internally. For these systems, changes in energy can only occur due to an external energy supply or dissipation effects. Unfortunately, dissipative properties tend to deteriorate in numerical computations, especially in nonlinear systems. Discrete gradient methods can help mitigate this problem. In this paper, we present a class of structure-preserving time discretization schemes based on discrete gradients for a special class of systems that are dissipative with respect to a quadratic supply rate.
Paper Structure (20 sections, 3 theorems, 47 equations, 1 figure)

This paper contains 20 sections, 3 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem setting
  4. Dynamical systems
  5. Dissipativity
  6. Characterizations of dissipative systems
  7. Time discretization
  8. Discrete gradients
  9. Proposed discretization scheme
  10. Numerical Experiments
  11. Implementation details
  12. Time discretization
  13. Nonlinear systems
  14. Convergence
  15. Power balance
  16. ...and 5 more sections

Key Result

Theorem 4

Let as:well-posedness hold with $T = \infty$, and let $\mathcal{H} \colon \mathbb{R}^n \to [0,\infty)$ be continuously differentiable. Then $\mathcal{H}$ is a storage function for eq:moylan-system with respect to the supply rate eq:hill-moylan-qsr-supply if and only if there exists $p \in \mathbb{N} for all $z \in \mathbb{R}^n$.

Figures (1)

  • Figure 1: Numerical results for \ref{['ex:pendulum', 'ex:qsr-hjb', 'ex:pi-controller', 'ex:qsr-hill-moylan']}.

Theorems & Definitions (12)

  • Definition 2: dissipative systems willems72-dissipative1
  • Definition 3: dissipation rate willems72-dissipative1
  • Theorem 4: hill75-cyclo
  • Definition 5
  • Theorem 8: well-posedness
  • proof
  • Proposition 9: discrete power balance
  • proof
  • Example 10: pendulum
  • Example 11: optimal control problem, vanderschaft17-l2gain
  • ...and 2 more