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Computing the nearest $Ω$-admissible descriptor dissipative Hamiltonian system

Vaishali Aggarwal, Nicolas Gillis, Punit Sharma

TL;DR

A dissipative Hamiltonian characterization is provided for the matrix pairs that are $\Omega$-admissible where $\Omega$ is an LMI region and the nearest $\Omega$-admissible matrix pair problem is solved.

Abstract

For a given set $Ω\subseteq \mathbb{C}$, a matrix pair $(E,A)$ is called $Ω$-admissible if it is regular, impulse-free and its eigenvalues lie inside the region $Ω$. In this paper, we provide a dissipative Hamiltonian characterization for the matrix pairs that are $Ω$-admissible where $Ω$ is an LMI region. We then use these results for solving the nearest $Ω$-admissible matrix pair problem: Given a matrix pair $(E,A)$, find the nearest $Ω$-admissible pair $(\tilde E, \tilde A)$ to the given pair $(E,A)$. We illustrate our results on several data sets and compare with the state of the art.

Computing the nearest $Ω$-admissible descriptor dissipative Hamiltonian system

TL;DR

A dissipative Hamiltonian characterization is provided for the matrix pairs that are -admissible where is an LMI region and the nearest -admissible matrix pair problem is solved.

Abstract

For a given set , a matrix pair is called -admissible if it is regular, impulse-free and its eigenvalues lie inside the region . In this paper, we provide a dissipative Hamiltonian characterization for the matrix pairs that are -admissible where is an LMI region. We then use these results for solving the nearest -admissible matrix pair problem: Given a matrix pair , find the nearest -admissible pair to the given pair . We illustrate our results on several data sets and compare with the state of the art.

Paper Structure

This paper contains 17 sections, 9 theorems, 47 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. LMI regions
  5. DH pairs and their relationship with $\Omega$-admissible pairs $(E,A)$
  6. Special LMI regions and $\Omega$-stability
  7. The nearest $\Omega$-admissible pair problem in DH form
  8. Fast gradient method for Hurwitz stability
  9. Initialization
  10. Block coordinate descent in the general case
  11. Numerical Experiments
  12. Hurwitz stability
  13. Schur stability
  14. Other sets $\Omega$: examples from ChouGS24
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(E,A)$ be a matrix pair, where $E,A \in {\mathbb R}^{n,n}$. Then the following are equivalent.

Figures (6)

  • Figure 1: Eigenvalues of $A$, of its $\Omega$-stable approximation $\tilde{A} = (J- R) Q$ChouGS24, and the $\Omega$-stable matrix pair $(\bar{E}, \bar{A})$ that approximates $(I_n,A)$ using our proposed algorithm. The set $\Omega$ is the intersection of a vertical strip, a horizontal strip, and left and right parabolic regions.
  • Figure 2: Eigenvalues of $A$, of its $\Omega$-stable approximation $\tilde{A} = (J- R) Q$ChouGS24, and the $\Omega$-stable matrix pair $(\bar{E}, \bar{A})$ that approximates $(I_n,A)$ using our proposed algorithm. The set $\Omega$ is the intersection of an ellipsoid, a left hyperbolic region, and a right conic sector.
  • Figure 3: Evolution of the relative error of the three algorithms for Grcar matrix pairs in case of Hurwitz stability: from top row to bottom row, $n=10,20,30$; from left to right, $k=1,2,3$.
  • Figure 4: Evolution of the relative error of the three algorithms for MSD systems in case of Hurwitz stability: from top row to bottom row, $n=10,20,30$; from left to right, $\epsilon=0.01,0.05,0.1$.
  • Figure 5: Evolution of the relative error of the three algorithms for Grcar matrix pairs in case of Schur stability: from top row to bottom row, $n=10,20,30$; from left to right, $k=1,2,3$.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Definition 1: $\Omega$-admissibility
  • Definition 2: DH matrix pair
  • Theorem 1
  • Lemma 1
  • proof
  • Definition
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • ...and 12 more