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Interpolatory $\mathcal{H}_2$-optimality Conditions for Structured Linear Time-invariant Systems

Petar Mlinarić, Peter Benner, Serkan Gugercin

TL;DR

It is shown that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

Abstract

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. Under certain diagonalizability assumptions, we show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

Interpolatory $\mathcal{H}_2$-optimality Conditions for Structured Linear Time-invariant Systems

TL;DR

It is shown that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.

Abstract

Interpolatory necessary optimality conditions for -optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on -optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for -optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. Under certain diagonalizability assumptions, we show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.
Paper Structure (17 sections, 13 theorems, 111 equations)

This paper contains 17 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. L2-optimal Reduced-order Modeling
  3. Setting
  4. Wirtinger Calculus and Gradients of the Squared L2 Error
  5. Necessary L2-optimality Conditions
  6. L2-optimality for Diagonal Structured Dynamics
  7. Interpolatory H2-optimality Conditions for Diagonal Structured Dynamics
  8. Second-order Systems
  9. Interpolatory Conditions
  10. Numerical Example
  11. Port-Hamiltonian Systems
  12. Interpolatory Conditions
  13. Numerical Example
  14. Time-delay Systems
  15. Interpolatory Conditions
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose that $\mathcal{P} \subseteq \mathbb{C}^{n_{\mathsf{p}}}$, $\mu$ is a measure over $\mathcal{P}$, the function $H$ is in $\mathcal{L}_{2}(\mathcal{P}, \mu; \mathbb{C}^{n_{\textnormal{o}} \times n_{\textnormal{i}}})$, functions $\widehat{\alpha}_i, \widehat{\beta}_j, \widehat{\gamma}_k \colon Let $\Sigma_{\textnormal{St}}$ denote the set of strom $(\widehat{A}_i, \widehat{B}_j, \widehat{C}_

Theorems & Definitions (27)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • proof
  • ...and 17 more