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Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems

Petar Mlinarić, Peter Benner, Serkan Gugercin

Abstract

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of $\mathcal{L}_2$-optimal reduced-order modeling of parametric stationary problems, we derive interpolatory $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.

Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems

Abstract

Interpolatory necessary optimality conditions for -optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of -optimal reduced-order modeling of parametric stationary problems, we derive interpolatory -optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.
Paper Structure (9 sections, 6 theorems, 66 equations)

This paper contains 9 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Background
  3. H2xL2-optimal Parametric Interpolation
  4. Parameters in Inputs and Outputs
  5. Parameter in Dynamics
  6. Numerical Experiments
  7. Synthetic Parametric Model
  8. Parametric Penzl's FOM
  9. Conclusions

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{\kappa}_i, \widehat{\zeta}_j, \widehat{\eta}_k \colon \m $\widehat{K}_i \in \mathbb{C}^{r \times r}$, $\widehat{F}_j \in \mathbb{C}^{r \times n_{\textnormal

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 5.1
  • ...and 1 more