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Structure-Preserving Linear Quadratic Gaussian Balanced Truncation for Port-Hamiltonian Descriptor Systems

Tobias Breiten, Philipp Schulze

TL;DR

A novel procedure that is based on a recently introduced Kalman-Yakubovich-Popov inequality for descriptor systems is provided, demonstrating how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.

Abstract

We present a new balancing-based structure-preserving model reduction technique for linear port-Hamiltonian descriptor systems. The proposed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We derive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically analyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel procedure that is based on a recently introduced Kalman-Yakubovich-Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.

Structure-Preserving Linear Quadratic Gaussian Balanced Truncation for Port-Hamiltonian Descriptor Systems

TL;DR

A novel procedure that is based on a recently introduced Kalman-Yakubovich-Popov inequality for descriptor systems is provided, demonstrating how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.

Abstract

We present a new balancing-based structure-preserving model reduction technique for linear port-Hamiltonian descriptor systems. The proposed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We derive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically analyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel procedure that is based on a recently introduced Kalman-Yakubovich-Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality.

Paper Structure

This paper contains 19 sections, 22 theorems, 159 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix Pencils
  4. Linear Descriptor Systems
  5. Port-Hamiltonian Descriptor Systems
  6. A Generalized KYP LMI
  7. Modified Generalized Algebraic Riccati Equations
  8. Structure-Preserving LQG Balanced Truncation
  9. LQG Balancing and Structure-Preserving Truncation
  10. A Priori Error Bound for Right Coprime Factors
  11. Improvement of the Error Bound by Replacing $Q$
  12. Procedure for Index-1 Systems
  13. Procedure for Systems with Arbitrary Index
  14. Numerical Examples
  15. Transport Network
  16. ...and 4 more sections

Key Result

Proposition 6

The port-Hamiltonian system eq:pHEQ is regular and strongly stabilizable and detectable if and only if the following conditions are satisfied:

Figures (2)

  • Figure 1: Comparison of the pH-structure-preserving LQG-BT method with the classical unstructured variant for a transport network. The $\mathcal{H}_{\infty}$-error is shown for different reduced system dimensions and different choices of the system Hamiltonian.
  • Figure 2: Comparison of the pH-structure-preserving LQG-BT method with the classical unstructured variant for a constrained mass-spring-damper system. The $\mathcal{H}_{\infty}$-error is shown for different reduced system dimensions and different choices of the system Hamiltonian.

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Remark 5
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 41 more