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On the balanced truncation error bound and sign parameters from arrowhead realizations

Sean Reiter, Tobias Damm, Mark Embree, Serkan Gugercin

TL;DR

This work analyzes when the balanced truncation $ ${ }H_ ${∞}$ error bound for linear time-invariant systems is attained with exact equality. It introduces a generalized state-space symmetry concept, showing that the sign parameters associated with Hankel singular values determine when equality holds for SISO systems; these signs are invariant across realizations satisfying the symmetry and can be read directly from arrowhead structures. For arrowhead realizations, the signs are read from off-diagonal entries, enabling direct prediction of equality in the error bound and extending to singular perturbation balancing via reciprocity. The results are illustrated with a power-system model, where an arrowhead realization naturally arises and the bound remains tight across truncation orders, providing practical criteria to certify exact reduced-order modeling. Overall, the paper broadens the class of systems for which balanced truncation yields exact error and offers concrete tools for identifying sign symmetry in network-inspired models.

Abstract

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced-order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling, and illustrate these results with a power system model that motivated this study.

On the balanced truncation error bound and sign parameters from arrowhead realizations

TL;DR

This work analyzes when the balanced truncation { }H_ error bound for linear time-invariant systems is attained with exact equality. It introduces a generalized state-space symmetry concept, showing that the sign parameters associated with Hankel singular values determine when equality holds for SISO systems; these signs are invariant across realizations satisfying the symmetry and can be read directly from arrowhead structures. For arrowhead realizations, the signs are read from off-diagonal entries, enabling direct prediction of equality in the error bound and extending to singular perturbation balancing via reciprocity. The results are illustrated with a power-system model, where an arrowhead realization naturally arises and the bound remains tight across truncation orders, providing practical criteria to certify exact reduced-order modeling. Overall, the paper broadens the class of systems for which balanced truncation yields exact error and offers concrete tools for identifying sign symmetry in network-inspired models.

Abstract

Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced-order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling, and illustrate these results with a power system model that motivated this study.

Paper Structure

This paper contains 14 sections, 9 theorems, 73 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Balanced truncation model reduction
  3. The sign symmetry of balanced realizations
  4. Error bound results
  5. Generalized conditions for the balanced truncation error bound to hold with equality
  6. An example with flipped signs and a strict bound
  7. Extension to singular perturbation balancing
  8. On the sign parameters of the Hankel singular values
  9. Systems with generalized state-space symmetry
  10. Systems with arrowhead realizations
  11. A motivating example from power systems modeling
  12. Arrowhead systems and preliminary tools
  13. A special case of arrowhead systems
  14. Conclusion

Key Result

Theorem 2.1

Let $\mathcal{G}$ be an order-$n$ minimal and asymptotically stable dynamical system having the balanced realization where the state space matrices are partitioned according to the system Gramian $\Sigma = \textrm{diag}(\Sigma_1,\Sigma_2)$, with where $r:=m_1+\cdots+m_k$. Then the $r$th order reduced model obtained via balanced truncation, having the transfer function is asymptotically stable

Figures (1)

  • Figure 1: An arrowhead network with $n=6$, with input $u$ and output $y$ restricted to state $x_1$.

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Definition 2.1
  • Theorem 3.1
  • Proof 3.1
  • Remark 3.1
  • Example 3.1
  • Theorem 3.2
  • Proof 3.2
  • ...and 11 more