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Model Reduction of Descriptor Systems by Interpolatory Projection Methods

Serkan Gugercin, Tatjana Stykel, Sarah Wyatt

arXiv:1301.4524v1math.NAcs.NAcs.SYeess.SYmath.DS

TL;DR

Standard state-space interpolation subspace conditions applied to descriptor systems can leave the polynomial parts of the full and reduced transfer functions mismatched, producing unbounded error. The paper modifies interpolation subspace conditions for bounded error, extends optimal H2 model reduction to descriptor systems, and gives index-1/index-2 approaches without explicitly computing deflating subspaces. Modified conditions enforce bounded error; balanced truncation and Algorithm 6.2 have respective H∞-errors of 3.3284×10−6 and 8.9663×10−6.

  • Problem

    Standard state-space interpolation subspace conditions applied to descriptor systems can leave the polynomial parts of the full and reduced transfer functions mismatched, producing unbounded error.

    Source: §1 · p. 2

  • Method

    The paper modifies interpolation subspace conditions for bounded error, extends optimal H2 model reduction to descriptor systems, and gives index-1/index-2 approaches without explicitly computing deflating subspaces.

    Source: §1 · p. 2

  • Results

    Modified conditions enforce bounded error; balanced truncation and Algorithm 6.2 have respective H∞-errors of 3.3284×10−6 and 8.9663×10−6.

    Source: §1 · p. 2, §6.2 · p. 20

  • Takeaways & Limitations

    For the Oseen model, Algorithm 6.2 produces a high-fidelity reduced model that nearly matches balanced truncation, using iteratively determined H2-optimal interpolation points.

    Source: §6.2 · p. 19

    The paper assumes equal numbers of left and right interpolation points (q = r).

    Source: §2 · p. 3

Abstract

from arXiv · show

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H-infinity errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-1 and index-2 descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the H2-norm by extending the Iterative Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.

1. Introduction.

The introduction frames descriptor-system model reduction as a way to reduce the computational burden of large models while retaining a high-fidelity approximation. It outlines projection-based interpolation and the paper’s extensions for bounded error and H2-optimal reduction.

1. Introduction.

  • Model reduction replaces large-scale dynamics with a much smaller model whose output remains a high-fidelity approximation over a wide range of bounded-energy inputs.The introduction cites electrical circuits, multibody dynamics, and semidiscretized partial differential equations as examples of applications with very large state spaces.Source: §1 · pp. 1–2
  • Projection-based reduction approximates the full state using V and enforces a Petrov–Galerkin condition with W to obtain a reduced-order model.The reduced matrices are formed from the projection matrices, and their subspaces can be computed in different ways.Source: §1 · p. 2
  • The paper’s interpolatory methods choose projection subspaces to enforce tangential interpolation of the original transfer function.The introduction situates this framework alongside earlier work and identifies Gallivan et al. as the source of its tangential rational interpolation framework.Source: §1 · p. 2
  • The paper modifies interpolatory subspace conditions for descriptor systems to address unbounded errors from the mismatch between full and reduced transfer-function polynomial parts.It uses deflating subspaces, extends IRKA for H2 reduction, and describes approaches for index-1 and index-2 systems that avoid computing those subspaces explicitly.Source: §1 · p. 2
  • For descriptor systems, the paper develops modified interpolation conditions and extends optimal H2 model reduction, with special treatments for index-1 and index-2 systems.These special-case approaches avoid explicit computation of deflating subspaces; the introduction also notes a numerical example where balanced truncation faces difficulty solving generalized Lyapunov equations.Source: §1 · p. 2

2. Model reduction by tangential rational interpolation.

Tangential rational interpolation can be enforced by projection, but applying standard state-space subspace conditions to descriptor systems can leave the approximation error unbounded. The section motivates modified conditions that also account for the transfer function’s polynomial part.

2. Model reduction by tangential rational interpolation.

  • The interpolation framework uses left and right tangential directions and allows matching higher-order derivatives; the paper assumes equal numbers of left and right interpolation points.The projection conditions yield tangential interpolation, and when the left and right points coincide they also yield higher-order derivative matching.Source: §2 · p. 3
  • Standard tangential interpolation conditions can hold for singular E, but the reduced transfer function may be proper when the full transfer function is improper.The polynomial parts must match exactly for bounded H2 or H-infinity error; otherwise the error grows unbounded regardless of interpolation-point quality.Source: §2 · p. 4
  • In an index-2 RLC example, Hermite interpolation at mirror images of dominant poles still produces error that grows unbounded with frequency and unbounded H2 and H-infinity errors.The example uses a full model of order 765 and a reduced model of order 20; Figure 2.1 plots the frequency responses and absolute error.Source: Figure 2.1 · p. 4, §2 · p. 4
  • The paper identifies spectral projectors onto finite-eigenvalue deflating subspaces as central to matching higher-order polynomial parts while interpolating at arbitrary complex-plane points.These subspaces also support the paper’s treatment of optimal interpolation points and tangential directions for descriptor systems.Source: §2 · p. 5

3. Interpolatory projection methods for descriptor systems.

The paper modifies interpolatory projection for descriptor systems by using deflating subspaces so the reduced model matches the full model’s polynomial part while satisfying tangential interpolation. It also describes a way to compute the subspaces for systems with uncontrollable and unobservable states at infinity.

3. Interpolatory projection methods for descriptor systems.

  • Matching the polynomial parts leaves the error without a polynomial part, addressing the source of unbounded error in descriptor-system reduction.The construction combines subspaces for the strictly proper part with subspaces associated with the eigenvalue at infinity.Source: §3 · pp. 5–6
  • The modified subspaces ensure the reduced model matches the polynomial part and satisfies left and right tangential interpolation, including derivative conditions.When the left and right interpolation points coincide, additional mixed tangential derivative conditions hold.Source: §3 · p. 5
  • The example’s maximum error is below 10^-2 and decays to zero as frequency approaches infinity because the polynomial part is captured exactly.This contrasts with the unbounded error under the earlier subspace conditions.Source: Figure 3.1 · p. 7, §3 · p. 7
  • When states are uncontrollable and unobservable at infinity, they can be removed without changing the transfer function, and projected Lyapunov equations can help compute the infinity subspaces.The paper notes that explicit projectors can be difficult to compute generally, though structured problems may admit explicit forms.Source: §3 · pp. 7–8

4. Interpolatory optimal H2 model reduction for descriptor systems.

The paper extends H2-optimal interpolatory model reduction to descriptor systems by separating the strictly proper and polynomial parts and adapting IRKA to the former. The resulting conditions preserve the polynomial part and impose H2-optimal interpolation on the strictly proper part, though general large-scale use may require difficult projector computations.

4. Interpolatory optimal H2 model reduction for descriptor systems.

  • Interpolatory H2 methods use sparse linear systems rather than the series of Lyapunov equations required by Lyapunov-based approaches.The paper describes interpolatory approaches as numerically effective for large-scale settings.Source: §4 · p. 8
  • An H2-optimal reduced descriptor model must match the full model’s polynomial part, leaving the strictly proper part to minimize H2 error.Otherwise the error norm is unbounded.Source: §4 · p. 9
  • The H2 optimality conditions require bi-tangential Hermite interpolation of the strictly proper transfer function at reflected reduced-model poles.The interpolation points and tangential directions depend on the reduced model being computed.Source: §4 · p. 9
  • The descriptor-system IRKA iteration updates interpolation data from the strictly proper intermediate model, then appends infinity subspaces to preserve the polynomial part.At convergence, points are reflected poles and directions are residue directions of the strictly proper reduced model.Source: §4 · pp. 9–10
  • The general construction requires explicit spectral projectors or deflating subspaces, which may be numerically infeasible for large-scale problems.The paper says special descriptor-system classes can avoid explicitly forming the projectors.Source: §4 · p. 10

5. Semi-explicit descriptor systems of index 1.

For index-1 descriptor systems, modified projection conditions match the polynomial part while retaining interpolation; incorporating that modification throughout IRKA yields first-order H2 conditions. The method avoids explicit spectral-projector computation and is illustrated on a supersonic-flow model.

5. Semi-explicit descriptor systems of index 1.

  • For index-1 systems, the shifted reduced matrices match the polynomial part and preserve bi-tangential Hermite interpolation, provided the reduced E matrix is nonsingular.The authors state that this nonsingularity holds generically when the projection matrices have full rank and rank(E) exceeds the reduced order.Source: §5 · p. 12
  • The cited result leads to Algorithm 5.1, which achieves bi-tangential Hermite interpolation of the semi-explicit descriptor system (5.1) without explicitly forming the spectral projectors.Source: §5 · p. 12

5.1. Optimal H2 model reduction for semi-explicit descriptor systems.

  • The eD modification must be applied throughout IRKA: adding it only at the end matches polynomial parts but changes poles and loses H2 first-order conditions.The intermediate model's H2 error can be unbounded when its polynomial part does not match the full model.Source: §5.1 · p. 13
  • Upon convergence, Algorithm 5.2 produces a reduced model satisfying the first-order necessary conditions for H2-optimal model reduction.The algorithm incorporates the eD shift during its iterations.Source: §5.1 · p. 13

5.2. Supersonic inlet flow example.

  • An order-11 H2-optimal reduced model accurately matches the low-frequency response of the supersonic-flow SISO subsystem.The original semi-explicit system has dimension 11,730; the method avoids spectral-projector computation.Source: §5.2 · p. 14

6. Stokes-type descriptor systems of index 2.

For index-2 descriptor systems, the paper derives interpolation conditions and a computational route that avoids explicit deflating-subspace bases. It extends IRKA to enforce first-order H2 conditions, with Oseen examples illustrating accurate reduced models.

6. Stokes-type descriptor systems of index 2.

  • The index-2 analysis does not require E11 to be symmetric positive definite or A21 to equal A12^T.The method extends the earlier analysis to optimal H2 reduction without computing deflating subspaces.Source: §6 · p. 15
  • For fixed nontrivial b ∈ C^m and c ∈ C^p, if (σE−A)I_Bb ∈ Im(V) ⊂ Im(Θr,1) and (µE^T−A^T)I_C^T c ∈ Im(W) ⊂ Im(Θl,2), then G(σ)b = eG(σ)b and c^T G(µ) = c^T eG(µ); if σ=µ, c^T G′(σ)b = c^T eG′(σ)b.Source: §6 · p. 16

6.1. Computational issues related to the reduction of index-2 descrip-

  • For index-2 systems, the required interpolation quantities can be computed by solving linear systems, so constructing the deflating-subspace basis matrices is unnecessary.This leads to Algorithm 6.1 for interpolatory model reduction.Source: §6.1 · p. 18
  • Upon convergence, index-2 IRKA yields a reduced model satisfying the first-order conditions for H2 optimality.The extension follows from the computationally effective bi-tangential Hermite interpolation framework.Source: §6.1 · pp. 18–19

6.2. Numerical results for Oseen equations.

  • 3.3284×10^-6 and 8.9663×10^-6 were the relative H∞ errors for balanced truncation and Algorithm 6.2, respectively.Both reduced models have order 20; the amplitude Bode plots show Algorithm 6.2 nearly matching the full model and balanced truncation.Source: §6.2 · pp. 19–20, Figure 6.1 · p. 20
  • For sin(6it) inputs, full and reduced outputs match and output error is four orders smaller; sin(it) trials show the same conclusions.The experiments use i = 1, . . . , 6 for the system's six inputs.Source: §6.2 · p. 20

7. Conclusions.

The authors acknowledge the provision of data and MATLAB files for the Oseen numerical example and identify research support.

7. Conclusions.

  • M. Heinkenschloss provided the data and MATLAB files for the Section 6.2 numerical example; the work also received NSF and BMBF support.The passage names NSF Grant DMS-0645347 and BMBF Grant 05M10WAB.Source: §8 · p. 20
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