Moment Matching for Descriptor Systems: A M\"obius Mapping Approach

A. Context

• Moment matching has developed from steady-state response matching and H2-based approximation toward data-driven solutions and discrete-time analogues.The passage describes the framework as originally proposed for linear and nonlinear systems and cites – for these developments.Source: A. Context · p. 1

B. Descriptor Systems and Centred Realisations

• Centred realisations and Möbius mappings allow classical theory to be applied to singular dynamics in both continuous- and discrete-time cases.The approach is described as time-domain-independent, unlike the descriptor realisations used in.Source: A. Context · p. 1, B. Descriptor Systems and Centred… · p. 2

B. Motivation

• Earlier moment-matching procedures for singular systems were limited to continuous time or lacked pole-zero placement and a unified treatment of both time settings.The passage identifies as continuous-time only and notes that also lacks pole-zero placement and a unified continuous/discrete framework.Source: B. Motivation · p. 1

C. Contribution

• The proposed procedure handles continuous- and discrete-time SISO systems, including improper transfer functions, and allows selecting reduced-model poles and zeros.The authors describe it as more flexible and reliable than when the transfer function is improper and cannot be represented via state-space systems.Source: C. Contribution · p. 1

A. Notation

• The notation defines a matrix pencil as M−λN and uses λ for either the Laplace variable s or the Z-transform variable z.The generalized eigenvalues of a matrix pencil are denoted Λ(M−λN).Source: A. Notation · p. 1

B. Descriptor Systems and Centred Realisations

• A centred realisation generalizes descriptor realisations, while setting E=I_n yields a standard state-space system.Its order is the realisation order, and A−λE is called its pole pencil.Source: B. Descriptor Systems and Centred… · p. 2
• A centred realisation is minimal when its order equals the transfer function’s McMillan degree, and the paper introduces moments from a frequency-based perspective.The minimality equivalence is stated in Proposition II.1; the moment definition is introduced in Definition II.5.Source: B. Descriptor Systems and Centred… · p. 2

C. M¨obius Mappings

• A Möbius mapping with ad−bc≠0 is bijective and has an inverse.The mapping and its inverse are given in equations (4) and the following lemma.Source: C. M¨obius Mappings · p. 2
• The specialized map f converts a centred realisation into a standard state-space realisation and preserves minimality.The result assumes A−λ0E is invertible and gives the transformed realization in terms of (A−λ0E)−1E and (A−λ0E)−1B.Source: C. M¨obius Mappings · p. 2
• Choosing a real centring point preserves real-valued realization matrices and can make A−λ0E well-conditioned for inversion.The passage gives these as the reasons for restricting λ0 to the real numbers and retaining freedom in its choice.Source: C. M¨obius Mappings · p. 2

A. Problem Statement

• The target model has order ν below the original order n and matches selected moments at distinct points, with specified multiplicities.The problem formulation also allows specified pole and zero constraints.Source: A. Problem Statement · p. 2

B. Reduced-order Models for Moment Matching

• At least one matched moment is assumed nonzero, ensuring that models satisfying the matching objective are not identically zero.The paper describes this assumption as practically nonrestrictive because moment matching typically seeks to preserve nonzero moments.Source: B. Reduced-order Models for Moment… · p. 3
• Theorem III.1 parametrizes moment-matching models using a matrix S similar to a specified Jordan matrix and an observable pair (L, S).A solution to the generalized Sylvester equation and a condition on G yield a well-defined transfer function satisfying the moment-matching objective.Source: B. Reduced-order Models for Moment… · p. 3
• Every model in the parametrized class has an invertible pole pencil and a McMillan degree no greater than the chosen order ν.The paper contrasts these guarantees with solutions in and notes that the degree bound is not always available for descriptor systems with poles at infinity.Source: B. Reduced-order Models for Moment… · p. 3

A. Problem Statement

• The reduced model can be constrained to have selected poles and zeros, alongside matching specified moments of the original transfer function.The pole and zero constraints may be omitted when their respective counts are zero.Source: A. Problem Statement · pp. 2–3, B. Reduced-order Models for Moment… · p. 3

C. Pole and Zero Constraints

• The pole-placement proposition guarantees nonsingular constraint matrices and a reduced realization satisfying the selected pole constraints; the associated matrix pairs can be computed in advance to check conditioning.Pole targets close to moment-matching locations make numerical accuracy especially sensitive.Source: C. Pole and Zero Constraints · p. 3
• The zero-placement theorem uses a generalized Sylvester equation, with rank and compatibility conditions required for the reduced realization to satisfy the zero constraints.The conditions include ΥzB = O and a linear equation in the gain matrix G.Source: C. Pole and Zero Constraints · p. 3

D. Numerical Considerations

• Pole placement amounts to solving linear equations in G once the generalized Sylvester equation has been solved and the reduced-model class is parameterized.The pole constraints are the equalities in (10).Source: C. Pole and Zero Constraints · p. 3, D. Numerical Considerations · p. 4
• A practical zero-placement route solves for W first, then checks the required rank and whether the resulting linear system for G is compatible.The text identifies simultaneous computation of W, G, and the free matrix Π as desirable but challenging because of nonlinear and discontinuous terms.Source: D. Numerical Considerations · p. 4
• The convex zero-placement procedure is considered successful when its objective falls below η2; termination by a small objective change instead suggests another initialization is needed.The procedure is described as guaranteed to converge, but the stated stopping criterion distinguishes success from likely convergence to a nonzero local optimum.Source: D. Numerical Considerations · p. 4
• Unconstrained eigenvalues near the mapped moment-matching points may produce a poorly conditioned solution, although this does not invalidate the theoretical results.Optimizing the conditioning of the solutions is outside the paper’s scope.Source: D. Numerical Considerations · p. 4

A. Modelling of the Descriptor System

• The example seeks a sixth-order continuous-time model with a triple pole at infinity and exactly one zero in the open right half-plane.The target is motivated by preserving structural elements of the original plant.Source: A. Modelling of the Descriptor System · p. 5
• The differentiator op-amp topology uses an input-path resistance and feedback inductance, and serves as the foundation for each of the three circuit stages.The passage notes that this implementation is not typically used in practice.Source: A. Modelling of the Descriptor System · p. 5
• Cascading three differentiator systems would introduce three zeros at zero, so an additional output stage is designed to avoid that mismatch with the target.The transfer function of the first op-amp topology has a zero at zero.Source: A. Modelling of the Descriptor System · p. 5
• An alternative output-stage topology guarantees that all circuit zeros lie in the open left half-plane by the Hurwitz criterion.The example uses selected component values to obtain a desirable finite pole-zero structure, then specifies a minimal realization centered at λ0 = −0.2.Source: A. Modelling of the Descriptor System · p. 6

B. Moment Matching with Pole-Zero Constraints

• The design preserves the triple pole at infinity and zero at 16, while matching zeroth-order moments at −0.1j, 0.1j, −j, and j.It also selects poles at λ0 + 5·10^4, λ0 + 10^5, and infinity, with λ0 = −0.2.Source: B. Moment Matching with Pole-Zero… · p. 6
• The distinct mapped pole locations 0, 10^-5, and 2·10^-5 permit a round-off approximation to a triple pole at s = 0.The passage says the effects of this approximation are addressed later.Source: B. Moment Matching with Pole-Zero… · p. 6
• The fourth-order approximant has the desired pole-zero structure, and its approximation only marginally degrades moment matching at ±0.1j and ±j.The passage reports this as confirming the practical validity of the approximation.Source: B. Moment Matching with Pole-Zero… · p. 6

• Möbius mappings with centred realisations adapt standard state-space theory to solve moment matching for descriptor systems.Source: §V · p. 6
• Because the mappings are reversible, imposing reduced-model pole-zero constraints amounts to remapping the extended complex plane.Source: §V · p. 6
• Promising results in the single-input single-output case point to multiple-input multiple-output systems as a focus for future research.Source: §V · p. 6

Proof of Theorem III.1

• Theorem III.1’s proof constructs a reduced transfer function that matches moments at the mapped interpolation points, then shows it equals the Möbius composition and has well-defined moments.Faà di Bruno’s formula transfers equality of the mapped function’s derivatives to equality of derivatives at the original interpolation points.Source: APPENDIX · p. 7

Proof of Proposition III.1

• The proof of Proposition III.1 uses distinct mapped interpolation points to establish matrix invertibility and verify the claimed pole-eigenvalue property.Source: APPENDIX · pp. 7–8

Proof of Theorem III.2

• Theorem III.2’s proof establishes a left inverse for the moment-equation solution and uses uniqueness of a Sylvester equation under a disjoint-eigenvalue condition.It also derives the stated constraint implications from the Sylvester equation and rank conditions.Source: APPENDIX · p. 8