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arxiv: 2604.01656 · v2 · submitted 2026-04-02 · 📡 eess.SY · cs.SY

Steady-state response assignment for a given disturbance and reference: Sylvester equation rather than regulator equations

Hyeonyeong Jang , Jin Gyu Lee This is my paper

Pith reviewed 2026-05-13 21:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords moment assignmentsteady-state responseSylvester equationoutput regulationdynamic compensatorlinear systemsclosed-loop interpolationmoment matching
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The pith

The closed-loop moment decomposes into open-loop moment plus a term set by the compensator moment, allowing any desired steady-state response to be assigned by solving a Sylvester equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reframes moment matching, long used for model reduction, as moment assignment for exact control of steady-state behavior in linear systems. It shows that interconnecting a plant with a dynamic compensator produces a closed-loop moment that equals the open-loop moment plus a linear function of the compensator moment. Necessary and sufficient conditions for assigning any target moment follow directly, together with a canonical compensator form whose parameters are found by solving one Sylvester equation. This construction recovers output regulation and closed-loop interpolation as special cases and dispenses with the full regulator equations. A reader would care because the approach reduces controller synthesis for prescribed steady-state responses to a standard linear equation whose solvability is easy to check.

Core claim

The closed-loop moment of an interconnected linear system equals the open-loop moment plus a term that is linear in the moment of the compensator. This identity yields necessary and sufficient conditions for assignability of an arbitrary prescribed moment, supplies a canonical dynamic compensator structure, and reduces synthesis to the solution of a Sylvester equation. The same procedure covers both output regulation and closed-loop interpolation problems.

What carries the argument

The linear decomposition of the closed-loop moment into open-loop moment plus compensator moment, which turns moment assignment into the solution of a Sylvester equation.

Load-bearing premise

The closed-loop moment always decomposes additively into the open-loop moment and a linear function of the compensator moment whenever the overall system is linear and time-invariant.

What would settle it

For a concrete linear plant, exosystem, and assigned target moment, construct the compensator via the Sylvester equation and measure whether the actual closed-loop steady-state response to the given reference or disturbance equals the assigned moment; any systematic mismatch falsifies the decomposition.

Figures

Figures reproduced from arXiv: 2604.01656 by Hyeonyeong Jang, Jin Gyu Lee.

Figure 1
Figure 1. Figure 1: Structural decomposition of interconnected dynamics on the invariant manifold (7). (a) The original interconnected system with the signal generator [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Disturbance rejection and tracking control of HiMAT aircraft model [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Conventionally, the concept of moment has been primarily employed in model order reduction to approximate system by matching the moment, which is merely the specific set of steady-state responses. In this paper, we propose a novel design framework that extends this concept from "moment matching" for approximation to "moment assignment" for the active control of steady-state. The key observation is that the closed-loop moment of an interconnected linear system can be decomposed into the open-loop moment and a term linearly parameterized by the moment of the compensator. Based on this observation, we provide necessary and sufficient conditions for the assignability of desired moment and a canonical form of the dynamic compensator, followed by constructive synthesis procedure of compensator. This covers both output regulation and closed-loop interpolation, and further suggests using only the Sylvester equation, rather than regulator equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the closed-loop moment of an interconnected linear system decomposes into the open-loop moment plus a term linearly parameterized by the compensator moment. It provides necessary and sufficient conditions for assignability of a desired moment, a canonical dynamic compensator form, and a constructive synthesis procedure based solely on the Sylvester equation. This is positioned as covering both output regulation and closed-loop interpolation for active steady-state response assignment, avoiding the full regulator equations.

Significance. If the decomposition and assignability results hold with stability, the framework could simplify controller synthesis for prescribed steady-state behavior in linear systems by reducing the problem to a Sylvester equation. This extends the moment concept from model reduction to control design and offers a potentially more direct algebraic route than classical regulator theory. The significance is limited by the absence of stability integration, which is required for the assigned moment to be the actual asymptotic response.

major comments (2)
  1. The synthesis procedure (via Sylvester equation) assigns the moment algebraically but omits closed-loop stability constraints. For the assigned moment to represent the attained steady-state response, the closed-loop dynamics must be asymptotically stable so that trajectories converge to the particular solution; without this, the decomposition does not guarantee the claimed steady-state behavior. This is load-bearing for the abstract's claim to cover output regulation, which standard theory requires jointly with internal stability.
  2. The necessary and sufficient conditions for moment assignability (stated after the decomposition) assume solvability of the Sylvester equation but do not address how the resulting compensator affects the closed-loop eigenvalues. If the compensator introduces unstable modes, the steady-state may not be reached even when the algebraic conditions hold.
minor comments (2)
  1. The introduction should explicitly recall the definition of system moment (as the steady-state response to exosystem signals) to clarify the shift from approximation to assignment.
  2. Notation for the compensator moment and interconnection matrices could be introduced with a small diagram or explicit block equations for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the role of stability in realizing the assigned steady-state response. The algebraic decomposition and Sylvester-based synthesis constitute the central contribution; we address the stability points directly below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The synthesis procedure (via Sylvester equation) assigns the moment algebraically but omits closed-loop stability constraints. For the assigned moment to represent the attained steady-state response, the closed-loop dynamics must be asymptotically stable so that trajectories converge to the particular solution; without this, the decomposition does not guarantee the claimed steady-state behavior. This is load-bearing for the abstract's claim to cover output regulation, which standard theory requires jointly with internal stability.

    Authors: We agree that asymptotic stability of the closed-loop system is required for trajectories to converge to the particular solution defined by the assigned moment. The manuscript derives the moment decomposition and the necessary-and-sufficient assignability conditions under the algebraic interconnection; it does not claim that the Sylvester solution automatically guarantees stability. In the revision we will add an explicit remark stating that the assigned moment is the asymptotic response only when the closed-loop is asymptotically stable, and we will note that the free parameters in the Sylvester solution (or the choice of compensator poles) can be used to enforce stability when the problem data permit. This clarifies the link to classical output regulation without altering the core algebraic result. revision: yes

  2. Referee: The necessary and sufficient conditions for moment assignability (stated after the decomposition) assume solvability of the Sylvester equation but do not address how the resulting compensator affects the closed-loop eigenvalues. If the compensator introduces unstable modes, the steady-state may not be reached even when the algebraic conditions hold.

    Authors: The assignability conditions are obtained directly from the linear parameterization of the closed-loop moment by the compensator moment; they guarantee existence of a compensator moment that realizes the desired value once the Sylvester equation is solved. The canonical compensator realization is given, yet the effect of that realization on the spectrum of the overall closed-loop matrix is not analyzed. We will revise the manuscript to include a brief discussion of this point, indicating that any solution of the Sylvester equation yields a valid compensator moment and that additional eigenvalue constraints (when feasible) can be imposed on the free parameters of the solution or on the choice of the compensator state matrix. This addition will be placed after the synthesis procedure. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on independent linear-algebra facts

full rationale

The central decomposition of closed-loop moment into open-loop moment plus compensator term follows directly from standard linear interconnection algebra and the Sylvester equation solvability condition, both of which are external to the paper and do not reduce to its own fitted quantities or prior self-citations. The assignability conditions and canonical compensator form are constructed from these independent facts rather than by re-labeling inputs as predictions. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard linear-systems assumptions; the compensator moment is a design choice rather than a fitted constant.

free parameters (1)
  • Compensator moment
    Chosen design parameter that directly sets the closed-loop moment; not obtained by fitting to data.
axioms (2)
  • domain assumption Systems are linear time-invariant and the interconnection is well-defined
    Moment decomposition requires linearity of the plant, exosystem, and compensator.
  • standard math Sylvester equation admits a unique solution when the assignability conditions hold
    Standard result invoked for the constructive procedure.

pith-pipeline@v0.9.0 · 5443 in / 1257 out tokens · 50535 ms · 2026-05-13T21:43:59.853328+00:00 · methodology

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Reference graph

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12 extracted references · 12 canonical work pages

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