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arxiv: 2605.13529 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.SY

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· Lean Theorem

Decentralized Frequency-Domain Conditions for D-Stability with Application to DC Microgrids

Zelin Sun , Shanshan Jiang , Xiaoyu Peng , Xiang Zhu , Xiuqiang He , Hua Geng
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Pith reviewed 2026-05-14 17:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords decentralizedmathcalmethodstabilitycommunicationfrequency-domainmicrogridsmodels
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The pith

Decentralized local frequency-domain criteria guarantee D-stability in networked linear systems and enable broadcastable parameter synthesis for DC microgrids.

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

The authors map any desired pole region D to an auxiliary left-half plane stability problem. They then introduce positive functions that certify stability for the resulting complex-coefficient system. Because the criteria depend only on local frequency responses, each subsystem can verify its contribution without knowing other models or exchanging data. For DC microgrids the method adds a loop transformation that moves the stability burden onto a form that can be expressed as a simple grid code. This code is broadcast once; each converter then tunes its own parameters locally while the overall network remains D-stable. Numerical examples illustrate the approach on typical microgrid topologies.

Core claim

We prove that D-stability is guaranteed via local frequency-domain criteria without requiring shared subsystem models or inter-subsystem communication.

Load-bearing premise

The target region D can be mapped to an auxiliary left-half plane such that positive functions exist to certify the resulting complex-coefficient dynamics; this mapping and function construction must hold for the specific network topology.

Figures

Figures reproduced from arXiv: 2605.13529 by Hua Geng, Shanshan Jiang, Xiang Zhu, Xiaoyu Peng, Xiuqiang He, Zelin Sun.

Figure 1
Figure 1. Figure 1: (a) Half plane D0(θ0, ω0, σ0), D0(−θ0, −ω0, σ0) and (b) their intersection D(θ0, ω0, σ0). -𝑗𝑗 Im 0 Re 𝑠-plane 𝑗 Im 0 𝑗𝑗 Im Re 𝑗 -𝑗 0 (a) (b) (c) Re 𝒟LHP 𝑗 𝒟SEC 𝑗 𝒟HS 𝑗 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Pole regions for various dynamic performance. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mapping D0 in the s-domain to the cLHP in the ν-domain. property, the subsequent analysis focuses on the single sym￾metric region D in (5). With the D-stability problem transformed into a standard cLHP stability problem in the ν-domain, we aim to use positive realness (PR) theory for decentralized certification. PR provides a compositional framework where the stability of an interconnected system can be de… view at source ↗
Figure 4
Figure 4. Figure 4: Loop transformation. distinction. While PR requires M(ν) to be a real matrix when ν is real positive [14], [22], [23], the mapped Gˆ k(ν) fails this due to its complex coefficients. A rational function satisfying the conditions in Lem. 1 without the real-coefficient constraint is formally termed a positive function in classical network theory [24]. Although both concepts are well-documented, to the best of… view at source ↗
Figure 5
Figure 5. Figure 5: (a) DCMG configuration. (b) Loop transformation via virtual admit [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Port converters and controllers for (a) ESS, (b) PV, and (c) CPL. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Under default parameters: (a) Distribution of the dominant closed [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

This paper proposes a decentralized method for regional pole placement, or $\mathcal{D}$-stability, in linearized networked systems. Existing LMI-based methods are hindered by confidentiality concerns regarding proprietary subsystem models and the absence of communication infrastructures. To overcome these barriers, we map the target region $\mathcal{D}$ of pole placement to an auxiliary left-half plane and introduce positive functions to handle the resulting complex-coefficient dynamics. We prove that $\mathcal{D}$-stability is guaranteed via local frequency-domain criteria without requiring shared subsystem models or inter-subsystem communication. This method is then tailored to DC microgrids, where a loop transformation is utilized to reallocate the burden of stability certification, deriving a broadcastable grid code for decentralized parameter synthesis. Numerical examples verify the efficacy of the proposed method.

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 manuscript proposes a decentralized frequency-domain method for D-stability (regional pole placement) in linearized networked systems. It maps the target region D to an auxiliary left-half plane, introduces positive functions to certify stability of the resulting complex-coefficient dynamics, and proves that local frequency-domain criteria suffice without shared subsystem models or communication. The approach is specialized to DC microgrids via a loop transformation that yields a broadcastable grid code for decentralized parameter synthesis, with numerical examples verifying performance.

Significance. If the central claims hold, the work offers a practical alternative to centralized LMI methods for systems where model confidentiality and lack of communication are constraints, such as DC microgrids. The derivation of a broadcastable grid code and the frequency-domain decentralization are notable strengths for implementation. The approach is general for networked linear systems and could enable scalable stability certification in power electronics and similar domains.

major comments (2)
  1. [Section 3 (mapping and positive-function construction)] The central proof relies on the existence and construction of positive functions that certify stability for the complex-coefficient dynamics after the D-to-LHP mapping. The manuscript does not provide explicit constructions, error bounds, or verification steps for these functions in the general case (see the derivation following the mapping step and the statement of the local criteria). This is load-bearing for the claim that D-stability is guaranteed via local criteria alone.
  2. [Section 5 (DC microgrid specialization)] In the DC microgrid application, the loop transformation reallocates the stability burden, but the resulting broadcastable grid code's dependence on the specific network topology and the handling of complex coefficients in the frequency-domain test are not fully detailed with respect to robustness margins or parameter ranges.
minor comments (2)
  1. [Section 2] Notation for the positive functions and the auxiliary LHP mapping could be clarified with a dedicated table or diagram to improve readability for readers unfamiliar with the complex-coefficient extension.
  2. [Section 6] The numerical examples would benefit from explicit comparison metrics (e.g., pole locations before/after synthesis) against a centralized LMI baseline to quantify the decentralization benefit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Section 3 (mapping and positive-function construction)] The central proof relies on the existence and construction of positive functions that certify stability for the complex-coefficient dynamics after the D-to-LHP mapping. The manuscript does not provide explicit constructions, error bounds, or verification steps for these functions in the general case (see the derivation following the mapping step and the statement of the local criteria). This is load-bearing for the claim that D-stability is guaranteed via local criteria alone.

    Authors: The manuscript establishes existence of the positive functions via the D-to-LHP mapping and derives the local criteria from that assumption. We acknowledge that explicit constructions, error bounds, and verification steps for the general case are not provided. In the revision we will add a dedicated subsection detailing constructive procedures for these functions, including verification steps and error bounds to support the local criteria. revision: yes

  2. Referee: [Section 5 (DC microgrid specialization)] In the DC microgrid application, the loop transformation reallocates the stability burden, but the resulting broadcastable grid code's dependence on the specific network topology and the handling of complex coefficients in the frequency-domain test are not fully detailed with respect to robustness margins or parameter ranges.

    Authors: The loop transformation is constructed precisely so that the resulting grid code depends only on local parameters and can be broadcast without topology-specific information. Complex coefficients are incorporated directly into the positive-function frequency-domain test. We agree that robustness margins and explicit parameter ranges merit further elaboration; we will expand Section 5 with sensitivity analysis to topology variations and concrete bounds on the admissible parameter sets. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions identified

full rationale

The paper maps the target D-region to an auxiliary LHP, constructs positive functions for the resulting complex-coefficient system, and derives local frequency-domain criteria from these steps. No equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation. The decentralization result follows directly from the mapping and positive-function certification without re-using the target stability property as an input. The approach is presented as a general proof for linearized networked systems, with the DC-microgrid application obtained via an explicit loop transformation; both steps are independent of the final stability certificate.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a suitable mapping from the D-region to an auxiliary left-half plane and on the construction of positive functions that certify stability for the resulting complex-coefficient system; these are domain assumptions in frequency-domain control theory rather than new entities.

axioms (2)
  • domain assumption Any target region D can be mapped to an auxiliary left-half plane while preserving the stability certification problem
    Invoked to convert regional pole placement into a standard stability question amenable to frequency-domain analysis.
  • ad hoc to paper Positive functions exist that certify stability for the complex-coefficient dynamics obtained after the mapping
    Introduced in the abstract to handle the non-real coefficients that arise from the transformation.

pith-pipeline@v0.9.0 · 5443 in / 1299 out tokens · 51059 ms · 2026-05-14T17:58:09.416183+00:00 · methodology

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Reference graph

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