Robust Distributed Sub-Optimal Coordination of Linear Agents with Uncertain Input Nonlinearities

Takumi Namba

arxiv: 2604.17934 · v1 · submitted 2026-04-20 · 📡 eess.SY · cs.SY

Robust Distributed Sub-Optimal Coordination of Linear Agents with Uncertain Input Nonlinearities

Takumi Namba This is my paper

Pith reviewed 2026-05-10 04:23 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords distributed coordinationsub-optimal optimizationinput nonlinearitiesrobust controlsector conditionsmatrix inequalitiesmulti-agent systemslinear agents

0 comments

The pith

A distributed control protocol steers linear agents with uncertain input nonlinearities to a neighborhood of the global optimizer by unifying nonlinearities and gradients through sector conditions.

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

This paper examines how linear agents connected over a network can approach the solution of a shared optimization problem even when their control inputs are distorted by unknown nonlinearities. The authors introduce a new protocol and recast the convergence question as a robust stability problem, folding both the input distortions and the objective-function gradients into sector bounds that are analyzed together. They then extract sufficient conditions for success in the form of solvable matrix inequalities. The result matters for multi-agent applications where perfect linearity cannot be assumed and the optimization goal is distributed across the network.

Core claim

We study robust distributed sub-optimal coordination of linear agents subject to input nonlinearities. We formulate a bounded distributed sub-optimal coordination problem in which each agent converges to a neighborhood of the optimizer of a global optimization problem defined over a communication network. We propose a novel control protocol and analyze convergence by employing a robust control approach in which both the input nonlinearities and the gradients of the objective functions are treated in a unified manner via sector conditions. We derive sufficient conditions for the solvability of the considered problem and characterize them in terms of matrix inequalities.

What carries the argument

A novel distributed control protocol whose stability is certified by a robust-control analysis that treats input nonlinearities and objective gradients uniformly through sector conditions, yielding matrix-inequality solvability criteria.

If this is right

The matrix inequalities supply a practical test that certifies whether a given network and set of sector bounds admit sub-optimal coordination.
Convergence occurs to a neighborhood whose size is governed by the sector bounds rather than to the exact optimizer.
The same sector-based treatment covers both actuator nonlinearities and the shape of the cost functions.
The protocol remains distributed, using only neighbor information over the given communication graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sector-condition approach could be reused for other bounded uncertainties, such as time delays or switching topologies, provided they also admit sector representations.
The framework suggests a route to sub-optimal coordination for agents whose dynamics are themselves nonlinear, if those dynamics can be absorbed into enlarged sector bounds.
Numerical verification of the matrix inequalities on realistic network sizes would indicate how conservative the sufficient conditions are in practice.

Load-bearing premise

Both the uncertain input nonlinearities and the gradients of the objective functions obey sector bounds that allow them to be handled together in a single robust-control framework.

What would settle it

A simulation or analytic counterexample in which the sector bounds hold and the matrix inequalities are satisfied yet the agents do not enter a neighborhood of the optimizer, or a case where the bounds are violated and the protocol still succeeds.

Figures

Figures reproduced from arXiv: 2604.17934 by Takumi Namba.

**Figure 1.** Figure 1: The topology of the graph  This nonlinearity satisfies the sector condition with 𝛼 = 0.6, 𝛽 = 1, and we set 𝛾 = 0.5, which is a conservative estimation of the uncertainties. The communication is executed along the graph  depicted in [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

**Figure 2.** Figure 2: Trajectories of 𝑥𝑖 (𝑡) 0 5 10 15 20 25 30 10-2 10-1 100 101 (a) Trajectories of ‖𝑥 − 1𝑁 ⊗ 𝑥⋆‖ 0 5 10 15 20 25 30 10-2 10-1 100 101 102 (b) Trajectories of |𝑓(𝑥) − 𝑓(𝑥 ⋆)| References [1] T. Yang, X. Yi, J. Wu, Y. Yuan, D. Wu, Z. Meng, Y. Hong, H. Wang, Z. Lin, K. H. Johansson, A survey of distributed optimization, Annual Reviews in Control 47 (2019) 278–305. [2] J. Wang, N. Elia, A control perspective for c… view at source ↗

read the original abstract

In this paper, we study robust distributed sub-optimal coordination of linear agents subject to input nonlinearities. Inspired by the robust agreement literature, we formulate a bounded distributed sub-optimal coordination problem, in which each agent converges to a neighborhood of the optimizer of a global optimization problem defined over a communication network. We propose a novel control protocol, and analyze convergence by employing a robust control approach, in which both the input nonlinearities and the gradients of the objective functions are treated in a unified manner via sector conditions. In particular, we derive sufficient conditions for the solvability of the considered problem and characterize them in terms of matrix inequalities. The effectiveness of the proposed method is demonstrated through a numerical simulation.

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.

Desk Editor's Note private letter to a colleague

LMI conditions for distributed sub-optimal coordination of linear agents with input nonlinearities via unified sector bounds, but the bounds must be known in advance.

read the letter

The main takeaway is that this paper gives a distributed protocol for linear multi-agent systems with uncertain input nonlinearities to converge to a neighborhood of the global optimizer of a networked cost function. They treat both the nonlinearities and the objective gradients as sector-bounded uncertainties in one robust-control analysis and turn that into sufficient LMI conditions for the problem to be solvable. A simulation example shows the agents settling near the optimum under the protocol. That unified sector treatment is the clearest new piece; it avoids handling the two sources of uncertainty separately and builds directly on existing robust agreement results. The matrix-inequality characterization is straightforward to state and the approach stays within standard Lyapunov-style arguments for distributed systems. The contribution is incremental rather than foundational, but the formulation of bounded sub-optimal coordination is cleanly posed. The soft spot is exactly the one flagged in the stress test: the sector slopes have to be known and valid a priori. The paper offers no procedure to compute or over-bound them from data or models when the nonlinearity is truly uncertain (saturation level, dead-zone width, etc.). Without that step the LMIs cannot be instantiated for a real system, so the robustness guarantee remains conditional on an unstated modeling assumption. The abstract claims convergence analysis, but the full derivations are not visible here, so it is hard to judge how conservative the bounds end up or whether the neighborhood size is characterized tightly. This work is for control engineers who already work with multi-agent coordination and actuator uncertainties and who are comfortable with LMI design. A reader looking for a practical extension of robust agreement techniques to approximate distributed optimization would find the protocol and conditions useful. I would bring the paper to a reading group to walk through the LMIs and the simulation details. I would not cite it in my own work in the next year because the advance is modest and the practicality gap on bound selection is real. A serious editor should still send it to peer review; the problem is well-posed, the method is grounded in established tools, and the gaps are fixable with revisions rather than fatal.

Referee Report

2 major / 2 minor

Summary. The paper studies robust distributed sub-optimal coordination for linear agents subject to uncertain input nonlinearities. It formulates a bounded coordination problem where agents converge to a neighborhood of the global optimizer of a networked sum of local costs. A novel distributed protocol is proposed and convergence is analyzed via a robust-control Lyapunov approach that treats both the input nonlinearities and the cost gradients as sector-bounded uncertainties in a unified manner. Sufficient conditions for solvability are derived and expressed as linear matrix inequalities; effectiveness is illustrated by a numerical simulation.

Significance. If the LMI conditions are feasible, the unified sector-bound treatment provides a systematic LMI-based design method for distributed approximate optimization under actuator nonlinearities, extending ideas from robust agreement to optimization settings. This could be useful for networked systems with uncertain actuators. The approach is technically standard but the unification of nonlinearity and gradient sectors is a modest positive contribution; practical impact is limited by the requirement that sector parameters be known.

major comments (2)

[Problem statement and main LMI conditions (likely §3–4)] Problem statement and main LMI conditions (likely §3–4): The sector bounds on the uncertain input nonlinearities are taken as known a priori and appear directly as parameters in the derived matrix inequalities. No procedure, over-bound, or data-driven method is supplied for obtaining or conservatively estimating these bounds when the nonlinearity (e.g., saturation level, dead-zone width) is truly uncertain. Because the feasibility of the LMIs and the size of the convergence neighborhood depend explicitly on these parameters, the sufficient conditions cannot be instantiated for real systems without an additional modeling step that the manuscript does not address.
[Convergence analysis] Convergence analysis: The neighborhood radius to which agents converge is characterized in terms of the sector parameters and the sub-optimality tolerance, but the manuscript does not provide an explicit scaling law or sensitivity result showing how the radius grows with increasing uncertainty in the input nonlinearity. Adding such a remark or corollary after the main theorem would strengthen the claim that the method is robust.

minor comments (2)

[Numerical simulation] The numerical example would be more informative if the specific LMI solver, feasibility margin, and condition number of the solved matrices were reported, allowing readers to assess numerical reliability.
[Preliminaries] Notation for the communication graph (Laplacian, adjacency matrix) should be introduced once in the preliminaries and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the practical aspects of our robust control design. We address each major comment below and will revise the manuscript accordingly to improve its completeness.

read point-by-point responses

Referee: Problem statement and main LMI conditions (likely §3–4): The sector bounds on the uncertain input nonlinearities are taken as known a priori and appear directly as parameters in the derived matrix inequalities. No procedure, over-bound, or data-driven method is supplied for obtaining or conservatively estimating these bounds when the nonlinearity (e.g., saturation level, dead-zone width) is truly uncertain. Because the feasibility of the LMIs and the size of the convergence neighborhood depend explicitly on these parameters, the sufficient conditions cannot be instantiated for real systems without an additional modeling step that the manuscript does not address.

Authors: We agree that the sector bounds are assumed known a priori, as is standard in the sector-bounded uncertainty framework used throughout the robust control literature. For common actuator nonlinearities such as saturation or dead-zones, these parameters are typically available from manufacturer specifications or can be conservatively over-estimated via simple experiments. To address the concern, we will insert a short remark in the problem statement section explaining how to select such bounds in practice, thereby providing the missing modeling guidance while preserving the LMI-based synthesis as the main contribution. revision: yes
Referee: Convergence analysis: The neighborhood radius to which agents converge is characterized in terms of the sector parameters and the sub-optimality tolerance, but the manuscript does not provide an explicit scaling law or sensitivity result showing how the radius grows with increasing uncertainty in the input nonlinearity. Adding such a remark or corollary after the main theorem would strengthen the claim that the method is robust.

Authors: We concur that an explicit sensitivity result would better highlight the robustness properties. We will add a corollary right after the main theorem that derives an explicit upper bound on the convergence neighborhood radius and shows its linear scaling with the sector parameters of the input nonlinearities. This addition will make the dependence on uncertainty transparent without requiring changes to the proof technique. revision: yes

Circularity Check

0 steps flagged

No circularity: standard sector-bound robust control applied to new coordination problem

full rationale

The derivation begins from a standard problem formulation of distributed sub-optimal coordination, proposes a protocol, and applies existing robust-control sector conditions simultaneously to input nonlinearities and cost gradients to obtain LMI feasibility conditions. No equation reduces to a prior fitted parameter or self-definition; the matrix inequalities are obtained via standard Lyapunov analysis or S-procedure on the closed-loop dynamics. No self-citations are invoked as load-bearing uniqueness theorems, and the sector bounds are treated as given modeling assumptions rather than derived outputs. The chain is therefore self-contained against external robust-control benchmarks and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from robust control and multi-agent coordination literature with no new free parameters or invented entities introduced.

axioms (2)

domain assumption Input nonlinearities and objective function gradients satisfy sector conditions.
Central to the unified robust control treatment of uncertainties.
domain assumption The communication network is connected and permits distributed protocol implementation.
Required for the coordination problem over the network.

pith-pipeline@v0.9.0 · 5408 in / 1392 out tokens · 61953 ms · 2026-05-10T04:23:26.910090+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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F. Zhang, H. L. Trentelman, J. M. A. Scherpen, Dynamic feedback synchronization of lur’e networks via incremental sector boundedness, IEEE Transactions on Automatic Control 61 (2016) 2579–2584

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