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arxiv: 2605.19003 · v1 · pith:MXXEGCPBnew · submitted 2026-05-18 · 🧮 math.OC · cs.SY· eess.SY

Scalable iterative Gramian synthesis for control-affine systems

Zongxi Yu , Cyprien Tamekue , Ruiqi Chen , ShiNung Ching This is my paper

Pith reviewed 2026-05-20 08:44 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords nonlinear control synthesisGramian-based controlcontrol-affine systemsiterative methodsscalable algorithmsminimum energy controlhigh-dimensional systemsunderactuated control
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The pith

A new iterative scheme scales nonlinear Gramian-based control synthesis to high-dimensional systems.

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

The paper introduces a scalable implementation of nonlinear Gramian-based control synthesis for control-affine systems. It tackles the main computational bottlenecks in the iterative synthesis map formulations to deliver rapid convergence and high precision. This is shown through applications to five canonical nonlinear systems and 100-dimensional recurrent neural network models, including underactuated cases. The results suggest that convergence is driven by the system's nonlinearity and controllability rather than its dimensionality, offering a route to apply nonlinear control theory in large-scale settings.

Core claim

By addressing key computational bottlenecks in iterative synthesis map formulations, the authors develop a computational scheme for nonlinear Gramian-based minimum energy control that exhibits rapid convergence and high precision, effective for both low-dimensional canonical systems and high-dimensional models up to 100 states.

What carries the argument

Iterative synthesis map formulation modified to resolve computational bottlenecks for scalable nonlinear Gramian synthesis.

If this is right

  • Rapid convergence is achieved across tested canonical and high-dimensional systems.
  • High precision holds for underactuated as well as fully actuated control-affine systems.
  • Convergence rate depends primarily on intrinsic properties like nonlinearity and controllability.
  • State-space dimensionality has limited impact on performance up to at least 100 dimensions.

Where Pith is reading between the lines

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

  • Similar bottleneck resolutions could be applied to other iterative control design methods for scalability.
  • Real-time implementation in engineering applications like autonomous vehicles or biological modeling may become feasible.
  • Further tests on systems beyond 100 dimensions or with varying degrees of controllability could confirm the scaling behavior.

Load-bearing premise

The computational bottlenecks of iterative synthesis maps for nonlinear Gramians can be addressed by targeted modifications to produce fast and precise results in high dimensions.

What would settle it

A high-dimensional control-affine system exhibiting slow convergence or poor precision despite moderate nonlinearity would indicate the method does not fully overcome the bottlenecks as claimed.

Figures

Figures reproduced from arXiv: 2605.19003 by Cyprien Tamekue, Ruiqi Chen, ShiNung Ching, Zongxi Yu.

Figure 1
Figure 1. Figure 1: End-point error convergence for benchmark systems [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Convergence of 100D mindy with 50D input [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fully-actuated 2D Hopfield Example mechanism: both Gramian syntheses maintain high cosine similarity between Bt(x)u(t) and Nt(x), indicating that the synthesized control aligns better with the natural drift. D. Controlling Underactuated 100D MINDy via Minimunm Energy Synthesis We additionally evaluate the synthesis on an underactuated 100D MINDy model with k = 50 input channels, a challeng￾ing example. The… view at source ↗
Figure 5
Figure 5. Figure 5: Runtime benchmark of General synthesis. The x-axis [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

This article presents a scalable implementation of nonlinear Gramian-based control synthesis for control-affine systems, including a minimum energy control construction. These synthesis advances are achieved by addressing key computational bottlenecks inherent to iterative synthesis map formulations, yielding a computational scheme that exhibits rapid convergence and high-precision. The efficacy of this synthesis framework is demonstrated across five canonical nonlinear control systems and 100-dimensional recurrent neural network models, including underactuated systems. Empirical scaling results further indicate that convergence is primarily governed by intrinsic system properties, such as nonlinearity and controllability, rather than by state-space dimensionality. This work provides a practical, scalable computational pathway for translating rigorous nonlinear synthesis theory into high-dimensional control applications.

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

1 major / 1 minor

Summary. The paper presents a scalable implementation of nonlinear Gramian-based control synthesis for control-affine systems, including a minimum energy control construction. Key computational bottlenecks in iterative synthesis map formulations are addressed to yield a scheme with rapid convergence and high precision. Efficacy is demonstrated on five canonical nonlinear control systems and 100-dimensional recurrent neural network models (including underactuated cases). Empirical scaling results are reported to indicate that convergence is governed primarily by intrinsic system properties such as nonlinearity and controllability, rather than state-space dimensionality, offering a practical pathway for high-dimensional applications.

Significance. If the scalability and dimension-independence claims are substantiated, the work would provide a valuable computational bridge between rigorous nonlinear Gramian synthesis theory and high-dimensional practical control problems. The demonstrations on 100-dimensional models and the focus on underactuated systems represent a concrete advance toward translating theoretical results into applications involving large-scale or learned dynamics.

major comments (1)
  1. Abstract and empirical scaling results: The central claim that 'convergence is primarily governed by intrinsic system properties, such as nonlinearity and controllability, rather than by state-space dimensionality' is not supported by controlled scaling experiments. Results are shown for five low-dimensional canonical systems plus separate 100-dimensional RNN examples, but no systematic study is described in which identical underlying dynamics are embedded into increasing state dimensions (e.g., via state augmentation or RNN width variation) while tracking iteration count, residual error, and wall-clock time. Without such isolation, the observed rapid convergence on the 100-dim cases cannot be confidently attributed to dimension independence rather than specific choices of controllability margins or model construction. This directly underpins the headline claim of a 'practical, scalable'
minor comments (1)
  1. The abstract states 'high-precision' results but the provided summary does not reference quantitative error bounds, residual norms, or convergence rate tables; these should be explicitly reported with respect to the synthesis map iterations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address the major comment regarding the empirical support for our scalability claims below, and we commit to revisions that strengthen the presentation without overstating the results.

read point-by-point responses

  1. Referee: Abstract and empirical scaling results: The central claim that 'convergence is primarily governed by intrinsic system properties, such as nonlinearity and controllability, rather than by state-space dimensionality' is not supported by controlled scaling experiments. Results are shown for five low-dimensional canonical systems plus separate 100-dimensional RNN examples, but no systematic study is described in which identical underlying dynamics are embedded into increasing state dimensions (e.g., via state augmentation or RNN width variation) while tracking iteration count, residual error, and wall-clock time. Without such isolation, the observed rapid convergence on the 100-dim cases cannot be confidently attributed to dimension independence rather than specific choices of controllability margins or model construction. This directly underpins the headline claim of a 'practical, scalable'

    Authors: We agree that a more controlled scaling study isolating dimensionality while holding underlying dynamics fixed would provide stronger evidence for the dimension-independence claim. Our current experiments demonstrate rapid convergence on both low-dimensional canonical systems and separately constructed 100-dimensional RNN models (including underactuated cases), with iteration counts and residuals appearing driven by nonlinearity and controllability margins rather than dimension. However, these do not constitute a single controlled family with systematic state augmentation or RNN width sweeps. To address this, we will revise the abstract and the empirical scaling discussion to qualify the claim as 'suggestive of dimension-independent behavior for the tested system classes' rather than a general assertion. We will also add a new set of controlled experiments (e.g., state-augmented versions of one canonical system and RNN width variation) tracking iteration count, residual, and runtime; these will be included in the revised manuscript and supplementary material. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and empirical claims are self-contained

full rationale

The paper introduces algorithmic improvements to iterative Gramian synthesis for control-affine systems by targeting specific computational bottlenecks, then validates rapid convergence and scaling behavior through direct numerical tests on five canonical nonlinear systems plus 100-dimensional RNN models. No derivation step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; the headline empirical observation that convergence depends primarily on intrinsic properties is presented as an outcome of those independent benchmark runs rather than an input that is renamed or presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard domain assumptions from nonlinear control theory for control-affine systems and iterative Gramian constructions; no free parameters or new invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The systems under consideration are control-affine
    Explicitly stated in the title and abstract as the class of systems for which the synthesis applies.

pith-pipeline@v0.9.0 · 5649 in / 1326 out tokens · 46823 ms · 2026-05-20T08:44:02.905181+00:00 · methodology

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Reference graph

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