Data-Driven Optimal Distributed Controller Synthesis via Spatial Regret

Alireza Karimi; Daniele Martinelli; Giancarlo Ferrari-Trecate; Luca Furieri; Vaibhav Gupta

arxiv: 2605.02506 · v1 · submitted 2026-05-04 · 📡 eess.SY · cs.SY

Data-Driven Optimal Distributed Controller Synthesis via Spatial Regret

Vaibhav Gupta , Daniele Martinelli , Giancarlo Ferrari-Trecate , Luca Furieri , Alireza Karimi This is my paper

Pith reviewed 2026-05-08 18:42 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords distributed controlspatial regretdata-driven synthesisfrequency response dataoptimal controlcommunication structurestabilitycontroller design

0 comments

The pith

An iterative algorithm synthesizes optimal distributed controllers from frequency-response data by minimizing spatial regret against a relaxed oracle.

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

The paper develops a method for designing structured distributed controllers that minimize spatial regret, the performance gap to an oracle allowed any enhanced communication topology. Relaxing the oracle structure requires an iterative procedure instead of one convex program, but the authors supply a tractable algorithm that operates directly on experimental frequency-response data. The procedure produces controllers that respect the prescribed communication pattern and maintain closed-loop stability. Readers would care because the approach enables high-performance control of networked systems using only measured data and without fixing the oracle topology in advance.

Core claim

Spatial regret measures the performance gap between a structured distributed controller and an oracle with enhanced communication topology. Relaxing assumptions so the oracle may adopt any enhanced structure leads to an iterative solution rather than a single convex program; nevertheless a tractable algorithm exists that synthesizes optimal controllers from frequency-response data while preserving stability and the desired communication structure.

What carries the argument

Spatial regret, the performance gap to an oracle with arbitrary enhanced communication; it relaxes topology constraints and motivates the iterative synthesis while the algorithm enforces structure and stability.

If this is right

Synthesis requires only experimental frequency-response data and no parametric plant model.
Stability and the prescribed communication structure are preserved throughout the iteration.
Numerical examples demonstrate better performance than classical H2 and H-infinity controllers.
The method replaces a single convex program with a convergent iterative procedure when the oracle is fully relaxed.

Where Pith is reading between the lines

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

The regret formulation may allow systematic reduction of communication links by quantifying their performance cost.
Similar iterative data-driven regret minimization could be examined for adaptive or online control settings.
The approach suggests a route to hybrid designs that blend measured data with partial model knowledge.

Load-bearing premise

The frequency-response data is rich enough to guarantee closed-loop stability and near-optimality of the iterative solution when the oracle communication structure is completely relaxed.

What would settle it

Run the algorithm on a plant using insufficiently rich or noisy frequency-response data and observe whether the resulting controller produces instability or performance far below a centralized reference design.

Figures

Figures reproduced from arXiv: 2605.02506 by Alireza Karimi, Daniele Martinelli, Giancarlo Ferrari-Trecate, Luca Furieri, Vaibhav Gupta.

**Figure 1.** Figure 1: Communication graph for Example 1. Yk ∈ ℝp×m contains the input and output at time k from one experiment, and G22(e jω) is evaluated at a discrete set of frequencies ω ∈ ΩNs ⊂ [0, π/Ts). Moreover, the synthesis process can account for estimation errors due to truncation and noise in the plant’s frequency response. G11(e jω), G12(e jω), and G21(e jω) encode the desired performance specifications and the di… view at source ↗

**Figure 2.** Figure 2: Interaction graph of the 5-bus power system model. view at source ↗

**Figure 4.** Figure 4: Plot of the squared 2-norm of view at source ↗

**Figure 5.** Figure 5: Plot of ∥zt∥ 2 as a function of time for the three controllers with a multi-sine disturbance with frequencies 8 rad/sec and 38 rad/sec on the first bus. sation and oracle topology. The cost of this generalisation is that the synthesis problem must be solved iteratively, and the completeness of the set of left-factorised distributed stabilising controllers is not guaranteed. By reformulating spatial regr… view at source ↗

read the original abstract

In this paper, we present a novel method for synthesising an optimal distributed spatial regret controller using experimentally obtained frequency-response data. Spatial regret provides a measure of the performance gap between a structured distributed controller and an oracle with enhanced communication topology. We relax assumptions on the communication topology, allowing the oracle to adopt any enhanced structure. While this generalisation requires an iterative solution in place of a single convex program, we provide a tractable algorithm that synthesises optimal controllers from frequency-response data while preserving stability and the desired communication structure. Through numerical examples, we illustrate the better performance of the spatial regret controller compared to classical H2/Hinf designs, underscoring the effectiveness of the proposed methodology.

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

The paper gives a data-driven way to synthesize distributed controllers that minimize spatial regret against a relaxed-oracle topology using frequency data and an iterative algorithm, but the stability and near-optimality claims rest on unstated richness conditions for that data.

read the letter

The main thing here is a method to design distributed controllers straight from frequency-response data by minimizing spatial regret to an oracle that can use any enhanced communication structure. This relaxation turns what used to be a single convex program into an iterative procedure, and the authors claim the algorithm stays tractable, keeps the closed loop stable, respects the desired structure, and beats standard H2/Hinf designs in their examples. That combination of relaxed oracle, spatial regret, and direct data synthesis is the actual new piece beyond the usual model-based distributed control results they cite. The numerical comparisons are a practical strength because they show concrete gains on the same plants, which makes the regret objective look useful when full communication is expensive. The soft spots sit around the data assumptions and the iteration. Stability preservation and near-optimality after the iterative steps appear to need the frequency data to be rich enough in coverage and excitation, yet the abstract gives no explicit conditions on frequency points, rank, or persistence of excitation. If those are not derived or tested in the full text, the guarantees could fail on limited or noisy real data, which undercuts the tractability claim. Convergence of the iteration and error bounds relative to the true oracle also look light on detail. This work is aimed at people doing control of networked systems who want data-driven methods without full plant models. A reader focused on distributed synthesis or regret formulations would get value from the relaxed-oracle idea and the algorithm sketch. It has enough of a distinct contribution and empirical support to deserve a serious referee, though the reviewers should press on the data conditions and convergence analysis. I would send it to peer review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a data-driven method for synthesizing optimal distributed controllers via spatial regret, using experimentally obtained frequency-response data. It relaxes the oracle communication topology to any enhanced structure (requiring an iterative algorithm in place of a single convex program) and claims to deliver a tractable procedure that preserves closed-loop stability and the desired controller structure, with numerical examples demonstrating superior performance relative to classical H2/H∞ designs.

Significance. If the stability-preservation and near-optimality guarantees hold, the work would provide a useful extension of structured optimal control to purely data-driven settings with flexible oracles, potentially aiding design of distributed controllers for systems where only frequency-response measurements are available. The numerical comparisons offer preliminary evidence of practical benefit, though the absence of explicit supporting analysis limits immediate applicability.

major comments (2)

Abstract: the central claim that the iterative algorithm 'synthesises optimal controllers from frequency-response data while preserving stability' when the oracle structure is fully relaxed rests on unstated richness conditions (e.g., frequency coverage, rank, or persistence-of-excitation requirements) on the data; without these, the stability guarantee cannot be verified and is load-bearing for the tractability assertion.
Algorithm and analysis sections: no derivation details, convergence proof for the iterative procedure, or error bounds relative to the oracle are provided, which directly undermines the claims of tractability and near-optimality under the generalized oracle.

minor comments (1)

The introduction would benefit from an explicit early definition of spatial regret and its relation to standard regret notions to improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment point by point below, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

Referee: Abstract: the central claim that the iterative algorithm 'synthesises optimal controllers from frequency-response data while preserving stability' when the oracle structure is fully relaxed rests on unstated richness conditions (e.g., frequency coverage, rank, or persistence-of-excitation requirements) on the data; without these, the stability guarantee cannot be verified and is load-bearing for the tractability assertion.

Authors: We agree that the stability preservation claim depends on suitable conditions on the frequency-response data. In the revised manuscript, we will explicitly state the required richness conditions (including sufficient frequency coverage, rank conditions, and persistence-of-excitation requirements) in both the abstract and the main text to make the assumptions transparent and verifiable. revision: yes
Referee: Algorithm and analysis sections: no derivation details, convergence proof for the iterative procedure, or error bounds relative to the oracle are provided, which directly undermines the claims of tractability and near-optimality under the generalized oracle.

Authors: We acknowledge that the current version lacks sufficient derivation details, a formal convergence proof, and error bounds. In the revision, we will expand the algorithm and analysis sections to include a complete derivation of the iterative procedure, a proof of convergence under the stated data conditions, and bounds quantifying the suboptimality gap relative to the oracle. These additions will directly support the tractability and near-optimality claims. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external frequency-response data as independent input

full rationale

The paper frames controller synthesis as solving an optimization problem whose inputs are experimentally obtained frequency-response data and whose outputs are structured controllers that minimize spatial regret relative to an oracle. No equation or step reduces the claimed optimal controller or its stability guarantee to a re-expression of the input data by construction; the iterative algorithm is presented as a tractable procedure that operates on the data without tautological fitting. Self-citations, if present, are not load-bearing for the central claim, which remains falsifiable against external closed-loop performance metrics. The method is therefore self-contained against the provided data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The method rests on the domain assumption that frequency-response data suffices for stability-certified synthesis and on the introduction of spatial regret as a performance metric; no free parameters or new physical entities are declared in the abstract.

axioms (1)

domain assumption Experimentally obtained frequency-response data is sufficient to certify stability and synthesize an optimal structured controller.
The entire synthesis pipeline operates directly on this data without an explicit plant model.

invented entities (1)

Spatial regret no independent evidence
purpose: Quantifies the performance gap between a distributed controller with prescribed communication structure and an oracle controller allowed any enhanced topology.
Introduced as the central objective; no independent falsifiable evidence outside the paper is provided in the abstract.

pith-pipeline@v0.9.0 · 5422 in / 1222 out tokens · 32165 ms · 2026-05-08T18:42:20.473436+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

min_{K∈Cstab∩K} ∥Tzw∥_{2/∞} ... Tzw := G11 + G12 K (I − G22 K)^{-1} G21
Foundation.BranchSelection branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

SpReg2(K, K̂) := max_{∥w∥2≤1} [J(w,K) − J(w,K̂)] = sup_ω λ_max(T*zw Tzw − T̂*zw T̂zw)
Foundation.AlphaCoordinateFixation J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Iterative convex relaxation: ΦΦ* ⪰ Φc Φ* + Φ Φc* − Φc Φc*, preserving closed-loop stability when initialised with stabilising Kc.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Anderson, J., Doyle, J.C., Low, S.H., and Matni, N. (2019). System level synthesis. Annual Reviews in Control, 47, 364--393

work page 2019
[2]

and Noll, D

Apkarian, P. and Noll, D. (2018). Structured H _ -control of infinite-dimensional systems. International Journal of Robust and Nonlinear Control, 28(9), 3212--3238

work page 2018
[3]

Boyd, S., Hast, M., and str \"o m, K.J. (2016). MIMO PID tuning via iterated LMI restriction. International Journal of Robust and Nonlinear Control, 26(8), 1718--1731

work page 2016
[4]

D \"o rfler, F., Jovanovi \'c , M.R., Chertkov, M., and Bullo, F. (2014). Sparsity-promoting optimal wide-area control of power networks. IEEE Transactions on Power Systems, 29(5), 2281--2291

work page 2014
[5]

Furieri, L., Zheng, Y., Papachristodoulou, A., and Kamgarpour, M. (2020). Sparsity invariance for convex design of distributed controllers. IEEE Transactions on Control of Network Systems, 7(4), 1836--1847

work page 2020
[6]

and Hassibi, B

Goel, G. and Hassibi, B. (2023). Regret-optimal estimation and control. IEEE Transactions on Automatic Control, 68(5), 3041--3053

work page 2023
[7]

Gupta, V., Klauser, E., and Karimi, A. (2026). Non-parametric IQC multipliers in data-driven robust controller synthesis. Automatica, 183, 112608

work page 2026
[8]

Hast, M., str \"o m, K., Bernhardsson, B., and Boyd, S. (2013). PID design by convex-concave optimization. In 2013 European Control Conference ( ECC ) , 4460--4465

work page 2013
[9]

and Galdos, G

Karimi, A. and Galdos, G. (2010). Fixed-order H _ controller design for nonparametric models by convex optimization. Automatica, 46(8), 1388--1394

work page 2010
[10]

and Kammer, C

Karimi, A. and Kammer, C. (2017). A data-driven approach to robust control of multivariable systems by convex optimization. Automatica, 85, 227--233

work page 2017
[11]

Madani, S.S., Kammer, C., and Karimi, A. (2021). Data- Driven Distributed Combined Primary and Secondary Control in Microgrids . IEEE Transactions on Control Systems Technology, 29(3), 1340--1347

work page 2021
[12]

Martinelli, D., Martin, A., Ferrari-Trecate, G., and Furieri, L. (2024). Closing the gap to quadratic invariance: a regret minimization approach to optimal distributed control. In 2024 European Control Conference (ECC), 756--761. IEEE

work page 2024
[13]

Martinelli, D., Martin, A., Ferrari-Trecate, G., and Furieri, L. (2025). A graph-informed regret metric for optimal distributed control. arXiv preprint arXiv:2511.14280

work page arXiv 2025
[14]

Naghnaeian, M., Voulgaris, P.G., and Elia, N. (2024). A Y oula operator state-space framework for stably implementable distributed control. IEEE Transactions on Automatic Control, 69(10), 6652--6667

work page 2024
[15]

and Schoukens, J

Pintelon, R. and Schoukens, J. (2012). System identification: a frequency domain approach. John Wiley & Sons

work page 2012
[16]

and Sorensen, N

Ren, W. and Sorensen, N. (2008). Distributed coordination architecture for multi-robot formation control. Robotics and Autonomous Systems, 56(4), 324--333

work page 2008
[17]

and Lall, S

Rotkowitz, M. and Lall, S. (2005). A characterization of convex problems in decentralized control. IEEE transactions on Automatic Control, 50(12), 1984--1996

work page 2005
[18]

Scattolini, R. (2009). Architectures for distributed and hierarchical model predictive control--a review. Journal of process control, 19(5), 723--731

work page 2009
[19]

Schuchert, P., Gupta, V., and Karimi, A. (2024). Data-driven fixed-structure frequency-based H _2 and H _ controller design. Automatica, 160, 111398

work page 2024
[20]

and Karimi, A

Schuchert, P. and Karimi, A. (2023). Achieving optimal performance with data-driven frequency-based control synthesis methods. IEEE Control Systems Letters, 7, 3567--3572. doi:10.1109/LCSYS.2023.3337592

work page doi:10.1109/lcsys.2023.3337592 2023
[21]

Witsenhausen, H.S. (1968). A counterexample in stochastic optimum control. SIAM Journal on Control, 6(1), 131--147

work page 1968
[22]

and Doyle, J.C

Zhou, K. and Doyle, J.C. (1998). Essentials of robust control, volume 104. Prentice hall Upper Saddle River, NJ

work page 1998

[1] [1]

Anderson, J., Doyle, J.C., Low, S.H., and Matni, N. (2019). System level synthesis. Annual Reviews in Control, 47, 364--393

work page 2019

[2] [2]

and Noll, D

Apkarian, P. and Noll, D. (2018). Structured H _ -control of infinite-dimensional systems. International Journal of Robust and Nonlinear Control, 28(9), 3212--3238

work page 2018

[3] [3]

Boyd, S., Hast, M., and str \"o m, K.J. (2016). MIMO PID tuning via iterated LMI restriction. International Journal of Robust and Nonlinear Control, 26(8), 1718--1731

work page 2016

[4] [4]

D \"o rfler, F., Jovanovi \'c , M.R., Chertkov, M., and Bullo, F. (2014). Sparsity-promoting optimal wide-area control of power networks. IEEE Transactions on Power Systems, 29(5), 2281--2291

work page 2014

[5] [5]

Furieri, L., Zheng, Y., Papachristodoulou, A., and Kamgarpour, M. (2020). Sparsity invariance for convex design of distributed controllers. IEEE Transactions on Control of Network Systems, 7(4), 1836--1847

work page 2020

[6] [6]

and Hassibi, B

Goel, G. and Hassibi, B. (2023). Regret-optimal estimation and control. IEEE Transactions on Automatic Control, 68(5), 3041--3053

work page 2023

[7] [7]

Gupta, V., Klauser, E., and Karimi, A. (2026). Non-parametric IQC multipliers in data-driven robust controller synthesis. Automatica, 183, 112608

work page 2026

[8] [8]

Hast, M., str \"o m, K., Bernhardsson, B., and Boyd, S. (2013). PID design by convex-concave optimization. In 2013 European Control Conference ( ECC ) , 4460--4465

work page 2013

[9] [9]

and Galdos, G

Karimi, A. and Galdos, G. (2010). Fixed-order H _ controller design for nonparametric models by convex optimization. Automatica, 46(8), 1388--1394

work page 2010

[10] [10]

and Kammer, C

Karimi, A. and Kammer, C. (2017). A data-driven approach to robust control of multivariable systems by convex optimization. Automatica, 85, 227--233

work page 2017

[11] [11]

Madani, S.S., Kammer, C., and Karimi, A. (2021). Data- Driven Distributed Combined Primary and Secondary Control in Microgrids . IEEE Transactions on Control Systems Technology, 29(3), 1340--1347

work page 2021

[12] [12]

Martinelli, D., Martin, A., Ferrari-Trecate, G., and Furieri, L. (2024). Closing the gap to quadratic invariance: a regret minimization approach to optimal distributed control. In 2024 European Control Conference (ECC), 756--761. IEEE

work page 2024

[13] [13]

Martinelli, D., Martin, A., Ferrari-Trecate, G., and Furieri, L. (2025). A graph-informed regret metric for optimal distributed control. arXiv preprint arXiv:2511.14280

work page arXiv 2025

[14] [14]

Naghnaeian, M., Voulgaris, P.G., and Elia, N. (2024). A Y oula operator state-space framework for stably implementable distributed control. IEEE Transactions on Automatic Control, 69(10), 6652--6667

work page 2024

[15] [15]

and Schoukens, J

Pintelon, R. and Schoukens, J. (2012). System identification: a frequency domain approach. John Wiley & Sons

work page 2012

[16] [16]

and Sorensen, N

Ren, W. and Sorensen, N. (2008). Distributed coordination architecture for multi-robot formation control. Robotics and Autonomous Systems, 56(4), 324--333

work page 2008

[17] [17]

and Lall, S

Rotkowitz, M. and Lall, S. (2005). A characterization of convex problems in decentralized control. IEEE transactions on Automatic Control, 50(12), 1984--1996

work page 2005

[18] [18]

Scattolini, R. (2009). Architectures for distributed and hierarchical model predictive control--a review. Journal of process control, 19(5), 723--731

work page 2009

[19] [19]

Schuchert, P., Gupta, V., and Karimi, A. (2024). Data-driven fixed-structure frequency-based H _2 and H _ controller design. Automatica, 160, 111398

work page 2024

[20] [20]

and Karimi, A

Schuchert, P. and Karimi, A. (2023). Achieving optimal performance with data-driven frequency-based control synthesis methods. IEEE Control Systems Letters, 7, 3567--3572. doi:10.1109/LCSYS.2023.3337592

work page doi:10.1109/lcsys.2023.3337592 2023

[21] [21]

Witsenhausen, H.S. (1968). A counterexample in stochastic optimum control. SIAM Journal on Control, 6(1), 131--147

work page 1968

[22] [22]

and Doyle, J.C

Zhou, K. and Doyle, J.C. (1998). Essentials of robust control, volume 104. Prentice hall Upper Saddle River, NJ

work page 1998