arxiv: 2605.07589 · v1 · submitted 2026-05-08 · 📡 eess.SY · cs.SY

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Distributionally Robust Data-Driven Predictive Control for Stochastic LTI Systems

Mirhan Urkmez , Shahab Heshmati-Alamdari

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Pith reviewed 2026-05-11 02:08 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords distributionally robust optimizationdata-driven predictive controlWasserstein ambiguity setsubspace predictive controlfinite-sample guaranteesstochastic LTI systemschance constraintsrobust control

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The pith

A distributionally robust framework uses one offline trajectory to guarantee bounds on expected cost and output constraints for unknown stochastic disturbances in linear systems.

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

The paper establishes a distributionally robust data-driven predictive control method for stochastic linear time-invariant systems with unknown dynamics and disturbances. Using data from one offline trajectory, it constructs an empirical distribution of prediction residuals from a subspace predictor and centers a Wasserstein ambiguity set on it to optimize the worst-case expected cost while enforcing probabilistic output constraints. The problem is reformulated into a tractable direct data-driven optimization without needing separate system identification. Finite-sample concentration inequalities determine the Wasserstein radius so that the guarantees hold for the true disturbance distribution with high probability. A sympathetic reader would care because this provides a practical way to achieve robust performance in uncertain stochastic environments using limited data.

Core claim

What carries the argument

Wasserstein ambiguity set centered at the empirical distribution of prediction residuals from least-squares fitting of the subspace predictive control predictor on offline data, enabling the tractable direct data-driven reformulation of the robust optimization.

If this is right

The optimization can be solved in a direct data-driven manner without identifying the underlying system matrices.
The chosen Wasserstein radius ensures high-probability bounds on the true expected cost and satisfaction of output constraints for the actual disturbance distribution.
The framework applies under different cost functions and various disturbance conditions as shown by numerical simulations.
Probabilistic constraints on system outputs are satisfied robustly over the entire ambiguity set.

Where Pith is reading between the lines

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

The direct data-driven form may reduce the need for separate identification steps in related predictive control schemes.
Similar residual-based ambiguity sets could support robustness analysis in other data-driven methods that produce prediction errors.
The radius selection rule offers a concrete way to adjust conservatism as more offline data becomes available.

Load-bearing premise

The empirical distribution of prediction residuals from a single offline trajectory is a sufficiently representative proxy for the unknown disturbance distribution so that the finite-sample concentration inequalities yield a valid Wasserstein radius.

What would settle it

A closed-loop experiment under a known disturbance distribution whose Wasserstein distance to the empirical residual distribution exceeds the selected radius, in which the realized expected cost exceeds the optimized objective or output constraints are violated, would disprove the high-probability guarantee.

Figures

Figures reproduced from arXiv: 2605.07589 by Mirhan Urkmez, Shahab Heshmati-Alamdari.

**Figure 1.** Figure 1: Parameter sweep results for εcon vs β using the DR-DDPC controller (18). Next, we compare tracking performance across varying noise levels using a sinusoidal reference yr,k = sin 2πk Trun . Under zero-mean Gaussian innovations ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Cost performance Jtest across covariance levels for different cost functions under zero-mean Gaussian innovation terms. 5 Conclusion In this work, we proposed a distributionally robust datadriven predictive control framework for stochastic LTI systems with unknown dynamics and disturbance distributions. We presented two equivalent formulations, one based on the SPC predictor and one in direct datadrive… view at source ↗

read the original abstract

We propose a distributionally robust data-driven predictive control framework for stochastic linear time-invariant systems with unknown dynamics and disturbance distributions. We use an offline trajectory to fit the subspace predictive control (SPC) predictor via least squares and construct an empirical distribution of the prediction residuals as a proxy for the unknown disturbance distribution. We then center a Wasserstein ambiguity set around this estimate and minimize the worst-case expected cost while enforcing probabilistic output constraint satisfaction over all distributions in the set. The resulting problem admits a tractable reformulation with an equivalent direct data-driven form, eliminating the need for explicit predictor identification. Using finite-sample concentration results, we provide a data-driven Wasserstein radius such that, with high probability, the true expected cost is bounded above by the tractable objective and output constraints are satisfied with respect to the true disturbance distribution. Numerical simulations validate the framework against existing methods under various disturbance conditions and cost functions.

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 combines subspace predictive control with Wasserstein DRO on single-trajectory residuals and claims tractable data-driven guarantees, but the concentration step for the radius looks vulnerable to dependence from the least-squares fit.

read the letter

This paper combines subspace predictive control with a Wasserstein ambiguity set built around prediction residuals from one offline trajectory. It fits the SPC predictor by least squares, treats the residuals as a proxy for the unknown disturbance, and then minimizes a worst-case cost over distributions in a ball of data-driven radius while enforcing output constraints in a probabilistic sense. The result is rewritten in direct data-driven form without needing explicit system matrices.

Referee Report

2 major / 2 minor

Summary. The paper proposes a distributionally robust data-driven predictive control framework for stochastic LTI systems with unknown dynamics and disturbance distributions. It fits the subspace predictive control (SPC) predictor via least squares on an offline trajectory, constructs an empirical distribution of prediction residuals as a proxy for the unknown disturbance distribution, and centers a Wasserstein ambiguity set around this estimate. The control problem minimizes the worst-case expected cost while enforcing probabilistic output constraint satisfaction over the ambiguity set. The resulting problem admits a tractable reformulation with an equivalent direct data-driven form. Finite-sample concentration results are invoked to select a data-driven Wasserstein radius guaranteeing that, with high probability, the true expected cost is bounded by the tractable objective and output constraints hold w.r.t. the true disturbance distribution.

Significance. If the finite-sample guarantees hold after accounting for dependence in the residuals, the work would provide a practical bridge between data-driven predictive control and distributionally robust optimization, with the direct data-driven reformulation and explicit finite-sample bounds as clear strengths that support implementation without full system identification.

major comments (2)

[Section deriving the data-driven Wasserstein radius and finite-sample guarantees] The central finite-sample guarantee (the data-driven Wasserstein radius selection and associated high-probability bounds on expected cost and output constraints) applies standard concentration inequalities to the empirical distribution of prediction residuals obtained after least-squares fitting of the SPC predictor on a single offline trajectory. These inequalities typically require i.i.d. draws independent of the center and exactly from the target distribution, but the residuals here are post-estimation quantities from the same data, inducing dependence and possible bias not addressed by a modified bound or citation in the manuscript. This directly undermines the high-probability claims.
[Section on tractable reformulation and direct data-driven form] The equivalence of the tractable reformulation to the original distributionally robust problem (including translation of the Wasserstein worst-case expectation and output constraints into direct data-driven form) requires explicit verification that the least-squares fitting step and multi-step prediction structure of the LTI system do not introduce additional conservatism or invalidate the reformulation steps.

minor comments (2)

[Abstract and Introduction] The abstract and introduction could more explicitly state the assumptions on the disturbance process (e.g., stationarity, independence from inputs) to clarify applicability of the concentration results.
[Numerical simulations] In the numerical simulations, include sensitivity plots or tables showing performance variation with the chosen Wasserstein radius to illustrate robustness of the data-driven selection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments on our manuscript. The points raised regarding the finite-sample guarantees and the tractable reformulation are important, and we address them point by point below. We plan to incorporate revisions to strengthen the presentation and address the concerns.

read point-by-point responses

Referee: [Section deriving the data-driven Wasserstein radius and finite-sample guarantees] The central finite-sample guarantee (the data-driven Wasserstein radius selection and associated high-probability bounds on expected cost and output constraints) applies standard concentration inequalities to the empirical distribution of prediction residuals obtained after least-squares fitting of the SPC predictor on a single offline trajectory. These inequalities typically require i.i.d. draws independent of the center and exactly from the target distribution, but the residuals here are post-estimation quantities from the same data, inducing dependence and possible bias not addressed by a modified bound or citation in the manuscript. This directly undermines the high-probability claims.

Authors: We agree that the prediction residuals are dependent due to the least-squares fitting on the same data. This is a valid observation. To address it, we will revise the manuscript to include a discussion on this dependence and provide a modified finite-sample bound that accounts for the estimation error in the SPC predictor. Specifically, we can use results from time-series analysis or add a term bounding the difference between the empirical residual distribution and the true one, ensuring the high-probability claims hold with an adjusted radius. We will cite relevant literature on concentration for dependent samples. revision: yes
Referee: [Section on tractable reformulation and direct data-driven form] The equivalence of the tractable reformulation to the original distributionally robust problem (including translation of the Wasserstein worst-case expectation and output constraints into direct data-driven form) requires explicit verification that the least-squares fitting step and multi-step prediction structure of the LTI system do not introduce additional conservatism or invalidate the reformulation steps.

Authors: The tractable reformulation is derived by substituting the data-driven predictor and expressing the worst-case expectation directly in terms of the data matrices, preserving equivalence. However, to make this explicit, we will add a detailed step-by-step verification in the supplementary material or an appendix, showing that the least-squares fitting does not introduce extra conservatism beyond the distributional robustness, and that the multi-step predictions are handled correctly via the subspace structure without invalidating the reformulation. This will confirm the direct data-driven form is equivalent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external concentration inequalities and algebraic reformulation

full rationale

The paper fits an SPC predictor via least squares on an offline trajectory and builds an empirical residual distribution, then centers a Wasserstein ball whose radius is set via cited finite-sample concentration results (external to the optimization). The tractable reformulation is presented as an algebraic equivalence that eliminates explicit model identification, not as a re-derivation of the fitted quantities themselves. No step reduces the claimed guarantee or objective to a fitted parameter by construction, and no self-citation chain is load-bearing for the central result. The derivation remains self-contained against the stated external inequalities and data-driven equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions of LTI dynamics, the existence of a finite offline trajectory, and the applicability of Wasserstein DRO concentration inequalities; no new entities are postulated and the only free parameter is the data-driven radius derived from concentration bounds.

axioms (2)

domain assumption The system is linear time-invariant and the disturbance is stationary.
Invoked when the SPC predictor is fitted and when the empirical residuals are treated as samples from the unknown disturbance distribution.
standard math Finite-sample concentration inequalities for Wasserstein distance apply to the empirical residual distribution.
Used to select the radius that delivers the high-probability bounds on true cost and constraints.

pith-pipeline@v0.9.0 · 5456 in / 1569 out tokens · 24569 ms · 2026-05-11T02:08:00.609604+00:00 · methodology

discussion (0)

Reference graph

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