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arxiv: 2502.09131 · v2 · submitted 2025-02-13 · 📡 eess.SY · cs.SY· math.OC

A Stochastic Fundamental Lemma with Reduced Disturbance Data Requirements

Ruchuan Ou , Guanru Pan , Timm Faulwasser This is my paper

Pith reviewed 2026-05-23 03:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords stochastic fundamental lemmadata-driven controlpolynomial chaos expansionsLTI systemsHankel matricesprocess disturbancescausality in stochastic control
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The pith

A variant of the stochastic fundamental lemma predicts future trajectories of stochastic LTI systems without past disturbance data in the Hankel matrices.

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

This paper proposes a modified stochastic fundamental lemma for linear time-invariant systems with random disturbances. The key advance is that predictions of future input and output trajectories can be made using Hankel matrices built only from input and output data, without including disturbance records. It achieves this by using polynomial chaos expansions to represent uncertainty and applying causality ideas from stochastic control, assuming the disturbance distribution is known. Readers interested in data-driven methods would care because it lowers the barrier for applying these techniques when disturbance measurements are not available or costly to collect. The paper demonstrates the idea with a numerical example in a control setting.

Core claim

By combining polynomial chaos expansions with causality concepts of stochastic control, the authors establish a stochastic fundamental lemma whose data matrices require only input-output trajectories, yet still enable accurate prediction of future behavior for systems subject to process disturbances whose distribution is known.

What carries the argument

Stochastic fundamental lemma variant using polynomial chaos expansions to eliminate disturbance data from Hankel matrices.

Load-bearing premise

The disturbance distribution must be known in advance and polynomial chaos expansions must sufficiently capture the stochastic dynamics.

What would settle it

Run a simulation of a stochastic LTI system, apply the lemma with and without disturbance data in the matrices, and check if the prediction error increases significantly when disturbance data is omitted.

Figures

Figures reproduced from arXiv: 2502.09131 by Guanru Pan, Ruchuan Ou, Timm Faulwasser.

Figure 1
Figure 1. Figure 1: Evolution of the PDFs of the outputs Y 1 and Y 2 over horizon N = 25. Deep blue-dashed line: Chance constraint. U(−0.8, 0.8), where Wi denotes the i-th element of W. Then we solve the following OCP min Uk ,Yk ,Gwk ,g,k∈I[1,N] X N k=1 E[Y ⊤ k QYk + U ⊤ k RUk] s.t. dynamics in form of Lemma 5, P[−0.349 ≤ Y 1 k ≤ 0.349] ≥ 0.8, k ∈ I[2,N] in the PCE framework. The weighting matrices in the stage cost are Q = d… view at source ↗
Figure 2
Figure 2. Figure 2: Aircraft example with Gaussian disturbance. Red-solid line: [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

Recently, the fundamental lemma by Willems et al. has been extended towards stochastic LTI systems subject to process disturbances. Using this lemma requires previously recorded data of inputs, outputs, and disturbances. In this paper, we exploit causality concepts of stochastic control to propose a variant of the stochastic fundamental lemma that does not require past disturbance data in the Hankel matrices. Our developments rely on polynomial chaos expansions and on the knowledge of the disturbance distribution. Similar to our previous results, the proposed variant of the fundamental lemma allows to predict future input-output trajectories of stochastic LTI systems. We draw upon a numerical example to illustrate the proposed variant in data-driven control context.

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 paper proposes a variant of the stochastic fundamental lemma for LTI systems subject to process disturbances. By combining causality concepts from stochastic control with polynomial chaos expansions (PCE) and an a priori known disturbance distribution, the approach removes the requirement for past disturbance trajectories in the data Hankel matrices while still enabling prediction of future input-output trajectories from input-output data alone. The claim is illustrated via a numerical example in a data-driven control setting.

Significance. If the derivation holds, the result meaningfully lowers the data-collection cost for stochastic data-driven control by replacing recorded disturbance samples with exact distributional knowledge (via PCE). This is potentially useful when disturbances are hard to measure directly but their statistics are available from first principles. The numerical example provides a concrete demonstration of the method in a control context, which strengthens the practical relevance.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (developments): The central claim that input-output Hankel matrices alone suffice for trajectory prediction rests on the assumption that the disturbance distribution is known exactly so that PCE coefficients can be computed without samples. The manuscript should provide a precise statement (e.g., a theorem or corollary) quantifying how the column-space guarantee degrades under distribution misspecification or finite-sample estimation of the distribution; without this, the claimed reduction in data requirements is conditional rather than unconditional.
  2. [Numerical example (§5)] Numerical example (§5): The example does not report whether the disturbance distribution used for PCE is the exact analytic form or an estimate obtained from additional data. If any estimation step is performed, the experiment does not isolate the claimed benefit of removing disturbance data from the Hankel matrices and therefore does not fully support the central claim.
minor comments (2)
  1. Notation for the reduced Hankel matrix and the PCE truncation order should be introduced with explicit definitions before their first use to improve readability.
  2. The relationship to the authors' prior stochastic fundamental lemma results could be stated more explicitly (e.g., which assumptions are relaxed and which are retained) to clarify the incremental contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript to improve clarity on assumptions and the example.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (developments): The central claim that input-output Hankel matrices alone suffice for trajectory prediction rests on the assumption that the disturbance distribution is known exactly so that PCE coefficients can be computed without samples. The manuscript should provide a precise statement (e.g., a theorem or corollary) quantifying how the column-space guarantee degrades under distribution misspecification or finite-sample estimation of the distribution; without this, the claimed reduction in data requirements is conditional rather than unconditional.

    Authors: The developments and central claim are derived under the explicit assumption of an exactly known disturbance distribution, as stated in the abstract and §3; the data reduction is therefore conditional on this premise, which is a core feature of the approach using PCE. We will revise the abstract and add a clarifying remark in §3 to emphasize the conditional nature of the guarantees. A full quantification of degradation under misspecification lies outside the paper's scope focused on the exact-distribution case. revision: partial

  2. Referee: [Numerical example (§5)] Numerical example (§5): The example does not report whether the disturbance distribution used for PCE is the exact analytic form or an estimate obtained from additional data. If any estimation step is performed, the experiment does not isolate the claimed benefit of removing disturbance data from the Hankel matrices and therefore does not fully support the central claim.

    Authors: The numerical example uses the exact analytic form of the disturbance distribution to compute the PCE coefficients, with no estimation from additional data. We will revise §5 to explicitly state this, thereby isolating the benefit of removing disturbance data from the Hankel matrices. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior results; derivation relies on explicit external assumptions of known disturbance distribution and PCE without reduction to fit or self-definition.

full rationale

The paper extends the stochastic fundamental lemma by invoking causality concepts and polynomial chaos expansions under the assumption of known disturbance distribution to eliminate disturbance data from Hankel matrices. The phrase 'Similar to our previous results' references prior work by the authors but does not bear the load of the central claim, which is the reduced-data variant itself. No equations or steps in the provided abstract reduce a prediction to a fitted parameter or import uniqueness via self-citation chain. The knowledge-of-distribution assumption is stated as an input rather than derived within the paper, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The central claim rests on unstated details of polynomial chaos expansions and disturbance distribution knowledge.

pith-pipeline@v0.9.0 · 5640 in / 1064 out tokens · 18122 ms · 2026-05-23T03:41:16.541890+00:00 · methodology

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Reference graph

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