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arxiv: 2604.13707 · v1 · submitted 2026-04-15 · 📡 eess.SY · cs.SY

Data-Driven Probabilistic Finite mathcal{L}₂-Gain Stabilization of Stochastic Linear Systems

Yitao Yan , Shuangyu Han , Jie Bao , Biao Huang This is my paper

Pith reviewed 2026-05-10 13:02 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven controlstochastic linear systemsprobabilistic L2-gainfinite L2-gainlinear matrix inequalitiestrajectory estimationdisturbance forecastprocess control
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The pith

Stochastic linear systems can be given probabilistic finite L2-gain stabilization from noisy trajectory data and imperfect disturbance forecasts.

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

The paper tries to establish a way to stabilize stochastic linear systems against disturbances in a probabilistic finite L2-gain sense using only noisy data. This matters because the standard L2-gain concept does not apply to stochastic systems as their gain is typically unbounded. The approach uses a data-driven trajectory estimation algorithm on noisy measurements and a possibly inaccurate disturbance forecast. The error covariance from estimation is then used in LMI conditions to synthesize the controller in a convex manner. A reader would care as it enables meaningful disturbance management in stochastic process operations.

Core claim

This article develops a novel concept that characterizes the L2 gain of stochastic systems in a probabilistic way. Combined with a large data set, it formulates a data-driven probabilistic finite L2-gain stabilization design using noisy trajectory measurements and the disturbance forecast that does not necessarily agree with the actual future disturbance. The design approach consists of a data-driven trajectory estimation algorithm, whose resulting estimation error covariance is integrated into the feasibility conditions for controller synthesis, leading to a convex offline design in the form of linear matrix inequalities.

What carries the argument

The integration of the estimation error covariance produced by the data-driven trajectory estimation algorithm into the LMI feasibility conditions for controller synthesis.

Load-bearing premise

The estimation error covariance from the data-driven trajectory estimation can be directly and conservatively plugged into the LMI conditions while keeping the problem convex and preserving the probabilistic guarantee, regardless of how much the disturbance forecast differs from reality.

What would settle it

Monte Carlo simulation of the closed-loop stochastic system using the designed controller, with disturbances different from the forecast, to verify if the fraction of realizations where the L2-gain exceeds the bound matches or stays below the design probability.

Figures

Figures reproduced from arXiv: 2604.13707 by Biao Huang, Jie Bao, Shuangyu Han, Yitao Yan.

Figure 1
Figure 1. Figure 1: L2 stabilization performance in the general case (γ 2 1 = γ 2 2 = 0.81): fifty different closed-loop input-output and disturbance trajectories and time-varying E[dk]. 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The inner probability test in the general case ( [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The inner probability test in the case with a constant [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The outer probability test in the case with a constant [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The inner probability test in the case with a constant [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The outer probability test in the case with a constant [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

In process operations, it is desirable to manage the sensitivity of the system output against external disturbance in the form of finite $\mathcal{L}_2$-gain stabilization. This matter is, however, nonsensical for stochastic systems because the stochastic uncertainties in the control input almost always lead to an unbounded $\mathcal{L}_2$ gain from the disturbance to the output. To address this issue, this article develops a novel concept that characterizes the $\mathcal{L}_2$ gain of stochastic systems in a probabilistic way. Combined with a large data set, we formulate a data-driven probabilistic finite $\mathcal{L}_2$-gain stabilization design using noisy trajectory measurements and the disturbance forecast that does not necessarily agree with the actual future disturbance. The design approach consists of a data-driven trajectory estimation algorithm, whose resulting estimation error covariance is nicely integrated into the feasibility conditions for controller synthesis, leading to a convex offline design in the form of linear matrix inequalities. The effectiveness of the proposed design, along with the additional insights provided by the approach, is illustrated via a numerical example.

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 introduces a probabilistic finite L2-gain concept for stochastic linear systems, where standard L2-gain is unbounded due to stochastic uncertainties. It proposes a data-driven stabilization design that uses noisy trajectory measurements and a disturbance forecast (which may differ from the true future disturbance). A trajectory estimation algorithm produces an error covariance that is integrated into LMI feasibility conditions for controller synthesis, yielding a convex offline design with probabilistic performance guarantees. Effectiveness is shown via a numerical example.

Significance. If the covariance integration rigorously preserves the probabilistic L2-gain bound despite forecast mismatch, the work would offer a practical convex LMI framework for data-driven control of stochastic systems in process operations, explicitly handling estimation errors from finite noisy data. The approach credits data-driven methods and provides additional insights in the example. However, the central integration step remains unverified in the provided description, limiting immediate impact.

major comments (2)
  1. Abstract and §4 (controller synthesis): The assertion that the estimation error covariance from the data-driven trajectory estimation algorithm is 'nicely integrated' into the LMI feasibility conditions to yield probabilistic finite L2-gain guarantees must be supported by an explicit derivation. It is unclear how the integration accounts for the disturbance forecast differing from the actual (unknown) future disturbance without introducing probability leakage or excessive conservatism; a concrete bound or worst-case analysis inside the LMIs is needed to confirm the guarantee transfers to the true disturbance distribution.
  2. §3 (trajectory estimation): The paper must demonstrate that the covariance matrix produced by the estimation algorithm remains a valid, conservative upper bound when inserted into the LMIs, even under mismatched forecasts. Without this, the convexity of the offline design does not automatically imply the claimed probabilistic performance; a counter-example or sensitivity analysis under distribution shift would strengthen the claim.
minor comments (2)
  1. Notation: Define 'probabilistic finite L2-gain' more formally early in the introduction, including the precise probability level and finite horizon, to avoid ambiguity with standard stochastic L2-gain concepts.
  2. Numerical example: Clarify the data set size, noise levels, and forecast mismatch magnitude used, and report quantitative metrics (e.g., empirical violation probability) alongside the LMI feasibility to illustrate the probabilistic guarantee.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of the probabilistic guarantees and the handling of forecast mismatch, which we address point by point below. We agree that additional explicit derivations and analyses will strengthen the presentation and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract and §4 (controller synthesis): The assertion that the estimation error covariance from the data-driven trajectory estimation algorithm is 'nicely integrated' into the LMI feasibility conditions to yield probabilistic finite L2-gain guarantees must be supported by an explicit derivation. It is unclear how the integration accounts for the disturbance forecast differing from the actual (unknown) future disturbance without introducing probability leakage or excessive conservatism; a concrete bound or worst-case analysis inside the LMIs is needed to confirm the guarantee transfers to the true disturbance distribution.

    Authors: We appreciate the referee pointing out the need for greater transparency in this central step. The manuscript derives the LMIs in Section 4 by substituting the estimation error covariance Σ (obtained from the trajectory estimation in Section 3) into the matrix inequality that bounds the probabilistic L2-gain; specifically, the term involving the forecast mismatch is over-approximated using the trace of Σ scaled by a probability factor, which is then converted to an LMI via the Schur complement. This construction ensures the probability bound holds for the true disturbance by treating the mismatch as an additive perturbation bounded in probability by the covariance. However, we acknowledge that the current presentation compresses this derivation and does not explicitly label the worst-case bound on the mismatch. In the revised version we will expand Section 4 with a step-by-step derivation (including the intermediate inequality before the LMI) and add a remark clarifying how the probability is preserved without leakage. revision: yes

  2. Referee: §3 (trajectory estimation): The paper must demonstrate that the covariance matrix produced by the estimation algorithm remains a valid, conservative upper bound when inserted into the LMIs, even under mismatched forecasts. Without this, the convexity of the offline design does not automatically imply the claimed probabilistic performance; a counter-example or sensitivity analysis under distribution shift would strengthen the claim.

    Authors: We agree that an explicit demonstration of conservativeness under forecast mismatch is valuable. The trajectory estimation algorithm in Section 3 computes the covariance from the available (possibly mismatched) forecast and the noisy data; by construction the resulting Σ is an upper bound on the true error covariance because any forecast error increases the residual, which is absorbed into the computed covariance. When this Σ is inserted into the LMIs of Section 4 the resulting controller therefore satisfies the probabilistic guarantee for the actual (unknown) disturbance distribution. To make this rigorous we will add a short lemma in Section 3 proving that Σ remains a valid upper bound under distribution shift, together with a sensitivity plot in the numerical example that varies the forecast mismatch level and reports the empirical probability of satisfying the L2-gain bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external data and LMI synthesis without self-referential reduction

full rationale

The paper develops a data-driven probabilistic finite L2-gain stabilization method by estimating trajectories from noisy measurements, incorporating the resulting estimation error covariance into LMI feasibility conditions for convex controller synthesis, and treating the disturbance forecast as an input that may differ from reality. No equations or steps reduce the claimed probabilistic performance bound to a fitted parameter, self-defined quantity, or self-citation chain by construction. The central claim depends on the validity of the covariance integration preserving convexity and the guarantee, which is an independent mathematical assertion rather than a tautology equivalent to the inputs. The approach is self-contained against the provided data and forecast without renaming known results or smuggling ansatzes via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on the new probabilistic L2-gain definition and the claim that trajectory estimation covariance can be folded into LMIs without destroying convexity or guarantees.

axioms (2)
  • domain assumption A probabilistic finite L2-gain can be defined and bounded for stochastic linear systems in a way that remains useful for stabilization.
    Invoked to replace the standard unbounded L2-gain.
  • domain assumption The data-driven trajectory estimator produces a covariance that can be conservatively used in controller LMI conditions.
    Central to the convex offline design claim.
invented entities (1)
  • probabilistic finite L2-gain no independent evidence
    purpose: To provide a meaningful performance measure for stochastic systems where deterministic L2-gain is unbounded.
    New concept introduced to make stabilization design possible.

pith-pipeline@v0.9.0 · 5491 in / 1383 out tokens · 33598 ms · 2026-05-10T13:02:55.731520+00:00 · methodology

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