A Modified Adaptive Data-Enabled Policy Optimization Control to Resolve State Perturbations

Alessandro V. Papadopoulos; Costanzo Manes; Mojtaba Kaheni; Niklas Persson; Vittorio De Iuliis

REVIEW 5 major objections 5 minor 23 references

A Modified Adaptive Data-Enabled Policy Optimization Control to Resolve State Perturbations

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A modified DeePO algorithm claims full-rank data and calm states without probing noise.

desk verdict A well-motivated heuristic for removing probing noise from DeePO, but the full-rank proof has a fatal gap and the stability theorem doesn't match the algorithm. read the letter →

arxiv 2507.20580 v1 pith:RODWGGK5 submitted 2025-07-28 eess.SY cs.SY

Mojtaba Kaheni , Niklas Persson , Vittorio De Iuliis , Costanzo Manes , Alessandro V. Papadopoulos This is my paper · ORCID

classification eess.SYcs.SY

keywords data-enabledpolicyoptimizationadaptiveLQRpersistencyofexcitationprobingnoisestateperturbationsdata-drivencontrolmultiplicativeminimumsingularvalue

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper targets a conflict at the heart of the data-enabled policy optimization (DeePO) algorithm for adaptive LQR control: DeePO needs persistently exciting inputs, but the linear state-feedback gains it produces cannot supply that excitation, so probing noise is normally added. That noise keeps the data informative but also rattles the states. The paper proposes Perturbation-Free DeePO (PFDeePO), which pauses gain updates when states are near equilibrium and applies a random multiplicative gain to the control signal only after the controller has converged. If the proposal works, adaptive LQR can keep learning from its own data without forever injecting dither noise, leaving states steady while still steering the gain toward the certainty-equivalence LQR solution.

What carries the argument

The central object is the data covariance Φ = (1/t)DD^T built from stacked input-state data; its full rank is exactly the persistency-of-excitation condition that lets DeePO invert Φ and update the gain. PFDeePO wraps this in two mechanisms: a pause rule keyed to ‖x_i‖ ≤ γ that prevents uninformative zero columns from entering the dataset, and a multiplicative random gain v_i on the control signal, triggered only when the gain update ‖ΔK‖ has fallen below δ. The stability guarantee rests on the interval [v,v] from Lemma 5 and the modified DARE (31), whose solution H yields the exponential bound ‖x_k‖ ≤ β^k $\sqrt$(λmax(H)/λmin(H))‖x_0‖ for any v_k in the interval.

What would settle it

Run Algorithm 2 on the paper's four-state system with zero initial gain, no process noise, and no external disturbance; if the state norm drops below γ before ‖K_i − K*‖ approaches zero, the gain error or σ(Φ_i) will plateau, showing that the pause rule can stop learning before convergence.

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Extended reading notes

Core claim

The paper claims that PFDeePO resolves the DeePO excitation dilemma by separating two failure regimes. When states converge to the equilibrium, appending them to the data matrix drives the minimum singular value of the sample covariance Φ to zero; PFDeePO stops updating the gain in this regime. When the gain has already converged but states are still far from equilibrium, the algorithm multiplies the control signal by a random scalar v_i drawn from an interval [v,v] with v < 1 < v, ensuring newly appended columns keep the data matrix full rank. The paper proves in Theorem 1 that Φ_i stays full rank under Algorithm 2, and in Theorem 2 that any sequence v_i in that interval preserves exponential stability of the closed loop via a modified DARE and a quadratic Lyapunov function. Simulations on a four-state system show states remaining at equilibrium after a disturbance, while σ(Φ_i) stays bounded away from zero.

Load-bearing premise

The method assumes that in the noiseless pre-convergence phase, state-feedback data remain informative enough for the DeePO gradient update to make progress before the pause rule triggers; if a stable plant drives the state to equilibrium first, learning can stall before the optimal gain is reached.

Editorial extensions

If this is right

Adaptive LQR can in principle run without continuous probing noise, eliminating the persistent dither that wears actuators and disturbs states.
The inverse Φ^{-1} in the DeePO update remains well-defined at every iteration because σ(Φ_i) is kept positive by the pause-and-scale logic.
When the random scaling is active, the closed loop stays exponentially stable for any sequence of gains in the allowed interval, so the added excitation does not sacrifice stability.
Near equilibrium, the gain update is frozen, so the algorithm stops spending control effort on data points that carry no new information.
The simulation comparison shows PFDeePO settling states after a disturbance while DeePO with probing noise keeps oscillating and increasing control effort.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An immediate testable extension is to replace the threshold heuristics γ and δ with a direct monitor of σ(Φ_i), pausing only when the data actually lose informativeness; this could remove two hand-tuned parameters.
If a stable open-loop plant drives the state to equilibrium before the gain has converged, the pause rule can stall learning altogether; the paper does not analyze that pre-convergence stall, and adding a bounded external disturbance or retaining a few persistently exciting offline columns would be a natural workaround.
The interval [v,v] could be selected online from a data-driven estimate of B via the pseudoinverse formula in Remark 2, making the stability certificate adaptive rather than fixed, at the price of needing an accurate B estimate.
The same pause-and-scale idea may carry over to output-feedback or nonlinear extensions of DeePO, but those settings would need their own Lyapunov or observability assumptions, not just the LTI DARE argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: this paper identifies a real practical problem—the probing noise that DeePO needs for persistent excitation causes state chatter—and proposes a plausible heuristic to avoid it. But the central proof that the new algorithm preserves the full-rank condition does not hold up, and the stability theorem doesn't actually apply to the algorithm as implemented.

What’s new: PFDeePO pauses gain updates when states are near equilibrium, and once the gain has (supposedly) converged it multiplies the control by a random scalar in [v,v] with mean one. That specific combination is new relative to DeePO. The authors also prove a stability bound for a fixed gain with random multiplicative scaling; that Lyapunov argument is standard and, on its own, works given the interval condition. The simulation clearly illustrates the problem they're trying to solve, and the writing is clear.

The soft spots are load-bearing. Theorem 1’s proof goes from the row-dependence equation to componentwise equalities that require the state vectors to span R^n; no such spanning assumption is stated or established. For a stable open-loop plant the states typically decay, so the inference fails. Worse, Lemma 4—which the paper itself states—shows that appending rank-deficient input/state data drives the minimum singular value of Φ to zero. The first branch of Algorithm 2 appends exactly that kind of data whenever the gain is still changing and the state is not too small, and no lower bound on σ(Φ) is supplied. The pause rule doesn't rescue this: for a stable plant, the state can cross the threshold γ while K_i is still far from optimal, freezing the adaptation. The simulation masks this by throwing in a disturbance at k=15.

Theorem 2 has a separate conflation: the gain K is first defined via the standard LQR DARE, then the theorem redefines H as the solution of a modified discounted DARE and uses the same K formula. Those are different H's, so the stability result is not connected to the implemented algorithm. Finally, the interval [v,v] that makes Lemma 5 hold is asserted to exist, but no constructive check is given, and the paper admits parameter selection is future work.

Net: the proposal is a reasonable heuristic, but the paper-as-written does not establish its main claims. I would not cite it, and I wouldn't bring it to a reading group except as an example of how easy it is to overclaim in adaptive data-driven control. Still, the underlying problem is real and the stability lemma could be repurposed. With a serious rewrite—either a correct full-rank proof or an honest framing as an empirical heuristic—the idea might be publishable. I'd let it go to review rather than desk-reject, because a referee's detailed comments could help the authors fix it.

Cheers.

Referee Report

5 major / 5 minor

Summary. The paper proposes PFDeePO, a modified version of the data-enabled policy optimization (DeePO) algorithm for adaptive LQR, with the goal of avoiding the state perturbations caused by probing noise. The algorithm combines a pause rule that stops gain updates when the state is near equilibrium with a multiplicative random gain applied once the controller has approximately converged. The paper claims two main theoretical results: Theorem 1 states that PFDeePO preserves the full rank of the data matrix Φ, and Theorem 2 establishes exponential stability of the closed loop under the random scaling. The claims are supported by a numerical example comparing PFDeePO with DeePO with and without probing noise.

Significance. If the central claims were established, the contribution would be practically useful: it would let DeePO-style adaptive LQR run without persistent input excitation after convergence, reducing state chattering and actuator wear. The paper correctly identifies a real limitation of DeePO, namely that LQR state feedback cannot generate persistently exciting inputs, and the simulation qualitatively illustrates the intended benefit. However, the two theoretical pillars are not sound as written. The proof of Theorem 1 contains an invalid inference that requires an unstated spanning condition, Lemma 4 points to a mechanism by which the first branch of Algorithm 2 can drive σ(Φ) to zero, and Theorem 2 conflates the standard LQR DARE gain with a different gain defined by a modified DARE. The stability analysis also relies on an interval [v,v] whose existence is asserted but not constructively verified for the actual system data. The paper does not provide reproducible code or machine-checked proofs, and the simulation alone does not compensate for the gaps in the theory.

major comments (5)

[Theorem 1, Section III] The inference from equations (23)-(24) to equation (25) is invalid. From (24) one obtains, for each k, the relation ∑_i (λ_i - ∑_j μ_j K_{j,i}^{(k)}) x_i^{(k)} = 0, which implies λ = (K^{(k)})^T μ only if the vectors x^{(k)} span R^n. No such spanning condition is stated or proven, and for a stable open-loop plant the states typically decay and may lie near a low-dimensional subspace. Moreover, the row dependence (23) involves all columns of X0 and U0, including the initial offline persistently exciting columns, whereas the substitution U0_{j,k} = ∑_i K_{j,i}^{(k)} x_i^{(k)} applies only to online columns generated by state feedback. The contradiction in Theorem 1 therefore does not follow.
[Lemma 4 and Algorithm 2, Section III] Lemma 4 shows that appending a rank-deficient block D2 drives σ(Φ) to zero. In Algorithm 2, whenever ∥ΔK∥ > δ and ∥x_i∥ > γ, the control is u_i = K_i x_i with no stochastic excitation, and the resulting column is appended to the dataset. By Lemma 3 these columns are rank-deficient, so σ(Φ_i) can be driven toward zero before the gain converges. Theorem 1 does not establish a positive lower bound on σ(Φ_i), and it cannot, because the first branch of Algorithm 2 is exactly the regime covered by Lemma 4. The pause rule does not fix this: it stops appending data only when ∥x_i∥ ≤ γ, not when ∥ΔK∥ > δ.
[Theorem 2, Equations (26)-(31)] Theorem 2 defines the converged gain K through the modified DARE (31) with β ∈ (0,1), but the DeePO/LQR gain referenced in (26)-(27) solves the standard DARE, which corresponds to β = 1. Unless β = 1, these gains are different, so the key identity (37), on which the Lyapunov decrease (38) relies, does not hold for the actual DeePO gain. The statement that 'K is equal to a certainly equivalent LQR gain' is therefore inconsistent with (31) for β < 1. The existence and uniqueness of a positive definite solution H to the modified DARE (31) is also assumed without proof.
[Algorithm 2, Remark 1, Section IV] The instruction 'Find v and v as in Theorem 2 for Ki' is not implementable as stated. It requires the solution H of the modified DARE (31) and verification of the matrix inequality (28), but no data-driven procedure is given to compute H or to certify (28) from measured data. Remark 1's iterative narrowing is not a concrete check, since no estimate of H is supplied. The simulation states that the interval [0.5,1.5] 'satisfies (28)' without presenting the verification, so the stability guarantee is not actually demonstrated for the reported experiment.
[Section IV, simulation setup] For the open-loop stable plant with initial gain K_t = 0, the state decays toward the equilibrium. If ∥x_i∥ ≤ γ is reached while ∥ΔK∥ > δ, the second if-else block freezes the gain K_{i+1} = K_i and stops appending data, so learning stalls permanently before the gain converges. The simulation avoids this by injecting a disturbance at sample k = 15, but Algorithm 2 contains no mechanism that guarantees the gain converges before the state enters the γ-ball. This is a load-bearing gap because the paper's central claim is that PFDeePO works without probing noise.

minor comments (5)

[Throughout] There are several language and typography issues: 'certainly equivalent' should be 'certainty-equivalent', 'Perturbations-free' should be 'Perturbation-free', and the author block contains a stray space in 'V .'.
[Equation (15)] The notation in (15) is unclear: the bar notation for U0, X0, and X1 is introduced informally, and the relationship between Φ = (1/t)DD^T and the subsequent X1 = X1D^T/t is not defined cleanly. Please define all overlined quantities explicitly.
[Section IV] The disturbance at sample k = 15 is described only as 'a uniform random value'; its dimension, range, and whether the reported results are from a single run or averaged over multiple runs should be specified.
[Figure 2] The curve labeled 'DeePO without e' is not covered by Algorithm 1, since Algorithm 1 requires a probing noise signal and Φ may become singular without it. The implementation of this benchmark should be described, particularly the matrix inversion step.
[Theorem 2, Equation (32)] The closed-loop stability result is stated for the estimated matrices (Â,B̂), but the relationship between these estimates and the true system (1) is not discussed. The practical relevance of the bound (33) for the true plant is therefore unclear.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PFDeePO's rank and stability arguments do not reduce to their own inputs or to load-bearing self-citations.

full rationale

The paper's central claims are that Algorithm 2 preserves the full rank of Φ and that the multiplicative-noise control v_k K_k x_k preserves closed-loop stability. Neither claim is circular. The rank preservation argument in Theorem 1 is a attempted proof by contradiction on rank(D); although the step from (24) to (25) requires the state samples to span R^n, which is not stated or proven, this is a proof gap and a correctness risk, not a definitional equivalence or a fitted input renamed as a prediction. The stability argument in Theorem 2 is a standard Lyapunov argument using the modified DARE (31) and the sufficient interval condition (28); the interval [v,v] is not fitted to simulation outcomes but is presented as satisfying a matrix inequality, and the theorem's conclusion does not coincide with its assumptions. The citation to the DeePO equivalence result (Lemma 1 of [16]) is to work by different authors (Zhao, Dörfler, Chiuso, You), not to the present authors, so it is not a load-bearing self-citation. The self-citations in the paper ([14], [18]) are contextual references to prior bicycle work and are not used to supply assumptions to the central theorems. The simulation parameters (γ=0.1, δ=0.1, v=0.5, v=1.5) are asserted to satisfy (28) without a shown verification, and the role of β in Algorithm 2 is unclear, but these are reproducibility and correctness concerns rather than instances where an output is equivalent to an input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several unverified premises: the modified DARE used in Theorem 2 is assumed to have a unique positive definite solution; the interval [v,v] is assumed to satisfy inequality (28) without a shown check; and the pre-convergence phase of Algorithm 2 is assumed to preserve the conditioning and convergence properties of DeePO without probing noise. The first two affect the stability guarantee; the third affects the correctness of the entire learning process.

free parameters (6)

gamma = 0.1
Threshold for pausing updates near equilibrium; chosen by hand in the simulation and not justified.
delta = 0.1
Threshold for declaring the gain converged; chosen by hand and determines when random scaling becomes active.
v and v (multiplicative scaling interval) = 0.5 and 1.5
Bounding the random gain scaling; asserted to satisfy inequality (28) without verification.
eta (learning rate) = 10^-4
Inherited from DeePO; chosen for the simulation and affects the convergence speed.
sigma_e (probing noise std for DeePO baseline) = 0.1
Used for the DeePO benchmark; larger than the state perturbations shown in the figures.
beta (discount factor in Theorem 2)
Introduced in the modified DARE (31) but never specified or used in the algorithm or the simulation.

assumptions (5)

ad hoc to paper There exists a unique positive definite solution H to the modified DARE (31) for the chosen beta in (0,1).
Theorem 2 states 'Let H be the unique solution' without proving existence or uniqueness; standard DARE results require stabilizability and detectability and do not directly cover the discounted term beta^2 H.
ad hoc to paper The interval [v,v] used in Algorithm 2 satisfies the matrix inequality (28) for the actual system data.
Lemma 5 only proves the existence of some interval around 1 for exactly known (A,B); the paper asserts the chosen interval [0.5,1.5] satisfies (28) but provides no verification, and Remark 1 gives only an iterative narrowing suggestion.
domain assumption The pre-convergence phase of Algorithm 2, with u_i = K_i x_i and no probing noise, preserves the conditioning of Phi and the descent property of the DeePO update.
This is assumed throughout Section III; the paper's own Lemmas 3 and 4 show that purely state-feedback data drive sigma(Phi) to zero in the limit, and Theorem 1's proof does not establish a positive lower bound during this phase.
standard math The offline data (U0, X0, X1) are generated by a persistently exciting input of order n+1, so that rank(D)=n+m initially.
This is a standard assumption from Willems' fundamental lemma, cited in Definition 1 and Lemma 1.
domain assumption The pair (A,B) is controllable and the noise omega_k is bounded.
Stated in Section II-A; needed for data-driven representation and stability.

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Cite this review

Pith. "Pith review of A Modified Adaptive Data-Enabled Policy Optimization Control to Resolve State Perturbations." pith.science (2026). https://pith.science/paper/RODWGGK5

@misc{pith2026250720580,
  author       = {Pith},
  title        = {Pith review of: A Modified Adaptive Data-Enabled Policy Optimization Control to Resolve State Perturbations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RODWGGK5}},
  note         = {Machine review of arXiv:2507.20580}
}

abstract

This paper proposes modifications to the data-enabled policy optimization (DeePO) algorithm to mitigate state perturbations. DeePO is an adaptive, data-driven approach designed to iteratively compute a feedback gain equivalent to the certainty-equivalence LQR gain. Like other data-driven approaches based on Willems' fundamental lemma, DeePO requires persistently exciting input signals. However, linear state-feedback gains from LQR designs cannot inherently produce such inputs. To address this, probing noise is conventionally added to the control signal to ensure persistent excitation. However, the added noise may induce undesirable state perturbations. We first identify two key issues that jeopardize the desired performance of DeePO when probing noise is not added: the convergence of states to the equilibrium point, and the convergence of the controller to its optimal value. To address these challenges without relying on probing noise, we propose Perturbation-Free DeePO (PFDeePO) built on two fundamental principles. First, the algorithm pauses the control gain updating in DeePO process when system states are near the equilibrium point. Second, it applies a multiplicative noise, scaled by a mean value of $1$ as a gain for the control signal, when the controller converges. This approach minimizes the impact of noise as the system approaches equilibrium while preserving stability. We demonstrate the effectiveness of PFDeePO through simulations, showcasing its ability to eliminate state perturbations while maintaining system performance and stability.

Figures

Figures reproduced from arXiv: 2507.20580 by the authors.

**Figure 1.** Evolution of the states x, using DeePO (top) and PFDeePO (bottom). 0 20 40 60 80 100 0 2 4 ·10−2 Time [k] σ(Φi) DeePO with e DeePO without e PFDeePO [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** σ(Φi) for PFDeePO and DeePO, with and without probing noise. contrast, the probing noise in DeePO continuously disturbs the states, inducing oscillations that increase control effort [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

A note on persistency of excitation,

J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor, “A note on persistency of excitation,” Systems & Control Letters, vol. 54, no. 4, pp. 325–329, 2005

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[2] [2]

Chen, Linear System Theory and Design , 3rd ed

C.-T. Chen, Linear System Theory and Design , 3rd ed. Oxford University Press, 1999

work page 1999

[3] [3]

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A. Cohen, T. Koren, and Y . Mansour, “Learning linear-quadratic regulators efficiently with only √ T regret,” in Proceedings of the 36th International Conference on Machine Learning , vol. 97, June 2019, pp. 1300–1309

work page 2019

[4] [4]

Certainty equivalence is efficient for linear quadratic control,

H. Mania, S. Tu, and B. Recht, “Certainty equivalence is efficient for linear quadratic control,” in 33rd International Conference on Neural Information Processing Systems (NIPS) , December 2019, pp. 10 154 – 10 164

work page 2019

[5] [5]

Learning robust LQ-controllers using application oriented explo- ration,

M. Ferizbegovic, J. Umenberger, H. Hjalmarsson, and T. B. Sch ¨on, “Learning robust LQ-controllers using application oriented explo- ration,” IEEE Control Systems Letters , vol. 4, no. 1, pp. 19–24, 2020

work page 2020

[6] [6]

On- policy data-driven linear quadratic regulator via combined policy iteration and recursive least squares,

L. Sforni, G. Carnevale, I. Notarnicola, and G. Notarstefano, “On- policy data-driven linear quadratic regulator via combined policy iteration and recursive least squares,” in 62nd IEEE Conference on Decision and Control (CDC) , 2023, pp. 5047–5052

work page 2023

[7] [7]

Formulas for data-driven control: Stabi- lization, optimality, and robustness,

C. De Persis and P. Tesi, “Formulas for data-driven control: Stabi- lization, optimality, and robustness,” IEEE Transactions on Automatic Control, vol. 65, no. 3, pp. 909–924, 2020

2020

[8] [8]

On the linear convergence of random search for discrete-time LQR,

H. Mohammadi, M. Soltanolkotabi, and M. R. Jovanovi ´c, “On the linear convergence of random search for discrete-time LQR,” IEEE Control Systems Letters , vol. 5, no. 3, pp. 989–994, 2021

work page 2021