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The Bias of Subspace-based Data-Driven Predictive Control

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Pith's one-line read For LTI systems, subspace-based data-driven predictive control (SPC, DeePC, γ-DDPC) is biased when training data come from a feedback loop; the paper derives the exact bias and shows a single-step-predictor alternative is consistent.

desk verdict Useful decomposition of closed-loop bias in subspace DDPC, but Theorem 2's finite-sample statement conflates sample and population quantities and needs revision. read the letter →

arxiv 2507.02468 v1 pith:WDUWFTBI submitted 2025-07-03 eess.SY cs.SY

Keith Moffat , Florian Dörfler , Alessandro Chiuso This is my paper · ORCID
classification eess.SYcs.SY MSC 93B3093C55
keywords data-drivenpredictivecontrolsubspaceDeePCclosed-loopidentificationpredictionbiasoptimismtransientpredictorLTIsystems
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 argues that the most popular data-driven predictive controllers built from subspace predictors—SPC, DeePC, and γ-DDPC—carry a statistical bias whenever the training data are collected under an existing feedback controller, and that the bias does not vanish as data grow. It traces the bias to a precise mechanism: in closed loop, future noise innovations correlate with current and past inputs, and the subspace predictor mistakes this correlation for a causal effect it can exploit. The paper quantifies the resulting Subspace Bias and, for DeePC and γ-DDPC, an additional Optimism Bias from the slack on the output trajectory, giving worst-case bounds in Theorems 1 and 2. It then shows that Transient Predictive Control, built by chaining single-step predictors, is consistent and immune to both biases, and demonstrates the performance gap on a double integrator. If correct, this gives practitioners a concrete reason to prefer single-step-predictor formulations when data must be collected in closed loop.

What carries the argument

The load-bearing object is the Subspace Bias Predictor $\beta = H_\epsilon\,\mathbb{E}[E_f U_f^T]M$, where $H_\epsilon$ maps future innovations to future outputs, $E_f$ is the matrix of training-data innovations, $U_f$ is the future input matrix, and $M$ is the weighting matrix from the inverse of the joint covariance of lead-in data and inputs. It captures the anti-causal misinterpretation: the subspace predictor sees that inputs are correlated with innovations and acts as if $u_f$ could set $\epsilon_f$, although the direction of causality is the reverse. The companion object is the Optimism Bias matrix $L_{YE}$ from the LQ decomposition of the training data, which lets DeePC and $\gamma$-DDPC hope that the unmodeled innovation $\gamma_\epsilon$ will improve the predicted trajectory. The Transient Predictor avoids $\beta$ by never forming a joint projection of outputs onto lead-in data and future inputs; it estimates single-step ARX predictors and chains them.

What would settle it

For a known LTI system, simulate closed-loop training data under a linear feedback law plus external excitation, estimate the Subspace Predictor $\hat{S}$ and the true predictor $H$ from the known model, and test whether $\hat{S}v - Hv$ converges to $\beta v$ with $\beta = H_\epsilon \mathbb{E}[E_f U_f^T]M$. If the mismatch does not converge to this expression (or converges to zero) as $N \to \infty$, the central bias claim fails.

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Extended reading notes

Core claim

For an LTI system with stationary noise, the paper's central discovery is that the subspace predictor $S = \mathbb{E}[Y_f V^T]\mathbb{E}[V V^T]^{-1}$ differs from the true certainty-equivalent multistep predictor $H = [H_p\ H_u]$ exactly when the training data come from closed-loop operation. Lemma 1 writes the Subspace Bias as $b_S(v) = \beta v$ with $\beta = H_\epsilon\,\mathbb{E}[E_f U_f^T] M$, where $M$ is the block-inverse weighting matrix defined in (16); the bias is nonzero precisely because $\mathbb{E}[E_f U_f^T] \neq 0$ under feedback. Theorem 1 bounds its worst-case size by $\sigma_{\max}(\beta_V)$ and gives the expected magnitude as a Frobenius norm of $H_\epsilon\,[0\ \ \mathbb{E}[E_f U_f^T]]\,\mathbb{E}[V V^T]^{-1/2}$. For DeePC and $\gamma$-DDPC, Lemma 3 shows the prediction bias decomposes into the same Subspace Bias plus an Optimism Bias carried by $L_{YE}\gamma_\epsilon$, and Theorem 2 bounds it by $\sigma_{\max}([\beta L_{VV}\ \ L_{YE}])$; even with open-loop data the Optimism Bias survives. Transient Predictive Control, which estimates the multistep predictor by chaining single-step ARX predictors, avoids both effects and is consistent under closed-loop data.

Load-bearing premise

The bias formulas and consistency claims rely on the future innovations being uncorrelated with the lead-in measurements and on the lead-in being long enough (at least the system order) that the finite-memory predictor equals the true optimal one.

Editorial extensions

If this is right

  • With open-loop training data, SPC is unbiased, but DeePC and γ-DDPC still carry the Optimism Bias, so their predictions are biased even when the subspace model is perfect (Theorem 2a).
  • With closed-loop training data, the Subspace Bias does not decay as the number of training samples grows; Theorem 1b expresses its worst-case size as $\sigma_{\max}(\beta_V)$ and its expected size as a Frobenius norm.
  • For DeePC and γ-DDPC, setting $\gamma_\epsilon = 0$ (or applying a strong projection regularizer) removes the Optimism Bias, leaving only the Subspace Bias; this matches a common practical heuristic.
  • Transient Predictive Control is consistent under both open- and closed-loop training data and, in the double-integrator experiments, tracks the reference well in both regimes.
  • The bias terms are estimable from the training data (Section V-A), so a practitioner can estimate $\beta$ before deploying the controller.

Reading between the lines

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

  • A direct corollary the authors do not spell out: any subspace-based predictor that forms a joint least-squares projection of future outputs onto past data and future inputs—not just SPC, DeePC, or γ-DDPC—will inherit the same closed-loop bias, so debiasing requires either subtracting $\hat{\beta} v$ or switching to chained single-step predictors.
  • The bias quantifications suggest a practical diagnostic: compute the sample correlation $\hat{\mathbb{E}}[E_f U_f^T]$ from closed-loop training data; a near-zero value indicates the Subspace Bias is negligible, while a large value warns that subspace-based DDPC will be optimistic about what inputs can achieve.
  • For high-order systems where the lead-in length is shorter than the system order, the single-step predictor bank may not be consistent, so the claimed advantage of Transient Predictive Control could shrink; testing this regime would clarify the practical limits.
  • The same optimism-bias mechanism may affect robust or stochastic variants of DeePC that keep a slack variable on the output; quantifying it for nonlinear systems would require going beyond the LTI setting.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper studies the bias of subspace-based data-driven predictive control (SPC, DeePC, and gamma-DDPC) for LTI systems. It introduces two bias notions: Subspace Bias, which arises when training data are gathered in closed loop, and Optimism Bias, which arises from the relaxed output constraint in DeePC/gamma-DDPC. The main theoretical results are Lemma 1 and Theorem 1 for the Subspace Predictor, and Lemma 3 and Theorem 2 for DeePC and gamma-DDPC. The paper also argues that the Transient Predictor, based on single-step predictors, is consistent regardless of how the training data are gathered, and presents double-integrator experiments supporting the bias analysis.

Significance. The paper addresses a practically important question: whether subspace-based DDPC methods are biased when training data are generated in closed loop, and whether an alternative predictor architecture avoids this bias. The explicit decomposition into Subspace Bias and Optimism Bias is a useful conceptual contribution, and the numerical experiments illustrate the predicted effect, including the poor performance of SPC with closed-loop data and the persistence of an optimism term for 2norm-DDPC even with open-loop data. The explicit formulas in terms of data correlations are valuable for diagnosing bias in practice. If the formal statements are corrected, the paper would provide a clear cautionary result for users of DeePC and gamma-DDPC and strengthen the case for single-step-predictor-based Transient Predictive Control. The paper does not provide code, and the experimental section would be stronger with error bars, but the qualitative conclusion is supported by the derivations modulo the issues detailed below.

major comments (3)
  1. Lemma 3 (Eq. (11)) and Theorem 2 (parts a and b) mix the population Subspace Bias Predictor beta defined in Eq. (7) with the sample quantity actually used in the proof. The proof of Lemma 3 substitutes S = L_YV L_VV^{-1}, which identifies S as the sample subspace predictor Yf V^T (V V^T)^{-1}, not the population predictor S = E[Yf V^T] E[V V^T]^{-1}. Therefore the beta appearing in Eq. (11) must be beta_sample = Yf V^T (V V^T)^{-1} - H, not beta = H_eps E[Ef Uf^T] M. Since beta_sample is nonzero for finite N even in open loop, Theorem 2a is false as stated: with open-loop training data the maximum of ||b_gamma(gamma)|| over ||gamma||=1 is sigma_max([beta_sample L_VV, L_YE]), not sigma_max(L_YE). The statement becomes correct only asymptotically (N to infinity) or after rephrasing with beta_sample and an explicit limit. This is a load-bearing issue because Theorem 2 is the paper's quantitative statement of the DeePC/gamma-DDPC bias.
  2. The displayed identity E[L_YV] E[L_VV]^{-1} = E[Yf V^T] E[V V^T]^{-1} = S is not valid as written: expectation does not factor through the LQ factors, and L_YV L_VV^{-1} is the sample subspace predictor, not the population S defined in Section II.C. This identity appears to be the source of the sample-versus-population ambiguity in Lemma 3 and Theorem 2. It should either be removed or replaced with a statement that clearly distinguishes the sample equality L_YV L_VV^{-1} = Yf V^T (V V^T)^{-1} from the population definition of S.
  3. Equation (14) has a dimension and notation problem: the expression for the estimated Subspace Bias Predictor is written as a vector in R^{q tau} and appears to depend on v, whereas beta from Eq. (7) is a matrix of size q tau x ((q+m)rho + m tau). The notation 'cM' is also undefined; it likely should be the estimated version of M. Please rewrite Equation (14) so that the estimator of beta is a matrix and the role of v is made explicit only when the bias beta v is formed.
minor comments (5)
  1. The figures report averages over 100 independent experiments but do not show error bars or confidence intervals; adding them would make the convergence claims in Fig. 2 more convincing.
  2. The consistency of the Transient Predictor is imported from reference [19]; the paper should state the precise theorem or conditions from [19] that guarantee consistency under closed-loop data, and clarify whether Assumption 1 is sufficient.
  3. There is a typo in 'Multisptep Predictor matrices'; it should be 'Multistep Predictor matrices'.
  4. No code or data are provided. For reproducibility, especially for the six-trajectory plots, the authors should include code or a detailed experimental protocol with random seeds and parameter values.
  5. The proof of Lemma 3 is very compressed; after the sample/population issue is resolved, a short derivation showing exactly which beta is used in Eq. (11) and Eq. (12) would improve readability.

Circularity Check

1 steps flagged · score 2.0 of 10

Bias formulas are derived from the definitions; only the Transient Predictor consistency claim is imported from the authors' prior work.

  1. self citation load bearing [Section V (Consistent Multistep Prediction), Eq. (13)]
    "The Transient Predictor Method for estimating the Multistep Predictor (3), introduced in [19], provides bH, a consistent estimate of H, regardless of whether the training data was gathered in open or closed-loop. ... Under Assumption 1, (13) is a consistent predictor of the true single-step predictor with l lead-in data."

    The paper's favorable claim that Transient Predictive Control is consistent and therefore avoids Subspace Bias and Optimism Bias is imported from reference [19], which is prior work by the same authors, and the proof of this consistency is not reproduced in this manuscript. This is a load-bearing self-citation for the TPC comparison and for contribution 3, though it does not enter the derivation of the subspace bias formulas (Lemmas 1 and 3, Theorems 1 and 2), which are derived algebraically from the definitions of S, H, and the LQ decomposition. Hence it is a minor self-citation rather than definitional circularity.

full rationale

The central derivation chain is self-contained. Lemma 1 computes S - H = H_epsilon E[Ef V^T] E[VV^T]^{-1} = H_epsilon E[Ef Uf^T] M using yf = Hv + H_epsilon Ef and E[Ef Zp^T] = 0; Lemma 3 obtains b_gamma(gamma) = [beta L_VV, L_YE] gamma from S = L_YV L_VV^{-1}; Theorems 1 and 2 are norm bounds on these algebraically defined matrices. No fitted parameter is renamed as a prediction, and no target quantity enters the definitions. The one notable self-citation is Section V's assertion that the Transient Predictor yields a consistent estimate of H, attributed to [19] by the same authors; since the proof is not included here, the TPC consistency claim is imported rather than independently established in this paper. This is a minor, ancillary self-citation because it does not set any constant used in the bias formulas. Separately, a reader's technical objection that Theorem 2(a) mixes the population beta with sample LQ blocks and is therefore not literally true for finite N is a correctness/quantifier concern, not a circularity, and does not change the circularity verdict.

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

No fitted parameters appear in the bias theorems. The derivations rely on standard innovation properties plus the persistent excitation assumption, and implicitly on a lead-in length that captures the system order. The main unstated caveat is the rho versus order condition; the sample versus population notation in Section IV also needs clarification.

assumptions (5)
  • domain assumption The joint input-output process is stationary and admits the infinite ARX representation (1) with white innovation e(t).
    Section II-B: the system class is LTI and the representation is used to define the multistep predictor H and all bias expressions.
  • domain assumption The training input is persistently exciting and the noise covariance is positive definite (Assumption 1).
    Section II-D: guarantees Hankel matrices are nonsingular and single-step predictors can be consistently estimated via (13).
  • domain assumption Future innovations are uncorrelated with the lead-in data, E[Ef Zp^T] = 0.
    Appendix I, proof of Lemma 1: this orthogonality is necessary to obtain bS(v) = beta v with the block inverse M.
  • domain assumption The feedback controller is external to the system and is not part of the predictor H.
    Remark 1: if feedback dynamics were folded into the predictor, the bias decomposition would change.
  • domain assumption The lead-in length rho is at least the system order, so H in (3) is the true multistep predictor.
    Section V uses consistency of single-step predictors with l lead-in data; without an order bound, finite-memory H may suffer truncation bias.

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Cite this review

Pith. "Pith review of The Bias of Subspace-based Data-Driven Predictive Control." pith.science (2026). https://pith.science/paper/WDUWFTBI

@misc{pith2026250702468,
  author       = {Pith},
  title        = {Pith review of: The Bias of Subspace-based Data-Driven Predictive Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDUWFTBI}},
  note         = {Machine review of arXiv:2507.02468}
}
abstract

This paper quantifies and addresses the bias of subspace-based Data-Driven Predictive Control (DDPC) for linear, time-invariant (LTI) systems. The primary focus is the bias that arises when the training data is gathered with a feedback controller in closed-loop with the system. First, the closed-loop bias of Subspace Predictive Control is quantified using the training data innovations. Next, the bias of direct, subspace-based DDPC methods DeePC and $\gamma$-DDPC is shown to consist of two parts--the Subspace Bias, which arises from closed-loop data, and an Optimism Bias, which arises from DeePC/$\gamma$-DDPC's "optimistic" adjustment of the output trajectory. We show that, unlike subspace-based DDPC methods, Transient Predictive Control does not suffer from Subspace Bias or Optimism Bias. Double integrator experiments demonstrate that Subspace and Optimism Bias are responsible for poor reference tracking by the subspace-based DDPC methods.

Figures

Figures reproduced from arXiv: 2507.02468 by the authors.

Figure 1
Figure 1. Double integrator input-innovation correlation averaged over 100 independent experiments. (The y-axis entries are duplicated because there are two outputs.) D. Data Collection We define the “training-time feedback controller” f(·) such that the closed-loop input u(t) = f [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. plots the bias-defining quantities from Theorem 1 as a function N for the same double integrator experiment as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Planned trajectories vs. open-loop realized trajectories (a) Open-loop data (b) Closed-loop data [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: includes the same six tests as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

22 extracted references · 19 canonical work pages · cited by 1 Pith paper

  1. [19]

    The Transient Predictor,

    K. Moffat, D. Florian, and A. Chiuso, “The Transient Predictor,” in 2024 IEEE 63rd Conference on Decision and Control (CDC) . IEEE, 2024

    work page 2024

  2. [1]

    Data-enabled predictive control: In the shallows of the DeePC,

    J. Coulson, J. Lygeros, and F. D ¨orfler, “Data-enabled predictive control: In the shallows of the DeePC,” in 2019 18th European Control Conference (ECC). IEEE, 2019, pp. 307–312

    work page 2019

  3. [2]

    Data-driven model predictive control with stability and robustness guarantees,

    J. Berberich, J. K ¨ohler, M. A. M ¨uller, and F. Allg ¨ower, “Data-driven model predictive control with stability and robustness guarantees,” IEEE Transactions on Automatic Control , vol. 66, no. 4, pp. 1702– 1717, 2020

    2020

  4. [3]

    Distributionally robust chance constrained data-enabled predictive control,

    J. Coulson, J. Lygeros, and F. D ¨orfler, “Distributionally robust chance constrained data-enabled predictive control,” IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3289–3304, 2021

    2021

  5. [4]

    Robust data-enabled predictive control: Tractable formulations and performance guaran- tees,

    L. Huang, J. Zhen, J. Lygeros, and F. D ¨orfler, “Robust data-enabled predictive control: Tractable formulations and performance guaran- tees,” IEEE Transactions on Automatic Control , vol. 68, no. 5, pp. 3163–3170, 2023

    work page 2023

  6. [5]

    Data-driven predictive control in a stochastic setting: a unified framework,

    V . Breschi, A. Chiuso, and S. Formentin, “Data-driven predictive control in a stochastic setting: a unified framework,” Automatica, vol. 152, no. 110961, 2023

    work page 2023

  7. [6]

    Harnessing uncertainty for a separation principle in direct data-driven predictive control,

    A. Chiuso, M. Fabris, V . Breschi, and S. Formentin, “Harnessing uncertainty for a separation principle in direct data-driven predictive control,” Automatica, vol. 173, p. 112070, 2025

    2025

  8. [7]

    SPC: Subspace predictive control,

    W. Favoreel, B. D. Moor, and M. Gevers, “SPC: Subspace predictive control,” IFAC Proceedings Volumes, vol. 32, no. 2, pp. 4004–4009, 1999, 14th IFAC World Congress 1999, Beijing, Chia, 5-9 July

    work page 1999

Show all 22 references
  1. [8]

    Closed-loop subspace predictive control for fault tolerant mpc design,

    J. Dong, M. Verhaegen, and E. Holweg, “Closed-loop subspace predictive control for fault tolerant mpc design,” IFAC Proceedings Volumes, vol. 41, no. 2, pp. 3216–3221, 2008

    2008

  2. [9]

    Huang and R

    B. Huang and R. Kadali, Dynamic modeling, predictive control and performance monitoring: a data-driven subspace approach. Springer, 2008

    2008

  3. [10]

    Closed-loop aspects of data-enabled predictive control,

    R. Dinkla, S. P. Mulders, J.-W. van Wingerden, and T. Oomen, “Closed-loop aspects of data-enabled predictive control,” IFAC- PapersOnLine, vol. 56, no. 2, pp. 1388–1393, 2023

    2023

  4. [11]

    Van Overschee and B

    P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems. Kluwer Academic Publications, 1996

    1996

  5. [12]

    The role of Vector AutoRegressive modeling in predictor based subspace identification,

    A. Chiuso, “The role of Vector AutoRegressive modeling in predictor based subspace identification,” Automatica, vol. 43, no. 6, pp. 1034– 1048, June 2007

    2007

  6. [13]

    Closed-loop subspace identification methods: an overview,

    G. Van der Veen, J.-W. van Wingerden, M. Bergamasco, M. Lovera, and M. Verhaegen, “Closed-loop subspace identification methods: an overview,” IET Control Theory & Applications , vol. 7, no. 10, pp. 1339–1358, 2013

    2013

  7. [14]

    Subspace identification from closed loop data,

    L. Ljung and T. McKelvey, “Subspace identification from closed loop data,” Signal Processing, vol. 52, no. 2, pp. 209–216, 1996

    1996

  8. [15]

    Closed loop subspace systems identification,

    P. Van Overschee and B. De Moor, “Closed loop subspace systems identification,” in Proc. of 36th IEEE Conference on Decision & Control, San Diego, CA, 1997, pp. pp. 1848–1853

    1997

  9. [16]

    Consistency analysis of some closed-loop subspace identification methods,

    A. Chiuso and G. Picci, “Consistency analysis of some closed-loop subspace identification methods,” Automatica, vol. 41, no. 3, pp. 377– 391, 2005

    2005

  10. [17]

    An indirect closed loop subspace identification method,

    M. Pouliquen, O. Gehan, and E. Pigeon, “An indirect closed loop subspace identification method,” in 49th IEEE Conference on Decision and Control (CDC) . IEEE, 2010, pp. 4417–4422

    2010

  11. [18]

    Closed-loop subspace identification with inno- vation estimation,

    S. Qin and L. Ljung, “Closed-loop subspace identification with inno- vation estimation,” in Proc. of SYSID 2003 , Rotterdam, 2003

    2003

  12. [20]

    Closed- loop data-enabled predictive control and its equivalence with closed- loop subspace predictive control,

    R. Dinkla, S. Mulders, T. Oomen, and J.-W. van Wingerden, “Closed- loop data-enabled predictive control and its equivalence with closed- loop subspace predictive control,” arXiv preprint arXiv:2402.14374 , 2024

    2024 arXiv

  13. [21]

    Bridging direct & indirect data-driven control formulations via regularizations and relaxations,

    F. Dorfler, J. Coulson, and I. Markovsky, “Bridging direct & indirect data-driven control formulations via regularizations and relaxations,” IEEE Transactions on Automatic Control , 2022

    2022

  14. [22]

    Uncertainty- aware data-driven predictive control in a stochastic setting,

    V . Breschi, M. Fabris, S. Formentin, and A. Chiuso, “Uncertainty- aware data-driven predictive control in a stochastic setting,” IFAC- PapersOnLine, vol. 56, no. 2, pp. 10 083–10 088, 2023. APPENDIX I PROOFS Proof of Lemma 1 β = E Yf V T E V VT −1 − H = E (HV + HϵEf ) V T E V...

    2023

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