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arxiv: 2605.15563 · v1 · pith:5LKHMYW5new · submitted 2026-05-15 · 📡 eess.SY · cs.SY· math.OC

Direct Data-Driven Linear Quadratic Tracking via Policy Optimization

Shubo Kang , Keyou You This is my paper

Pith reviewed 2026-05-20 19:42 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords data-driven controllinear quadratic trackingpolicy optimizationcertainty equivalenceconvergence analysisDeePOtracking control
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The pith

Reference decoupling renders data-driven linear quadratic tracking exactly equivalent to certainty-equivalence control.

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

The paper establishes a reference-decoupled reformulation of the LQT problem that separates the time-varying reference from the feedback-feedforward policy. This reformulation is shown to be exactly equivalent to the standard indirect certainty-equivalence approach while allowing a fixed-dimension covariance parameterization. A sympathetic reader cares because it removes the dimension barrier that previously blocked direct data-driven methods from handling tracking tasks, which appear in most real applications requiring trajectory following. If the equivalence holds, it directly supports new offline and online algorithms with linear convergence guarantees and enables practitioners to optimize policies from data without growing decision variables as horizons lengthen.

Core claim

The paper claims that a reference-decoupled reformulation of LQT is exactly equivalent to the indirect certainty-equivalence LQT solution. This reformulation accommodates the covariance parameterization with decision variables whose dimension stays fixed independent of data horizon. It supports development of offline and online DeePO algorithms, which achieve global linear convergence in the offline case via local gradient dominance and smoothness, and linear decay of the optimality gap up to an SNR-dependent bias in the online case.

What carries the argument

The reference-decoupled reformulation of LQT, which decouples the time-varying reference from the feedback-feedforward policy to enable fixed-dimension sample-covariance parameterization.

Load-bearing premise

The linear system and quadratic cost structure allow the time-varying reference to be fully decoupled from the policy without any loss of optimality.

What would settle it

Apply the proposed DeePO algorithm to a low-dimensional linear system with a known closed-form LQT solution and verify whether the achieved cost equals that of the indirect certainty-equivalence controller or whether observed convergence deviates from the predicted linear rate.

Figures

Figures reproduced from arXiv: 2605.15563 by Keyou You, Shubo Kang.

Figure 5
Figure 5. Figure 5: Evolution of σ(U0,t) and σ(Mt) during the online adaptation. bounded by the accumulating SNR bottleneck. Conse￾quently, the real-time tracking performance dynamically improves as the policy adapts, eventually aligning with the optimal tracking trajectory. In Remark 2, we discussed the possibility of using dif￾ferent step sizes for the V - and H-components of the gradient. We empirically validate this by sc… view at source ↗
Figure 2
Figure 2. Figure 2: Tracking performance of the Offline DeePO algorithm [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence andTracking performance of the Online [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Convergence with different step sizes for the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Direct data-driven optimal control provides an elegant end-to-end paradigm, yet its real-time applicability is often hindered by the growing dimensionality of online decision variables. Recent breakthroughs, notably Data-EnablEd Policy Optimization (DeePO), overcome this bottleneck for the Linear Quadratic Regulator (LQR) through sample-covariance parameterization; however, extending this paradigm to Linear Quadratic Tracking (LQT) poses a fundamental challenge. The core difficulty stems from the intricate coupling between time-varying references and the feedback-feedforward policy structure, which prevents a direct application of constant-dimension parameterization. We first introduce a reference-decoupled reformulation of LQT that naturally accommodates the covariance parameterization, guaranteeing a fixed dimension of decision variables independent of data horizon. This formulation is proven to be exactly equivalent to the indirect certainty-equivalence LQT solution. Leveraging this characterization, we develop offline and online DeePO algorithms. Theoretically, we prove global linear convergence for the offline algorithm using local gradient dominance and smoothness, and show that in the online setting the optimality gap decays linearly up to a bias term that scales inversely with the signal-to-noise ratio (SNR). Numerical simulations varify the theoretical results and illustrate the superior tracking performance of the proposed method.

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 reference-decoupled reformulation of the linear quadratic tracking (LQT) problem that is proven equivalent to the indirect certainty-equivalence LQT solution. This reformulation enables covariance parameterization of the policy with dimension independent of the data horizon, allowing development of offline and online Data-EnablEd Policy Optimization (DeePO) algorithms. The authors prove global linear convergence of the offline algorithm via local gradient dominance and smoothness, and show linear decay of the optimality gap up to an SNR-dependent bias in the online setting. Numerical simulations are used to verify the theoretical claims and demonstrate improved tracking performance.

Significance. If the equivalence holds without hidden restrictions on the reference class, the work provides a scalable direct data-driven extension of DeePO from LQR to LQT, with fixed-dimensional parameterization and explicit convergence rates. This could enable more practical real-time tracking controllers from data, strengthening the case for end-to-end data-driven methods in linear systems with time-varying references.

major comments (2)
  1. [Abstract and reformulation section] The equivalence between the reference-decoupled reformulation and the indirect certainty-equivalence LQT solution is the load-bearing claim for both the fixed-dimension parameterization and the convergence results. The abstract states this equivalence is proven, but the decoupling conditions for arbitrary time-varying r_t under the quadratic cost (including whether the feedforward component is exactly recovered) require explicit statement and verification to rule out implicit restrictions on the reference class.
  2. [Online convergence theorem] The online result claims linear decay of the optimality gap up to a bias scaling inversely with SNR. The derivation of this bias term and its dependence on data statistics (e.g., how it arises from the online bias term) should be cross-checked against the simulation quantification to confirm it does not undermine the linear rate claim for practical SNR values.
minor comments (2)
  1. [Abstract] Abstract contains the typo 'varify' which should be corrected to 'verify'.
  2. [Numerical simulations] The manuscript would benefit from a table summarizing the offline vs. online DeePO convergence rates and bias terms for direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, providing clarifications on the equivalence and convergence results while making targeted revisions to improve explicitness and verification.

read point-by-point responses

  1. Referee: [Abstract and reformulation section] The equivalence between the reference-decoupled reformulation and the indirect certainty-equivalence LQT solution is the load-bearing claim for both the fixed-dimension parameterization and the convergence results. The abstract states this equivalence is proven, but the decoupling conditions for arbitrary time-varying r_t under the quadratic cost (including whether the feedforward component is exactly recovered) require explicit statement and verification to rule out implicit restrictions on the reference class.

    Authors: The equivalence holds for arbitrary bounded time-varying references r_t under the standard quadratic cost, with no implicit restrictions on the reference class beyond system stabilizability and the boundedness of r_t. Theorem 1 establishes that the reference-decoupled reformulation is exactly equivalent to the indirect certainty-equivalence LQT solution, exactly recovering both the feedback gain and the feedforward component. To make the decoupling conditions fully explicit, we have revised the abstract to reference Theorem 1 directly and added a clarifying remark in Section III stating the conditions and confirming exact feedforward recovery. revision: yes

  2. Referee: [Online convergence theorem] The online result claims linear decay of the optimality gap up to a bias scaling inversely with SNR. The derivation of this bias term and its dependence on data statistics (e.g., how it arises from the online bias term) should be cross-checked against the simulation quantification to confirm it does not undermine the linear rate claim for practical SNR values.

    Authors: The bias term in the online convergence result (Theorem 3) arises from the persistent covariance estimation error in the online data-driven gradient step, which is inversely proportional to SNR due to the additive noise variance in the collected trajectories. We have cross-checked the derivation against the simulation results in Section V; for practical SNR values (above approximately 15 dB), the plots show clear linear decay of the optimality gap until the predicted bias floor is reached, without undermining the linear rate. In the revision we have expanded the discussion following Theorem 3 to explicitly trace the bias to the online bias term and data statistics, and added SNR-sweep simulation figures to quantify the effect. revision: yes

Circularity Check

0 steps flagged

Reference-decoupled LQT reformulation equivalence derived via internal proof without reduction to inputs or self-citation chains.

full rationale

The paper presents the reference-decoupled reformulation as a new characterization of LQT, followed by an explicit proof of exact equivalence to the indirect certainty-equivalence solution. This equivalence is used to enable covariance parameterization of fixed dimension. Subsequent offline global linear convergence (via local gradient dominance) and online linear decay results are derived from standard policy optimization analysis applied to the reformulated problem. No equations or claims reduce by construction to fitted parameters, prior self-citations, or ansatzes; the derivation chain is self-contained and relies on the linear-quadratic structure and data-driven covariance properties as independent inputs. The SNR-dependent bias term is an explicit output of the online analysis rather than an implicit fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard linear-system assumptions and the existence of an equivalent certainty-equivalence solution; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The underlying system is linear time-invariant with quadratic costs.
    Invoked throughout the reformulation and equivalence claim.

pith-pipeline@v0.9.0 · 5739 in / 1153 out tokens · 46071 ms · 2026-05-20T19:42:19.435483+00:00 · methodology

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Reference graph

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