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Direct and Indirect Data-Driven Control with Prior Information about the Equilibrium Manifold

Yi-Chun Liao, Valentina Breschi, Marco M. Nicotra

arXiv:2609.28883v1eess.SY

TL;DR

Many data-driven control methods do not use available or readily inferred system priors. This paper incorporates equilibrium-manifold information into indirect and direct LQR, reporting improved model and closed-loop results while also identifying a limitation on direct use.

  • Problem

    Many data-driven control approaches assume a fully unknown system and do not leverage available or readily inferable priors.

    Source: §I · p. 1

  • Method

    For indirect control, the paper constrains identification using the equilibrium subspace; for direct control, it uses that information to recover open-loop matrices from the closed-loop matrix.

    Source: §IV · p. 3, A. Exploiting the Equilibrium Subspace… · p. 4

  • Results

    The equilibrium-informed indirect method better approximates the true system model than the classic indirect approach, and experiments show improved indirect closed-loop performance.

    Source: §VII · p. 5, §VIII · p. 5

  • Takeaways & Limitations

    Numerical and experimental results provide evidence of potential advantages and drawbacks of using equilibrium-manifold knowledge in direct and indirect data-driven methods.

    Source: §IX · p. 6

    In the direct method, equilibrium-subspace information cannot explicitly guide controller design, though it can recover open-loop matrices from the closed-loop matrix.

    Source: A. Exploiting the Equilibrium Subspace… · p. 4

Abstract

from arXiv · show

By hinging on the assumption that a system to be controlled is fully unknown, many data-driven control approaches do not leverage available or readily inferable priors. In contrast to this viewpoint, this paper analyzes the impact of using the system's equilibrium subspace to inform direct and indirect linear quadratic regulation. For the indirect case, we show how including a constraint on the equilibrium subspace in the identification problem changes the statistical properties of the learned model. In particular, we show that enforcing consistency with respect to the equilibrium subspace leads to a reduction in the estimator variance that, in turn, enhances model-based control performance. In the direct case, we show how this prior can be leveraged to gain insight into the controlled system without requiring an explicit identification step. These results are supported by both numerical and experimental evidence, showcasing the advantages of explicitly leveraging the equilibrium manifold as a prior in data-driven control.

I. INTRODUCTION

Data-driven control can identify a model before designing a controller or synthesize controller gains directly, while many approaches assume the system is fully unknown. This paper studies using prior knowledge of the equilibrium manifold in LQR design for unknown systems.

I. INTRODUCTION

  • Indirect methods identify a system model before computing feedback gains, whereas direct methods synthesize gains from data without explicit identification.Source: §I · p. 1
  • Prior work incorporated several system properties into indirect methods, while direct-method priors had focused on known or bounded system parameters.The paper instead investigates equilibrium-subspace information, which may be available a priori or estimated from steady-state measurements.Source: §I · p. 1

II. SETTING & GOAL

  • The study examines whether equilibrium-manifold knowledge can inform indirect and direct LQR for unknown systems.Source: §I · p. 1, §II · p. 2

III. PRELIMINARIES

The preliminaries represent steady-state equilibria through an equilibrium subspace and describe how that information relates to feedback control. They also establish when the equilibrium relation can constrain identification.

A. Equilibrium Subspace

B. Linear Quadratic Regulator

The LQR preliminaries express the design in error coordinates around an equilibrium. The resulting design is independent of which equilibrium point is stabilized.

B. Linear Quadratic Regulator

IV. INDIRECT DATA-DRIVEN CONTROL WITH INSIGHTS ON EQUILIBRIUM MANIFOLDS

The indirect approach imposes equilibrium-subspace consistency in system identification, addressing possible inconsistency with known equilibria and changing the learned model’s statistical properties.

IV. INDIRECT DATA-DRIVEN CONTROL WITH INSIGHTS ON EQUILIBRIUM MANIFOLDS

  • The conventional indirect pipeline estimates A and B from data, then computes the LQR gain using the model.The gain can be designed under certainty equivalence or with regularization for modeling errors.Source: §IV · p. 2
  • Adding the equilibrium condition to least squares addresses the possibility that standard estimates conflict with an available equilibrium subspace.The unconstrained problem does not account for the available equilibrium information.Source: §IV · p. 3
  • The equilibrium prior always affects parameter variance and changes bias when noise signals are correlated with the measured state and input.Under the stated assumption, estimates are unbiased when the noise is uncorrelated with those signals.Source: §IV · p. 3

V. DIRECT DATA-DRIVEN CONTROL

The direct approach designs the LQR without a preliminary identification step, using data to parameterize closed-loop dynamics. The resulting solution provides the feedback gain and an approximate closed-loop state matrix.

V. DIRECT DATA-DRIVEN CONTROL

  • Direct design uses data to parameterize closed-loop dynamics without first identifying the open-loop model.The approach uses covariance transform matrices to build the data-driven closed-loop representation.Source: §V · pp. 3–4
  • For sufficiently large T such that W̃0≈0, the paper recasts the LQR design problem using the data-driven representation.The passage states this condition but does not specify a numerical threshold for T.Source: §V · p. 4
  • The solution V gives the feedback gain K=Ũ0V and approximate closed-loop state matrix Aπ≈X̃1V.Source: §V · p. 4

A. Exploiting the Equilibrium Subspace & Opening the Loop

The equilibrium subspace does not add information to the direct controller optimization, but it can be used afterward to recover open-loop dynamics from the closed-loop matrix.

A. Exploiting the Equilibrium Subspace & Opening the Loop

VI. EQUILIBRIUM SUBSPACE IDENTIFICATION

The equilibrium map may be known beforehand or estimated from static experiments. The procedure uses reference-driven closed-loop responses to estimate equilibrium points and recover the map matrices.

VI. EQUILIBRIUM SUBSPACE IDENTIFICATION

  • Equilibrium-map matrices may be known a priori or estimated using static experiments when unavailable.Mechanical systems can have trivial equilibria, such as position at the reference and zero velocity, regardless of their dynamics.Source: §VI · p. 4
  • The procedure uses at least m references spanning an m-dimensional reference space and runs independent experiments with inputs uk|i=Kxk|i+ri.The closed-loop responses satisfy input-to-state stable equilibrium conditions for each experiment.Source: §VI · p. 4
  • Equilibrium-map matrices are obtained from the equilibrium state and input data, with equilibrium points estimated from response data.τ must be large enough for transients to deplete, and Teq is the averaging window.Source: §VI · p. 4

VII. NUMERICAL VALIDATION

The numerical validation compares equilibrium-informed approaches with a baseline. The equilibrium prior improves indirect model approximation, while opening the direct method yields a lower-quality model estimate than indirect methods.

VII. NUMERICAL VALIDATION

  • The numerical test used system parameters k = 100 and c = 2, then discretized the continuous-time model with a zero-order hold at τs = 0.01.Source: §VII · p. 5
  • The approaches were compared over 1,000 Monte Carlo simulations with white Gaussian process noise of standard deviation 0.015.The comparison used the error metric referenced in the passage.Source: §VII · p. 5
  • Equilibrium-informed indirect identification approximates the true system better than the classic indirect approach.Table I summarizes error statistics across 1,000 Monte Carlo trials.Source: Table I · p. 5, §VII · p. 5
  • Opening the direct method estimates the open-loop system, but its estimate is lower quality than indirect methods because direct control prioritizes the control objective over fitting.The direct-method gain was used for a fair comparison.Source: §VII · p. 5

VIII. EXPERIMENTAL VALIDATION

In the rotary flexible-joint experiment, the equilibrium-prior indirect method improves the closed-loop response to be comparable to direct control. Its transient is faster, but requires more aggressive action.

VIII. EXPERIMENTAL VALIDATION

  • The test uses a rotary flexible-joint system without input u2; its second degree of freedom uses error coordinates, and the controllers run at 500 Hz.Initial data were collected over 20 seconds.Source: §VIII · p. 5
  • The classic indirect method produces a significant steady-state mismatch between measured and identified-model responses.Source: §VIII · p. 5, Figure 3 · p. 6
  • The covariance-based direct method achieves virtually no steady-state error, even without an integrator.Source: §VIII · p. 5, Figure 4 · p. 6
  • The prior-augmented indirect response has negligible steady-state error, lower overshoot than direct control, and a faster transient, at the cost of more aggressive action.Its closed-loop response is comparable to the direct method.Source: §VIII · pp. 5–6, Figure 5 · p. 6

IX. CONCLUSION

The conclusion describes how equilibrium-manifold knowledge can inform both indirect and direct data-driven control. It summarizes the indirect identification constraint, direct open-loop recovery, and evidence of potential advantages and drawbacks.

IX. CONCLUSION

  • For indirect methods, equilibrium-manifold knowledge can be embedded directly in system identification.Source: §IX · p. 6
  • For direct methods, the prior can recover an estimate of open-loop dynamics from the closed-loop model.Source: §IX · p. 6
  • Simulation and experimental results provide evidence of potential advantages and drawbacks of the proposed procedure.Source: §IX · p. 6
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