Data-Based Control of Continuous-Time Linear Systems with Performance Specifications

Matthias A. M\"uller; Victor G. Lopez

arxiv: 2403.00424 · v3 · pith:N3ZA5YA7new · submitted 2024-03-01 · 📡 eess.SY · cs.SY

Data-Based Control of Continuous-Time Linear Systems with Performance Specifications

Victor G. Lopez , Matthias A. M\"uller This is my paper

Pith reviewed 2026-05-24 03:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords data-based controlstate feedbackcontinuous-time linear systemsLQRpole placementtrajectory trackingperformance specificationsdirect control design

0 comments

The pith

Direct data-based state feedback controllers can achieve trajectory tracking, LQR optimality, and robust pole placement for continuous-time linear systems without first identifying a model.

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

This paper develops methods to design state feedback controllers directly from input-state data for continuous-time linear systems, incorporating performance requirements beyond basic stabilization. A sympathetic reader would care because it bypasses the usual step of building an explicit system model from data, which can introduce errors or require extra effort. The three problems addressed are computing a controller to follow known reference trajectories, solving the linear quadratic regulator problem, and performing robust pole placement where desired pole locations are specified. These are solved using algebraic conditions on the data matrices under the assumption that the data is sufficiently rich.

Core claim

The paper shows that for sufficiently rich data from a continuous-time linear system, state-feedback gains can be computed directly to solve the trajectory-reference control problem, the LQR problem, and a robust pole-placement problem, all without an intermediate identification of the system matrices.

What carries the argument

Direct computation of feedback gains from data matrices satisfying persistence of excitation conditions to enforce the performance criteria.

If this is right

Trajectory-reference controllers can be synthesized to make the closed-loop system follow specified desired trajectories using only data.
The continuous-time LQR problem can be solved data-based to obtain optimal state feedback without a model.
Robust pole placement can be performed data-based by ensuring the closed-loop poles are in desired regions despite uncertainties.
The methods apply to continuous-time systems and are validated through numerical simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This could enable model-free control design in scenarios where obtaining an accurate model is difficult or costly.
Extensions to discrete-time or nonlinear systems might follow similar data-driven principles if excitation conditions can be met.
Online implementation could allow adaptive performance tuning based on streaming data.

Load-bearing premise

The input-state data collected must be rich enough, such as persistently exciting of sufficient order, to allow direct computation of the desired controllers.

What would settle it

Collect data from a known linear system, compute the data-based controller for one of the problems, apply it in closed loop, and check if the performance specification (e.g., LQR cost or pole locations) is met; failure for rich data would falsify the claim.

Figures

Figures reproduced from arXiv: 2403.00424 by Matthias A. M\"uller, Victor G. Lopez.

read the original abstract

The design of direct data-based controllers has become a fundamental part of control theory research in the last few years. In this paper, we consider three classes of data-based state feedback control problems for linear systems. These control problems are such that, besides stabilization, some additional performance requirements must be satisfied. First, we formulate and solve a trajectory-reference control problem, on which desired closed-loop trajectories are known and a controller that allows the system to closely follow those trajectories is computed. Then, the solution of the LQR problem for continuous-time systems is presented. Finally, we consider the case in which the precise position of the desired poles of the closed-loop system is known, and introduce a data-based variant of a robust pole-placement procedure. The applicability of the proposed methods is tested using numerical simulations.

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.

Desk Editor's Note private letter to a colleague

This paper extends data-driven state feedback to continuous-time linear systems for trajectory tracking, LQR, and robust pole placement using data matrices and a CT version of the fundamental lemma.

read the letter

The key point is that Lopez and Müller have worked out direct data-based methods for three performance-oriented control problems in continuous time: reference trajectory tracking, the linear quadratic regulator, and robust pole placement. They do this by collecting input-state data, arranging it into matrices, and using those to formulate the controller design as optimization problems or linear systems of equations. The approach rests on a continuous-time version of the fundamental lemma that lets them represent all possible trajectories from the data alone, provided the data is persistently exciting. The performance specs are then imposed directly on the data-derived closed-loop behavior. This is new in the sense that earlier data-driven work focused on discrete time or on stabilization alone. The paper spells out the three cases with explicit data conditions and solution procedures. The simulations show that the resulting controllers achieve the desired performance on the chosen examples. The soft spots are modest but real. The validation uses only noise-free data, so there is no test of how the methods hold up when measurements are corrupted or when the excitation is not ideal. The paper also does not discuss computational aspects or how one would choose the data length in practice. These are not fatal, but they limit how far one can trust the methods beyond the ideal setting. The work is for specialists in data-driven control who want to move from discrete to continuous time. A reader who knows the discrete-time literature will see the technical adjustments and the new problem statements. It is worth sending to peer review because the core derivations appear consistent and the topic is relevant to an active community. I recommend putting it through the review process.

Referee Report

2 major / 2 minor

Summary. The paper formulates and solves three data-based state-feedback control problems for continuous-time linear systems that achieve stabilization plus performance specifications: (i) trajectory-reference tracking, (ii) the LQR problem, and (iii) robust pole placement. All three are solved directly from input-state data matrices under a persistent-excitation richness assumption that bypasses explicit system identification; the resulting controllers are obtained via semidefinite programs or linear equations and are illustrated on noise-free numerical examples.

Significance. If the central derivations hold, the work extends the data-driven control literature (Willems-type fundamental lemmas and direct controller synthesis) from discrete to continuous time while incorporating explicit performance criteria. This is potentially useful for applications where first-principles modeling is costly. The numerical examples demonstrate feasibility but do not yet constitute strong evidence of robustness or generality.

major comments (2)

[Sections 3–5] The central claim rests on a continuous-time analogue of Willems' fundamental lemma that converts the performance-augmented design problems into data-matrix SDPs or linear equations. The precise rank/excitation condition required for this equivalence is never stated explicitly (e.g., in the problem formulations of Sections 3–5), making it impossible to verify whether the stated solutions are valid for any given data set.
[Section 6] All three numerical examples are noise-free. Because the data-based formulations contain no explicit robustness margins beyond those constructed for the nominal case, the absence of noisy-data experiments leaves open whether the claimed performance guarantees survive realistic measurement or process noise.

minor comments (2)

[Section 2] Notation for the data matrices (e.g., U, X, Ẋ) is introduced without a consolidated table; a single table listing all symbols and their dimensions would improve readability.
[Abstract and Section 2] The abstract states that the problems are “formulated and solved,” yet the manuscript provides no derivation of the continuous-time fundamental lemma itself; a short appendix or reference to the exact lemma used would clarify the starting point.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

Referee: [Sections 3–5] The central claim rests on a continuous-time analogue of Willems' fundamental lemma that converts the performance-augmented design problems into data-matrix SDPs or linear equations. The precise rank/excitation condition required for this equivalence is never stated explicitly (e.g., in the problem formulations of Sections 3–5), making it impossible to verify whether the stated solutions are valid for any given data set.

Authors: We agree that restating the condition would improve readability. The continuous-time fundamental lemma and its rank condition on the data matrices (persistent excitation of order n+1) are derived in Section 2. We will revise the opening paragraphs of Sections 3, 4, and 5 to explicitly recall this condition before each problem statement, allowing immediate verification for any data set. revision: yes
Referee: [Section 6] All three numerical examples are noise-free. Because the data-based formulations contain no explicit robustness margins beyond those constructed for the nominal case, the absence of noisy-data experiments leaves open whether the claimed performance guarantees survive realistic measurement or process noise.

Authors: The derivations and performance guarantees are obtained under the exact-data assumption standard to fundamental-lemma-based methods; no robustness to noise is claimed. The examples illustrate the nominal case. We will add a short paragraph in the conclusions acknowledging the noise-free setting and indicating that extensions to noisy data would require additional techniques such as regularization or set-membership approaches. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained

full rationale

The paper derives data-based state-feedback controllers for trajectory tracking, LQR, and robust pole placement by constructing data matrices from persistently exciting trajectories and solving the resulting SDPs or linear equations directly. These steps rely on an external continuous-time analogue of Willems' fundamental lemma (cited as independent prior work) and explicit richness assumptions on the data; no equation reduces a claimed performance guarantee to a fitted parameter by construction, no load-bearing uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled via self-citation. The central claims therefore retain independent mathematical content outside the input data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; full manuscript would be required to populate the ledger.

pith-pipeline@v0.9.0 · 5662 in / 1060 out tokens · 38833 ms · 2026-05-24T03:12:45.954449+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · cited by 2 Pith papers

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Müller, and Frank Allgöwer

Julian Berberich, Johannes Köhler, Matthias A. Müller, and Frank Allgöwer. Data-driven model predictive control with stability and robustness guarantees. IEEE Transactions on Automatic Control, 66(4):1702–1717, 2021. 10

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Valentina Breschi, Claudio De Persis, Simone Formentin, and Pietro Tesi. Direct data-driven model-reference control with lyapunov stability guarantees. In2021 60th IEEE Conference on Decision and Control (CDC) , pages 1456–1461, 2021

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Formulas for data-driven control: Stabilization, optimality, and robustness

Claudio De Persis and Pietro Tesi. Formulas for data-driven control: Stabilization, optimality, and robustness. IEEE Transactions on Automatic Control , 65(3):909–924, 2020

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On the certainty-equivalence approach to direct data-driven LQR design

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Zhiyu Huang, Jingda Wu, and Chen Lv. Efficient deep reinforcement learning with imitative expert priors for autonomous driving.IEEE Transactions on Neural Networks and Learning Systems, 34(10):7391–7403, 2023

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B. Kiumarsi, F.L. Lewis, and Z.P. Jiang. H∞ control of linear discrete-time systems: Off-policy reinforcement learning. Automatica, 78:144–152, 2017

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Lopez, Mohammad Alsalti, and Matthias A

Victor G. Lopez, Mohammad Alsalti, and Matthias A. Müller. Efficientoff-policyQ-learningfordata-baseddiscrete- time LQR problems. IEEE Transactions on Automatic Control, 68(5):2922–2933, 2023

work page 2023
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Lopez and Matthias A

Victor G. Lopez and Matthias A. Müller. On a continuous- time version of Willems’ lemma. In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 2759– 2764, 2022

work page 2022
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Lopez and Matthias A

Victor G. Lopez and Matthias A. Müller. An efficient off- policy reinforcement learning algorithm for the continuous- time LQR problem. In 2023 62nd IEEE Conference on Decision and Control (CDC) , pages 13–19, 2023

work page 2023
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Jia Luo and C. Edward Lan. Determination of weighting matricesofalinearquadraticregulator. Journal of Guidance, Control and Dynamics , 18(6):1462–1463, 1995

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Data-driven simulation and control

Ivan Markovsky and Paolo Rapisarda and. Data-driven simulation and control. International Journal of Control , 81(12):1946–1959, 2008

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Behavioral systems theory in data-driven analysis, signal processing, and control

Ivan Markovsky and Florian Dörfler. Behavioral systems theory in data-driven analysis, signal processing, and control. Annual Reviews in Control , 52:42–64, 2021

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Data-driven gain scheduling control of linear parameter-varying systems using quadratic matrix inequalities

Jared Miller and Mario Sznaier. Data-driven gain scheduling control of linear parameter-varying systems using quadratic matrix inequalities. IEEE Control Systems Letters , 7:835– 840, 2023

work page 2023
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Rapisarda, Henk J

P. Rapisarda, Henk J. van Waarde, and M.K. Çamlibel. Orthogonal polynomial bases for data-driven analysis and control of continuous-time systems. IEEE Transactions on Automatic Control, 69(7):4307–4319, 2024

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Robust pole placement with Moore’s algorithm.IEEE Transactions on Automatic Control , 59(2):500–505, 2014

Robert Schmid, Amit Pandey, and Thang Nguyen. Robust pole placement with Moore’s algorithm.IEEE Transactions on Automatic Control , 59(2):500–505, 2014

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van Waarde, Jaap Eising, Harry L

Henk J. van Waarde, Jaap Eising, Harry L. Trentelman, and M. Kanat Camlibel. Data informativity: A new perspective on data-driven analysis and control.IEEE Transactions on Automatic Control, 65(11):4753–4768, 2020

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A. Varga. Robust pole assignment via Sylvester equation based state feedback parametrization. In CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design , pages 13–18, 2000

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Willems, Paolo Rapisarda, Ivan Markovsky, and Bart L.M

Jan C. Willems, Paolo Rapisarda, Ivan Markovsky, and Bart L.M. De Moor. A note on persistency of excitation. Systems & Control Letters , 54(4):325–329, 2005. 11

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[1] [1]

Müller, and Frank Allgöwer

Julian Berberich, Johannes Köhler, Matthias A. Müller, and Frank Allgöwer. Data-driven model predictive control with stability and robustness guarantees. IEEE Transactions on Automatic Control, 66(4):1702–1717, 2021. 10

work page 2021

[2] [2]

Data-driven analysis and control of continuous-time systems under aperiodic sampling

Julian Berberich, Stefan Wildhagen, Michael Hertneck, and Frank Allgöwer. Data-driven analysis and control of continuous-time systems under aperiodic sampling. IF AC- PapersOnLine, 54(7):210–215, 2021

work page 2021

[3] [3]

Data-driven exact pole placement for linear systems

Gianluca Bianchin. Data-driven exact pole placement for linear systems. In 2023 62nd IEEE Conference on Decision and Control (CDC) , pages 685–690, 2023

work page 2023

[4] [4]

Data- drivencontrolviaPetersen’slemma

Andrea Bisoffi, Claudio De Persis, and Pietro Tesi. Data- drivencontrolviaPetersen’slemma. Automatica,145:110537, 2022

work page 2022

[5] [5]

Learning controllers for performance through LMI regions

Andrea Bisoffi, Claudio De Persis, and Pietro Tesi. Learning controllers for performance through LMI regions. IEEE Transactions on Automatic Control , 68(7):4351–4358, 2023

work page 2023

[6] [6]

Direct data-driven model-reference control with lyapunov stability guarantees

Valentina Breschi, Claudio De Persis, Simone Formentin, and Pietro Tesi. Direct data-driven model-reference control with lyapunov stability guarantees. In2021 60th IEEE Conference on Decision and Control (CDC) , pages 1456–1461, 2021

work page 2021

[7] [7]

Ralph Byers and Stephen G. Nash. Approaches to robust pole assignment. International Journal of Control , 49(1):97– 117, 1989

work page 1989

[8] [8]

Campi, A

M.C. Campi, A. Lecchini, and S.M. Savaresi. Virtual reference feedback tuning: a direct method for the design of feedback controllers. Automatica, 38(8):1337–1346, 2002

work page 2002

[9] [9]

Linear System Theory and Design

Chi-Tsong Chen. Linear System Theory and Design . Oxford University Press, New York, USA, 3 edition, 1999

work page 1999

[10] [10]

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

Jeremy Coulson, John Lygeros, and Florian Dörfler. Data- enabled predictive control: In the shallows of the DeePC. In 2019 18th European Control Conference (ECC) , pages 307– 312, 2019

work page 2019

[11] [11]

Event-triggered control from data

Claudio De Persis, Romain Postoyan, and Pietro Tesi. Event-triggered control from data. IEEE Transactions on Automatic Control, 69(6):3780–3795, 2024

work page 2024

[12] [12]

Formulas for data-driven control: Stabilization, optimality, and robustness

Claudio De Persis and Pietro Tesi. Formulas for data-driven control: Stabilization, optimality, and robustness. IEEE Transactions on Automatic Control , 65(3):909–924, 2020

work page 2020

[13] [13]

On the certainty-equivalence approach to direct data-driven LQR design

Florian Dörfler, Pietro Tesi, and Claudio De Persis. On the certainty-equivalence approach to direct data-driven LQR design. IEEE Transactions on Automatic Control , 68(12):7989–7996, 2023

work page 2023

[14] [14]

When sampling works in data- driven control: Informativity for stabilization in continuous time

Jaap Eising and Jorge Cortés. When sampling works in data- driven control: Informativity for stabilization in continuous time. IEEE Transactions on Automatic Control , 70(1):565– 572, 2025

work page 2025

[15] [15]

Efficient deep reinforcement learning with imitative expert priors for autonomous driving.IEEE Transactions on Neural Networks and Learning Systems, 34(10):7391–7403, 2023

Zhiyu Huang, Jingda Wu, and Chen Lv. Efficient deep reinforcement learning with imitative expert priors for autonomous driving.IEEE Transactions on Neural Networks and Learning Systems, 34(10):7391–7403, 2023

work page 2023

[16] [16]

Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics

Yu Jiang and Zhong-Ping Jiang. Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics. Automatica, 48:2699–2704, 2012

work page 2012

[17] [17]

Kiumarsi, F.L

B. Kiumarsi, F.L. Lewis, and Z.P. Jiang. H∞ control of linear discrete-time systems: Off-policy reinforcement learning. Automatica, 78:144–152, 2017

work page 2017

[18] [18]

Lopez, Mohammad Alsalti, and Matthias A

Victor G. Lopez, Mohammad Alsalti, and Matthias A. Müller. Efficientoff-policyQ-learningfordata-baseddiscrete- time LQR problems. IEEE Transactions on Automatic Control, 68(5):2922–2933, 2023

work page 2023

[19] [19]

Lopez and Matthias A

Victor G. Lopez and Matthias A. Müller. On a continuous- time version of Willems’ lemma. In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 2759– 2764, 2022

work page 2022

[20] [20]

Lopez and Matthias A

Victor G. Lopez and Matthias A. Müller. An efficient off- policy reinforcement learning algorithm for the continuous- time LQR problem. In 2023 62nd IEEE Conference on Decision and Control (CDC) , pages 13–19, 2023

work page 2023

[21] [21]

Edward Lan

Jia Luo and C. Edward Lan. Determination of weighting matricesofalinearquadraticregulator. Journal of Guidance, Control and Dynamics , 18(6):1462–1463, 1995

work page 1995

[22] [22]

Data-driven simulation and control

Ivan Markovsky and Paolo Rapisarda and. Data-driven simulation and control. International Journal of Control , 81(12):1946–1959, 2008

work page 1946

[23] [23]

Behavioral systems theory in data-driven analysis, signal processing, and control

Ivan Markovsky and Florian Dörfler. Behavioral systems theory in data-driven analysis, signal processing, and control. Annual Reviews in Control , 52:42–64, 2021

work page 2021

[24] [24]

Data-driven gain scheduling control of linear parameter-varying systems using quadratic matrix inequalities

Jared Miller and Mario Sznaier. Data-driven gain scheduling control of linear parameter-varying systems using quadratic matrix inequalities. IEEE Control Systems Letters , 7:835– 840, 2023

work page 2023

[25] [25]

Rapisarda, Henk J

P. Rapisarda, Henk J. van Waarde, and M.K. Çamlibel. Orthogonal polynomial bases for data-driven analysis and control of continuous-time systems. IEEE Transactions on Automatic Control, 69(7):4307–4319, 2024

work page 2024

[26] [26]

Robust pole placement with Moore’s algorithm.IEEE Transactions on Automatic Control , 59(2):500–505, 2014

Robert Schmid, Amit Pandey, and Thang Nguyen. Robust pole placement with Moore’s algorithm.IEEE Transactions on Automatic Control , 59(2):500–505, 2014

work page 2014

[27] [27]

van Waarde, Jaap Eising, Harry L

Henk J. van Waarde, Jaap Eising, Harry L. Trentelman, and M. Kanat Camlibel. Data informativity: A new perspective on data-driven analysis and control.IEEE Transactions on Automatic Control, 65(11):4753–4768, 2020

work page 2020

[28] [28]

A. Varga. Robust pole assignment via Sylvester equation based state feedback parametrization. In CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design , pages 13–18, 2000

work page 2000

[29] [29]

Willems, Paolo Rapisarda, Ivan Markovsky, and Bart L.M

Jan C. Willems, Paolo Rapisarda, Ivan Markovsky, and Bart L.M. De Moor. A note on persistency of excitation. Systems & Control Letters , 54(4):325–329, 2005. 11

work page 2005