A data-based image representation for continuous-time LTI systems
Pith reviewed 2026-05-16 09:50 UTC · model grok-4.3
The pith
A numerically stable method computes an image representation of unknown continuous-time linear systems solely from input-output data.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By leveraging a continuous-time fundamental lemma and algebraic differentiators for derivative approximation, one can form a data matrix whose image directly provides a representation of the system's behavior, eliminating the need to identify system parameters or solve DAEs explicitly.
What carries the argument
The data-based image representation formed from stacked matrices of inputs, outputs, and their derivatives approximated by algebraic differentiators.
If this is right
- The method remains numerically stable for continuous-time systems.
- Computational complexity drops because redundant degrees of freedom tied to unknown parameters are removed.
- The image representation can be obtained without explicitly solving differential-algebraic equations.
- The approach retains effectiveness under severe measurement disturbances.
Where Pith is reading between the lines
- The same data-matrix construction could support direct data-driven controller synthesis for continuous-time plants.
- Hardware experiments would expose how derivative-approximation errors propagate into the image subspace.
- Similar matrix constructions might extend to switched linear systems where mode-dependent images are needed.
Load-bearing premise
The underlying system is exactly linear and time-invariant with sufficiently accurate derivative approximations from the data.
What would settle it
For a known LTI system whose true image representation is computable, apply the data method with noisy measurements and check whether the computed image subspace matches the true one within expected noise bounds.
read the original abstract
We derive a numerically stable method to compute an image representation of an unknown linear system only from data, leveraging a continuous-time version of Willems et al.'s fundamental lemma. To this end, we use derivatives approximated by algebraic differentiators. Our novel image representation avoids solving differential-algebraic equations and significantly reduces computational complexity by eliminating redundant degrees of freedom corresponding to the number of unknown quantities to be identified. Simulation results confirm the effectiveness of the proposed approach, even in the presence of severe measurement disturbances.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a data-driven approach to obtain an image representation for continuous-time linear time-invariant systems using a continuous-time version of the fundamental lemma and algebraic differentiators for derivative approximation. It claims this yields a numerically stable method that avoids solving differential-algebraic equations and reduces computational complexity by removing redundant degrees of freedom, with simulations validating performance under noise.
Significance. If the theoretical guarantees hold despite derivative approximations, the method could provide an efficient tool for data-based analysis and control of continuous-time systems, particularly in noisy environments. The avoidance of explicit system identification and DAE solving is a notable advantage, but the lack of error bounds means the significance is currently limited to empirical evidence.
major comments (3)
- [§3 (derivation of image representation)] The application of the continuous-time fundamental lemma to trajectories with approximated derivatives does not include an analysis of how the approximation errors from algebraic differentiators affect the validity of the image representation. This is load-bearing for the claim of an 'exact' representation from noisy data.
- [Abstract and §4 (simulations)] No persistency-of-excitation condition is provided for the noisy, differentiated data, which is necessary for the fundamental lemma to hold. The simulation results demonstrate effectiveness but do not quantify the impact of noise on the representation accuracy or compare computational complexity to alternative methods.
- [§2 (algebraic differentiators)] The choice of differentiator parameters appears tuned for the simulations; without a general bound on the approximation error and its propagation, the numerical stability claim lacks rigorous support.
minor comments (2)
- [Notation] The notation for the image representation could be clarified to distinguish between true and approximated trajectories.
- [References] Ensure all references to algebraic differentiators and the continuous-time fundamental lemma are complete and up-to-date.
Simulated Author's Rebuttal
We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment point by point below, clarifying our approach and outlining planned revisions to strengthen the presentation.
read point-by-point responses
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Referee: [§3 (derivation of image representation)] The application of the continuous-time fundamental lemma to trajectories with approximated derivatives does not include an analysis of how the approximation errors from algebraic differentiators affect the validity of the image representation. This is load-bearing for the claim of an 'exact' representation from noisy data.
Authors: We clarify that the manuscript does not claim an 'exact' representation from noisy data; rather, the image representation is computed exactly from the trajectories obtained after algebraic differentiation, to which the continuous-time fundamental lemma applies directly. The approximation errors are acknowledged as perturbations that affect practical accuracy. In the revised manuscript, we will add a dedicated paragraph in §3 discussing error propagation, drawing on the known convergence rates of algebraic differentiators (error bounds decrease with filter window size). This will make explicit that the representation remains valid for sufficiently accurate differentiators while remaining numerically stable. revision: partial
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Referee: [Abstract and §4 (simulations)] No persistency-of-excitation condition is provided for the noisy, differentiated data, which is necessary for the fundamental lemma to hold. The simulation results demonstrate effectiveness but do not quantify the impact of noise on the representation accuracy or compare computational complexity to alternative methods.
Authors: We agree that a persistency-of-excitation (PE) condition is required. The continuous-time fundamental lemma demands that the input be PE of sufficient order on the underlying signals; the algebraic differentiators are applied afterward and provide inherent noise robustness. In the revision, we will explicitly state this PE requirement for the noise-free trajectories in §3 and add to §4 quantitative noise-impact metrics (e.g., Frobenius-norm errors between the data-based image and a reference representation across noise levels) together with runtime and flop-count comparisons against DAE-based alternatives. These additions will directly address the referee's concerns. revision: yes
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Referee: [§2 (algebraic differentiators)] The choice of differentiator parameters appears tuned for the simulations; without a general bound on the approximation error and its propagation, the numerical stability claim lacks rigorous support.
Authors: The differentiator parameters follow the standard tuning rules from the algebraic-differentiation literature (window length and filter order chosen relative to noise intensity and sampling rate). We will revise §2 to include a concise summary of the existing error bounds for these differentiators, which establish that the approximation error converges to zero as the window size increases. This reference to established bounds will supply the missing rigorous support for the numerical-stability claim without requiring new derivations. revision: yes
Circularity Check
No circularity: derivation applies external continuous-time fundamental lemma to algebraic-differentiator data
full rationale
The paper constructs its image representation by invoking the continuous-time fundamental lemma (Willems et al., independent prior work) on trajectories whose derivatives are obtained from algebraic differentiators. No equation reduces the claimed representation or complexity reduction to a fitted parameter, self-defined quantity, or self-citation chain by construction. The elimination of redundant degrees of freedom follows directly from the lemma's rank conditions applied to the data matrix; algebraic differentiators are treated as an external approximation tool without the paper re-deriving or fitting their properties to match the target result. Simulation results serve only as empirical confirmation, not as the source of the guarantees.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption A continuous-time version of Willems et al.'s fundamental lemma holds for LTI systems.
- domain assumption Algebraic differentiators can approximate derivatives accurately enough from noisy measurements.
discussion (0)
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