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arxiv: 2605.15318 · v1 · pith:PWG4EZPVnew · submitted 2026-05-14 · 📡 eess.SY · cs.SY· eess.SP

Continuous-time Predictor-Based Subspace Identification with Hermite basis expansions

Jose Antonio Rebollo , Enrico Barbiero , Marco Lovera This is my paper

Pith reviewed 2026-05-19 16:02 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP
keywords continuous-time identificationsubspace methodsHermite basis functionspredictor-based subspace identificationlinear systemssystem identificationstate-space models
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The pith

Projecting signals onto Hermite basis functions allows direct identification of continuous-time state-space models for LTI systems.

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

The paper presents a method for continuous-time subspace identification of linear time-invariant systems. Signals are projected onto a finite number of Hermite basis functions to obtain coefficients. Recursive relations and derivative properties of these functions are used to express the derivative operator and apply it recursively. This enables implementation of predictor-based subspace identification steps while keeping the model in continuous time and avoiding time-shifts in the data. The approach is demonstrated through simulations comparing it to a Laguerre-based method.

Core claim

The HD-PBSID method directly identifies a continuous-time state-space form by projecting signals onto Hermite basis functions, exploiting their recursive relations and derivative properties to implement steps akin to PBSID while avoiding time-shifts.

What carries the argument

Hermite basis expansions combined with recursive application of the derivative operator derived from the basis properties, which preserves the continuous-time nature of the state-space matrices.

If this is right

  • The method provides accurate estimates of continuous-time system matrices from input-output data.
  • It eliminates the need for time-shifting signals that is typical in discrete-time approaches.
  • Performance is comparable to existing continuous-time methods based on Laguerre projections.
  • Finite-order projections suffice for practical identification accuracy in simulated LTI systems.

Where Pith is reading between the lines

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

  • Extending the order of Hermite projections could further reduce any residual approximation error in the identified models.
  • This technique might apply to other basis functions with similar recursive derivative properties for continuous-time identification.
  • Real-world applications in control systems could benefit from the direct continuous-time output without post-processing discretization.

Load-bearing premise

Finite-order projection onto Hermite basis functions combined with recursive derivative application preserves sufficient information for accurate subspace identification of the underlying continuous-time LTI system without introducing significant approximation bias or loss of observability.

What would settle it

Running the HD-PBSID algorithm on data from a known continuous-time system and finding that the estimated matrices do not converge to the true values as the number of basis functions increases would falsify the claim of sufficient information preservation.

Figures

Figures reproduced from arXiv: 2605.15318 by Enrico Barbiero, Jose Antonio Rebollo, Marco Lovera.

Figure 1
Figure 1. Figure 1: Hermite basis up to order 10. The normalized Hermite functions, hn(t), are defined using the Hermite polynomials 1 Hn(t) as hn(t) = e −t 2/2 Hn(t) p 2 n √ πn! . (13) The sequence {hn(t)}n∈N is an orthonormal Schauder 1 In the literature, the polynomials used in this paper are sometimes referred to as the physicist’s Hermite polyno￾mials. An explicit expression for the n-th Hermite poly￾nomial is the partic… view at source ↗
Figure 2
Figure 2. Figure 2: Hermite basis of order 200, with related numerical [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stability region in the complex plane for the modified [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bode diagram comparison between the real system [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bode diagram comparison between the real system [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: Error of the eigenvalues identified by the HD-PBSID [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: Eigenvalues comparison between the real system [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Error of the eigenvalues identified by the HD-PBSID [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Error of the eigenvalues identified by the HD-PBSID [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Eigenvalues comparison between the CT-PBSID [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
read the original abstract

In this paper the problem of continuous-time subspace identification for Linear Time Invariant (LTI) systems is considered and a method which directly identifies a continuous-time state-space form is proposed. First, Hermite basis functions are used to project signals and obtain a finite number of Hermite coefficients. By exploiting recursive relations and time derivative properties of the Hermite basis functions, an expression of the derivative operator is obtained. The latter is then recursively applied, ensuring that the state-space matrices remain in continuous-time form and making the system suitable for the implementation of steps which are akin to those of the Predictor-Based Subspace IDentification (PBSID) method. This new method, hereby called the Hermite-Domain PBSID (HD-PBSID) method, has the further advantage of avoiding time-shifts by properly scaling the input and output signals. The performance of the proposed approach is illustrated in a simulation study aimed at showing the accuracy of the estimates and at comparing the HD-PBSID method and the Laguerre-projections based Continuous-Time PBSID (CT-PBSID) algorithm.

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 proposes the Hermite-Domain Predictor-Based Subspace Identification (HD-PBSID) method for continuous-time LTI systems. Input and output signals are projected onto a finite set of Hermite basis functions to obtain coefficients. Recursive relations and derivative properties of the Hermite functions are exploited to obtain an expression for the derivative operator, which is applied recursively so that the identified state-space matrices remain in continuous-time form. This enables PBSID-like steps while avoiding explicit time-shifts through appropriate scaling of the signals. The approach is illustrated and compared to a Laguerre-projection-based CT-PBSID method in a simulation study.

Significance. If the finite-order Hermite projection and recursive derivative application can be shown to recover the continuous-time observability range without introducing non-orthogonal residuals, the method would provide a direct continuous-time subspace identification route that sidesteps discretization and time-shift artifacts. The simulation study supplies empirical evidence of estimate accuracy, which is a positive feature for an identification paper.

major comments (2)
  1. [§3] §3 (derivation of the derivative operator): the finite truncation replaces the exact infinite-dimensional relation d/dt ψ_n = √(n/2) ψ_{n-1} − √((n+1)/2) ψ_{n+1} with a banded matrix on R^N. The manuscript must show that the resulting O(1/N) residual in the projected state equation is orthogonal to the column space of the future outputs (or otherwise bounded so that it does not bias the singular-value gap and the subsequent least-squares estimates of A, B, C). Without this argument the central claim of unbiased direct continuous-time identification is not yet established.
  2. [§4] §4 (simulation study): the reported comparison with CT-PBSID shows qualitative agreement but does not include quantitative metrics (e.g., pole estimation error or variance of the estimated matrices) across repeated Monte-Carlo runs or varying noise levels. This weakens the claim that HD-PBSID achieves comparable or superior accuracy.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the Hermite order N used in each experiment and whether the same N is employed for both HD-PBSID and the Laguerre comparator.
  2. [Notation] The notation for the projected coefficient vectors and the scaling factors applied to inputs/outputs should be introduced once and used consistently; a small table summarizing the symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the contributions and limitations of our work. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the derivative operator): the finite truncation replaces the exact infinite-dimensional relation d/dt ψ_n = √(n/2) ψ_{n-1} − √((n+1)/2) ψ_{n+1} with a banded matrix on R^N. The manuscript must show that the resulting O(1/N) residual in the projected state equation is orthogonal to the column space of the future outputs (or otherwise bounded so that it does not bias the singular-value gap and the subsequent least-squares estimates of A, B, C). Without this argument the central claim of unbiased direct continuous-time identification is not yet established.

    Authors: We agree that an explicit treatment of the truncation residual is necessary to support the claim of direct continuous-time identification. In the revised manuscript we will add a dedicated paragraph in §3 that expresses the residual term arising from the finite banded approximation and shows, via the orthogonality of the Hermite basis, that this residual is orthogonal to the column space of the future outputs in the projected domain. We will also supply a simple O(1/N) bound on its norm and argue that it therefore does not perturb the singular-value gap or bias the subsequent least-squares steps for A, B and C. This addition will be accompanied by a short numerical check confirming the orthogonality property for increasing N. revision: yes

  2. Referee: [§4] §4 (simulation study): the reported comparison with CT-PBSID shows qualitative agreement but does not include quantitative metrics (e.g., pole estimation error or variance of the estimated matrices) across repeated Monte-Carlo runs or varying noise levels. This weakens the claim that HD-PBSID achieves comparable or superior accuracy.

    Authors: The current simulation section indeed presents only single-run trajectory plots and qualitative agreement. We will revise §4 to include a Monte-Carlo study (100 runs) reporting mean and standard deviation of the pole estimation error, as well as the Frobenius norm of the estimated (A,B,C) matrices, for several noise-to-signal ratios. These quantitative results will be tabulated and compared directly with the Laguerre-based CT-PBSID method, thereby strengthening the empirical evidence for the accuracy claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation uses external Hermite properties and PBSID steps

full rationale

The paper's chain begins with standard projection onto Hermite functions, invokes their known recursive derivative relation (a classical L2 fact independent of the paper), obtains a finite coefficient-domain operator, and then applies predictor-based subspace steps that are already established in the literature. None of these reductions are self-definitional, fitted-to-target, or dependent on a load-bearing self-citation whose validity is assumed only inside the present work. The simulation comparison to CT-PBSID supplies an external benchmark. The method therefore remains self-contained against external mathematical facts and prior algorithms.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method depends on standard properties of orthogonal polynomials and subspace identification assumptions; no new entities are introduced.

free parameters (1)
  • Hermite expansion order
    Finite number of coefficients chosen for projection; selection criterion not specified in abstract.
axioms (1)
  • domain assumption Hermite basis functions admit recursive relations and time-derivative properties that can be applied to projected signals without loss of system information.
    Invoked to obtain the derivative operator expression and enable continuous-time PBSID steps.

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Reference graph

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