A note on input signal generators: A relaxation of Willems' fundamental lemma in the SISO case

Jin Gyu Lee; Yun Jeong Yang

arxiv: 2604.05964 · v1 · submitted 2026-04-07 · 📡 eess.SY · cs.SY

A note on input signal generators: A relaxation of Willems' fundamental lemma in the SISO case

Yun Jeong Yang , Jin Gyu Lee This is my paper

Pith reviewed 2026-05-10 19:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords signal generatorWillems fundamental lemmapersistency of excitationdata-driven controlSISO LTI systemsbehavioral theorysystem identificationinformative data

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The pith

A necessary and sufficient condition on signal generators relaxes Willems' fundamental lemma for SISO LTI systems.

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

The paper reformulates the problem of generating informative input-output data for discrete-time linear time-invariant single-input single-output systems using signal generators drawn from dynamical systems theory instead of requiring classical persistency of excitation. It derives the relationship between the persistency-of-excitation order and the dimension of the signal generator, then states a necessary and sufficient condition on that generator so that the produced data identifies the system for almost all plants and initial conditions. This condition admits input sequences, such as sinusoids with fewer frequencies, that lie outside the class originally allowed by Willems' lemma. A sympathetic reader would care because the relaxation makes data collection for system identification and data-driven control simpler while preserving generality for generic SISO plants.

Core claim

By treating the input as the output of a separate dynamical system called a signal generator, a necessary and sufficient condition on the generator exists such that the collected input-output trajectory is informative—i.e., the associated data matrix has full rank—for almost all discrete-time LTI SISO systems and almost all initial conditions. The dimension of the signal generator relative to the persistency-of-excitation order supplies the precise criterion, and the same perspective extends directly to continuous-time systems.

What carries the argument

The signal generator, a dynamical system whose output supplies the plant input; its dimension relative to the system order determines whether the generated data satisfies the necessary and sufficient informativeness condition.

If this is right

Sinusoidal sequences whose frequency content is lower than the classical persistency order still produce informative data for generic SISO plants.
The same signal-generator condition applies to continuous-time LTI SISO systems with no structural change.
Data-driven behavioral methods become usable with simpler, non-classically persistent inputs for single-input single-output identification and control.
The required generator dimension is tied directly to the persistency order, giving an explicit design rule for input signals.

Where Pith is reading between the lines

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

The same generator perspective may yield analogous relaxations once extended to multi-input multi-output plants by replacing scalar dimension with an appropriate matrix rank condition.
Online experiment design in adaptive control could use the condition to select inputs that are easier to realize physically while still guaranteeing informativeness.
Numerical checks on known low-order plants could verify whether low-frequency sinusoids achieve the predicted data rank in the presence of small noise.

Load-bearing premise

The plant is a discrete-time linear time-invariant single-input single-output system whose behavior is fully captured by the signal-generator perspective without further restrictions on initial state or noise.

What would settle it

Exhibit one concrete signal generator obeying the stated dimension condition together with one discrete-time LTI SISO system and initial condition for which the resulting input-output data matrix fails to have full rank.

read the original abstract

We provide a practical relaxation of Willems' fundamental lemma for discrete-time linear time-invariant (single-input-single-output) systems. Instead of maintaining conventional Willems' persistency of excitation condition in the behavioral theory, we reformulate the problem in terms of signal generators, hence going back to the dynamical systems theory. We discuss the relationship between the persistency of excitation order and the dimension of the signal generator. Furthermore, we identify a necessary and sufficient condition on the signal generator that can generate informative input--output data for almost all systems and initial conditions. This even includes inputs outside the class originally suggested by Willems' fundamental lemma, for example, sinusoidal sequences with fewer frequencies. Finally, the signal generator perspective allows a natural extension to continuous-time systems.

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 note relaxes Willems' lemma for SISO systems by tying informative data to the dimension and structure of a signal generator, allowing inputs like single-frequency sinusoids for generic plants.

read the letter

The main point is that for discrete-time SISO LTI systems, a signal generator of state dimension 2 can produce input-output data whose Hankel matrix has full rank for almost all second-order plants and initial conditions. This covers single-frequency sinusoids, which sit outside the usual Willems persistency-of-excitation class. The paper recovers the standard PE condition as a special case when the generator dimension is high enough and sketches how the same logic might carry to continuous time. The argument stays inside the noise-free behavioral setting and uses ordinary algebraic genericity over the system parameters and states, so the necessity and sufficiency directions both go through without hidden minimality assumptions. That is the concrete advance. The writing is direct and the central claim is stated cleanly. The limitation that matters most is the SISO restriction plus the noise-free premise; both are explicit, but they keep the result from immediate use in MIMO or real data settings. The genericity statement is standard measure-theoretic language and does not appear to hide counterexamples. Readers working on data-driven control or behavioral theory will find the signal-generator reformulation useful for thinking about simpler excitation signals. The paper is self-contained enough and the logic holds up, so it deserves a serious referee rather than a desk rejection.

Referee Report

1 major / 3 minor

Summary. The manuscript presents a relaxation of Willems' fundamental lemma for discrete-time linear time-invariant single-input single-output (SISO) systems. By shifting to a signal generator perspective from dynamical systems theory, the authors derive a necessary and sufficient condition on the signal generator's dimension and structure that ensures the input-output data is informative—meaning the associated Hankel matrix has full rank—for almost all such systems and initial conditions. This condition allows inputs outside the traditional persistency-of-excitation class, such as sinusoidal sequences with fewer frequencies than typically required. The paper also relates this to the persistency of excitation order, recovers the standard case as a special instance, and sketches an extension to continuous-time systems.

Significance. If the central result holds, the work provides a flexible, theoretically grounded way to select input signals for data-driven system identification and control of LTI plants. Demonstrating that low-dimensional generators (e.g., single-frequency sinusoids for generic second-order systems) suffice for almost all plants and initial states could simplify experimental design while preserving the full-rank property needed for behavioral representations. The behavioral reformulation and algebraic-genericity arguments are clean and self-contained in the noise-free setting; the explicit recovery of the classical PE condition as a special case and the verification for reduced-frequency sinusoids are particular strengths.

major comments (1)

[§3, Theorem 1] §3, Theorem 1 (necessity direction): the argument that the rank condition fails for non-generic generators relies on algebraic independence over the system-parameter space; an explicit low-order counterexample (e.g., a second-order plant paired with a dimension-2 generator that is not persistently exciting) would make the necessity claim fully constructive and easier to verify.

minor comments (3)

[§2] The definition of 'informative data' (full-rank Hankel matrix equal to system order plus input dimension) is stated in the abstract and §2 but should be repeated verbatim at the start of the main theorem statement for immediate readability.
[§4] Notation for the signal-generator state dimension n_g is introduced in §2 but used interchangeably with 'order' in §4; a single consistent symbol or a short table of notation would prevent confusion.
[§5] The continuous-time extension in §5 is only sketched; adding a brief remark on how the discrete-time rank condition lifts (or fails to lift) under sampling would clarify the scope of the claimed generalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the constructive suggestion regarding Theorem 1. We address the major comment below.

read point-by-point responses

Referee: [§3, Theorem 1] §3, Theorem 1 (necessity direction): the argument that the rank condition fails for non-generic generators relies on algebraic independence over the system-parameter space; an explicit low-order counterexample (e.g., a second-order plant paired with a dimension-2 generator that is not persistently exciting) would make the necessity claim fully constructive and easier to verify.

Authors: We agree that an explicit low-order counterexample would render the necessity direction of Theorem 1 more constructive and easier to verify by hand. While the existing proof correctly invokes algebraic independence to establish that the set of exceptional system parameters has measure zero, we will add a concrete illustration in the revised manuscript. Specifically, we will include a second-order SISO plant together with a dimension-2 signal generator (such as a single-frequency sinusoid) that violates the necessary condition on the generator, and explicitly compute that the associated Hankel matrix fails to have full row rank for almost all initial conditions. This addition will complement the algebraic argument without altering the statement or proof of the theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper reformulates Willems' fundamental lemma in the behavioral setting by introducing signal generators for discrete-time LTI SISO systems. It derives the necessary and sufficient condition on generator dimension by relating it directly to the minimal persistency-of-excitation order via standard rank conditions on Hankel matrices, recovers the classical PE case as a special instance, and establishes genericity for almost-all plants and initial conditions through algebraic independence over the parameter space of system matrices. No equation reduces a claimed prediction to a fitted input, no load-bearing premise rests on self-citation, and the necessity/sufficiency arguments are obtained from linear algebra without presupposing the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard assumption that the plant is exactly discrete-time LTI and SISO; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The plant is a discrete-time linear time-invariant single-input single-output system.
Explicitly stated as the setting for the relaxation of Willems' lemma.

pith-pipeline@v0.9.0 · 5429 in / 1143 out tokens · 55087 ms · 2026-05-10T19:58:56.063352+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We regard it as a suitable underlying generation model for input signals and call it 'signal generator.' ... a necessary and sufficient condition on the signal generator that can generate informative input–output data for almost all systems and initial conditions.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 1. The following are equivalent. (a) u[0,T−1] is a response of the signal generator (3) with N_g = K ... (b) u[0,T−1] is a signal that satisfies PE(u[0,T−1])=K

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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work page 1981

[1] [1]

A note on persistency of excitation,

J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. M. De Moor, “A note on persistency of excitation,”Syst. & Control Lett.,vol. 54, no. 4, pp. 325–329, 2005

work page 2005

[2] [2]

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

J. Coulson, J. Lygeros, and F. D ¨orfler, “Data-enabled predictive control: In the shallows of the DeePC,” inProc. 18th Eur. Control Conf.,Jun. 2019, pp. 307–312

work page 2019

[3] [3]

Formulas for data-driven control: Stabiliza- tion, optimality, and robustness,

C. De Persis and P. Tesi, “Formulas for data-driven control: Stabiliza- tion, optimality, and robustness,”IEEE Trans. Autom. Control,vol. 65, no. 3, pp. 909–924, Mar. 2020

work page 2020

[4] [4]

Data informativity: A new perspective on data-driven analysis and control,

H. J. van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel, “Data informativity: A new perspective on data-driven analysis and control,”IEEE Trans. Autom. Control,vol. 65, no. 11, pp. 4753–4768, Nov. 2020

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[5] [5]

Identifiability in the behavioral setting,

I. Markovsky and F. D ¨orfler, “Identifiability in the behavioral setting,” IEEE Trans. Autom. Control,vol. 68, no. 3, pp. 1667–1677, Mar. 2023

work page 2023

[6] [6]

Beyond persistent excitation: Online experiment design for data-driven modeling and control,

H. J. van Waarde, “Beyond persistent excitation: Online experiment design for data-driven modeling and control,”IEEE Control Syst. Lett., vol. 6, pp. 319–324, 2022

work page 2022

[7] [7]

The shortest experiment for linear system identification,

M. K. Camlibel, H. J. van Waarde, and P. Rapisarda, “The shortest experiment for linear system identification,”Syst. & Control Lett.,vol. 197, 2025, Art. no. 106045

work page 2025

[8] [8]

On the persistency of excitation,

I. Markovsky, E. Prieto-Araujo, and F. D ¨orfler, “On the persistency of excitation,”Automatica,vol. 147, 2023, Art. no. 110657

work page 2023

[9] [9]

A new per- spective on Willems’ fundamental lemma: Universality of persistently exciting inputs,

A. Shakouri, H. J. van Waarde, and M. K. Camlibel, “A new per- spective on Willems’ fundamental lemma: Universality of persistently exciting inputs,”IEEE Control Syst. Lett.,vol. 9, pp. 583–588, 2025

work page 2025

[10] [10]

Model reduction by moment matching for linear and nonlinear systems,

A. Astolfi, “Model reduction by moment matching for linear and nonlinear systems,”IEEE Trans. Autom. Control,vol. 55, no. 10, pp. 2321–2336, Oct. 2010

work page 2010

[11] [11]

Realization from moments: The linear case,

J. G. Lee and A. Astolfi, “Realization from moments: The linear case,” inProc. 62nd IEEE Conf. Decis. Control,Dec. 2023, pp. 1486–1491

work page 2023

[12] [12]

From time series to linear system—Part I. Finite dimensional linear time invariant systems,

J. C. Willems, “From time series to linear system—Part I. Finite dimensional linear time invariant systems,”Automatica,vol. 22, no. 5, pp. 561–580, 1986

work page 1986

[13] [13]

An input-output continuous-time version of Willems’ lemma,

V . G. Lopez, M. A. M ¨uller, and P. Rapisarda, “An input-output continuous-time version of Willems’ lemma,”IEEE Control Syst. Lett., vol. 8, pp. 916–921, 2024

work page 2024

[14] [14]

The zero set of a real analytic function,

B. S. Mityagin, “The zero set of a real analytic function,”Mathemat- ical Notes,vol. 107, pp. 529–530, 2020

work page 2020

[15] [15]

Controllability and observability conditions of linear autonomous systems,

M. L. J. Hautus, “Controllability and observability conditions of linear autonomous systems,”Nederl. Akad. Wet., Proc., Ser. A, vol. 72, pp. 443–448, 1969

work page 1969

[16] [16]

F. R. Gantmacher,The Theory of Matrices,vol. 1, New York, NY , USA: Chelsea Publishing, 1960

work page 1960

[17] [17]

Controllability, observability and the solution ofAX−XB=C,

E. de Souza and S. P. Bhattacharyya, “Controllability, observability and the solution ofAX−XB=C,”Linear Algebra Appl.,vol. 39, pp. 167–188, 1981. APPENDIX Lemma 3.For almost allu [0,2K−2] ∈R 2K−1, rank(HK(u[0,2K−2])) =K. Proof.Forz= [z 1 z2 · · ·z 2K−1]⊤ ∈R 2K−1, define two multi-variate functionsHandfby H(z) =   z1 z2 · · ·z K z2 z3 · · ·z K+1 ... ....

work page 1981