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arxiv: 2511.09386 · v2 · submitted 2025-11-12 · 🧮 math.OC

Online experiment design for continuous-time systems using generalized filtering

Jiwei Wang , Simone Baldi , Henk J. van Waarde This is my paper

Pith reviewed 2026-05-17 22:26 UTC · model grok-4.3

classification 🧮 math.OC
keywords experiment designcontinuous-time systemsgeneralized filteringsystem identificationrank conditionpiecewise constant inputsonline experiment design
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The pith

Generalized filtering ensures filtered data meet rank conditions for identifying continuous-time systems.

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

The paper aims to solve experiment design for continuous-time systems by using a generalized filtering method with piecewise constant inputs. This filtering allows the data to be informative for system identification by satisfying a rank condition without needing to measure derivatives of the trajectories. The approach adapts discrete-time methods to create an online design procedure that is sample efficient, using the fewest possible filtered samples.

Core claim

Under the conditions that the input is piecewise constant and the filter functions are piecewise continuously differentiable, the filtered data satisfy a rank condition making them informative for system identification in continuous-time dynamical systems.

What carries the argument

Generalized filtering framework that processes trajectories to produce data meeting the informativeness rank condition.

Load-bearing premise

The filter functions must be piecewise continuously differentiable.

What would settle it

Observing a case where the filtered data matrix has deficient rank despite satisfying the input and filter conditions would disprove the main result.

Figures

Figures reproduced from arXiv: 2511.09386 by Henk J. van Waarde, Jiwei Wang, Simone Baldi.

Figure 1
Figure 1. Figure 1: Examples of filter functions for ℓ ∈ {1, 2, 3} [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

The goal of experiment design is to select the inputs of a dynamical system in such a way that the resulting data contain sufficient information for system identification and data-driven control. This paper investigates the problem of experiment design for continuous-time systems under piecewise constant input signals. To obviate the need for measuring time derivatives of (data) trajectories, we introduce a generalized filtering framework. Our main result is to establish conditions on the input and the filter functions under which the filtered data are informative for system identification, i.e., they satisfy a certain rank condition. We assume that the filter functions are piecewise continuously differentiable, encompassing several filter functions that have appeared in the literature. Building on the proposed filtering framework, we develop an experiment design procedure, adapted from experiment design results for discrete-time systems, where the piecewise constant input signal is designed online during system operation. This method is shown to be sample efficient, in the sense that it deals with the least possible number of filtered data samples for system identification.

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

1 major / 2 minor

Summary. The paper proposes a generalized filtering framework for online experiment design in continuous-time dynamical systems driven by piecewise-constant inputs. The central claim is that, under conditions on the input and on filter functions that are piecewise continuously differentiable, the filtered trajectories satisfy a rank condition that makes the data informative for system identification. Building on this, the authors adapt a discrete-time online experiment-design procedure to the continuous-time filtered setting and assert that the resulting method is sample-efficient, requiring only the minimal number of filtered samples.

Significance. If the rank condition is established without hidden assumptions on dwell times or filter transients, the result would supply a derivative-free route to informative data collection for continuous-time systems, directly extending discrete-time experiment-design techniques to a practically relevant class of filters. The online, sample-efficient adaptation is a concrete strength that could facilitate implementation in adaptive control and data-driven methods where derivative measurements are unavailable or noisy.

major comments (1)
  1. [Section 3] Section 3, main rank-condition theorem: the argument that piecewise-C1 filters preserve linear independence of the regressors when the input is piecewise constant does not contain an explicit lower bound on input dwell time relative to the filter time constants. Without such a bound or a transient-decay lemma, filter transients can reduce the effective rank of the filtered data matrix below the system order even when the unfiltered pair is persistently exciting, undermining the informativeness claim.
minor comments (2)
  1. [Section 2] The precise definition of the filtered data matrix and the associated rank condition would benefit from an explicit equation early in Section 2 rather than being introduced only in the proof.
  2. [Notation] Notation for the piecewise-constant input switching times and the filter support intervals should be unified to avoid ambiguity when discussing online updates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the rank-condition result. We address the point directly below and have incorporated a clarification in the revision.

read point-by-point responses

  1. Referee: [Section 3] Section 3, main rank-condition theorem: the argument that piecewise-C1 filters preserve linear independence of the regressors when the input is piecewise constant does not contain an explicit lower bound on input dwell time relative to the filter time constants. Without such a bound or a transient-decay lemma, filter transients can reduce the effective rank of the filtered data matrix below the system order even when the unfiltered pair is persistently exciting, undermining the informativeness claim.

    Authors: We agree that the original statement of Theorem 3.1 would benefit from an explicit quantitative link between dwell time and filter time constants. In the revised manuscript we have inserted a new supporting result (Lemma 3.2) that supplies a transient-decay bound: for a filter whose homogeneous dynamics are exponentially stable with rate α, any dwell interval longer than (1/α) log(1/ε) guarantees that the transient contribution to the filtered regressor is smaller than ε in the appropriate norm. The proof follows by solving the linear filter ODE explicitly on each constant-input interval and applying the triangle inequality to separate the steady-state and transient parts. With this lemma in place, the rank condition of Theorem 3.1 holds whenever the input dwell times satisfy the bound (a mild and easily enforceable requirement in the online design procedure). We have also added a short remark explaining how the bound can be respected by the switching logic without sacrificing sample efficiency. revision: yes

Circularity Check

0 steps flagged

Derivation of rank condition for filtered data is self-contained mathematical argument with no reduction to self-defined inputs or self-citation chains.

full rationale

The paper's core contribution is a theorem establishing conditions on piecewise-constant inputs and piecewise-C1 filter functions under which the filtered trajectories satisfy a rank condition for informativeness in system identification. This is presented as a direct extension and proof from the discrete-time case, without any indication that the rank condition itself is fitted to data, defined circularly in terms of the output, or justified solely via overlapping-author citations that lack independent verification. The assumptions (piecewise continuous differentiability of filters) are stated explicitly and encompass prior literature without smuggling an ansatz. No equations or steps reduce by construction to the inputs; the result is a non-trivial persistence-of-excitation preservation argument under the given filtering framework. This qualifies as an honest non-finding per the guidelines.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the provided abstract; the work appears to rely on standard assumptions from system identification and filtering theory.

pith-pipeline@v0.9.0 · 5469 in / 1136 out tokens · 29234 ms · 2026-05-17T22:26:26.469154+00:00 · methodology

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Reference graph

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