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

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Pith's one-line read Data informativity gives exact conditions for when measured data, rich or not, suffice for certifying controllability, designing stabilizing or deadbeat feedback, or solving LQR from data.

arxiv 1908.00468 v3 pith:U7NH5CAJ submitted 2019-08-01 math.OC math.DS

Henk J. van Waarde , Jaap Eising , Harry L. Trentelman , M. Kanat Camlibel This is my paper · ORCID
classification math.OCmath.DS
keywords controldataanalysisdata-drivensystemexcitingidentificationpersistently
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The pith

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

The reading

This paper is about making decisions from measured data about a dynamical system when you do not have a model. In control engineering, the usual recipe is to first identify a model from data, then design a controller. The authors ask: when can you skip the identification step and still be sure the controller works?

Their framework collects every system that could have produced the observed data. Data are 'informative' for a property, such as stability or stabilizability, if every system in that collection has the property. For controlling the system, data are informative if there is one controller that works for every system in the collection.

The main results are exact conditions on the data matrices. For example, to know whether the true system is controllable, it suffices to check the rank of a certain matrix pencil built only from state measurements; the paper calls this a data-driven Hautus test. For stabilization by state feedback, the data only need to provide a full-row-rank state matrix and a right inverse that makes a certain product stable. These conditions are weaker than the usual persistently exciting conditions needed to identify the system uniquely.

For linear quadratic regulation, the situation changes. The authors prove that, except for a special zero-cost case, the data must be rich enough to identify the system uniquely before an optimal LQ controller can be certified from data. This explains why earlier work needed persistent excitation: in that problem, it is genuinely necessary.

Extended reading notes

Core claim

The central claim is that a single notion of data informativity yields necessary and sufficient conditions for several data-driven analysis and control problems. For controllability/stabilizability analysis and state-feedback stabilization, these conditions are strictly weaker than the rank condition (8) needed for system identification (Theorems 8, 16, 17). For linear quadratic regulation, informativity for system identification is essentially necessary: except for a pathological case with K=0, the data must satisfy rank [X-; U-] = n+m (Theorem 26).

Load-bearing premise

The framework presupposes that the measured data are generated exactly by a discrete-time LTI system in the assumed model class, with no noise or disturbances. This is stated at the end of Section II: 'we stress that throughout the paper it is assumed that the data are given and are not corrupted by noise'. All theorems are proved under this premise; if data are noisy, the set of consistent systems as defined in (6) and (42) is empty or ill-posed, and the informativity framework does not apply.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard linear systems theory (Hautus, Lyapunov, Riccati) and on the assumption that the data are exact and generated by an LTI system of known dimensions. No free parameters are fitted to data. No new entities are introduced.

assumptions (5)
  • domain assumption The true system is a discrete-time linear time-invariant system x(t+1)=As x(t)+Bs u(t) of known state dimension n and input dimension m.
    Central model class, Section II Example 2 and Section III Eq. (5).
  • domain assumption Data D are generated by the true system and are not corrupted by noise.
    Explicitly assumed before Section III and in Section VI; the consistency set Sigma_D is defined by exact equations X+ = [A B][X-; U-].
  • standard math System-theoretic properties (stabilizability, controllability, detectability) are characterized by Hautus rank conditions and Lyapunov/Riccati theory.
    Used throughout: Theorem 8 uses Hautus; Lemma 25 uses Lyapunov; Theorem 23 uses discrete-time ARE, cited from [45].
  • standard math The discrete-time algebraic Riccati equation has the stated maximal solution properties (Theorem 23).
    Invoked for LQR solvability and optimal gain formulas; standard result.
  • domain assumption For input/output data, the system is minimal and the state dimension n is known or computable a priori.
    Footnote 5 in Section V-B mentions subspace identification, and Corollary 41 assumes minimality and k>n.

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Pith. "Pith review of Data informativity: a new perspective on data-driven analysis and control." pith.science (2026). https://pith.science/paper/U7NH5CAJ

@misc{pith2026190800468,
  author       = {Pith},
  title        = {Pith review of: Data informativity: a new perspective on data-driven analysis and control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7NH5CAJ}},
  note         = {Machine review of arXiv:1908.00468}
}
read the original abstract

The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.

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