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Data-Driven Control of Continuous-Time LTI Systems via Non-Minimal Realizations

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Pith's one-line read A single noise-free trajectory of a continuous-time LTI plant suffices to design a stabilizing output-feedback controller, and the same construction solves output regulation when all outputs share one observability index.

desk verdict Solid extension of the authors' prior SISO work to MIMO output-feedback stabilization and regulation, but the MIMO claim narrows substantially under the uniform observability index assumption. read the letter →

arxiv 2505.22505 v1 pith:RTFAKGB7 submitted 2025-05-28 eess.SY cs.SY

Alessandro Bosso , Marco Borghesi , Andrea Iannelli , Giuseppe Notarstefano , Andrew R. Teel This is my paper
classification eess.SYcs.SY MSC 93B5293B3093C0593C3593D15
keywords data-drivencontrolcontinuous-timeLTIsystemsoutputfeedbacknon-minimalrealizationslinearmatrixinequalitiesregulationobservabilityindicesMIMO
verification ladder T0 review T1 audit T2 compute T3 formal

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 claims that output-feedback controllers for unknown continuous-time linear time-invariant systems can be designed directly from one input-output experiment, without measuring states or derivatives. The key move is to use a canonical non-minimal realization: the user chooses stable filter dynamics with the unknown plant confined to the output equation, then reconstructs its state by filtering the measured signals. From sampled batches of filtered data, a linear matrix inequality computes a gain that provably makes the closed loop exponentially stable, and the same construction, augmented with an internal model driven by the known exosystem frequencies, solves output regulation. The method works for MIMO plants whose observability indices are all equal, and the paper provides a data-based test to estimate that common index. If true, the result removes derivative measurements—the main obstacle to continuous-time data-driven output feedback—from both stabilization and regulation problems.

What carries the argument

The central object is the canonical non-minimal realization, a lifted system $\dot\zeta=(F+LH)\zeta+Gu$, $y=H\zeta$ in which $F,G,L$ are user-chosen and $F$ is Hurwitz, while the unknown plant parameters are absorbed into the output map $H$, which must satisfy the matrix equations (13) with some $\Pi$. The non-minimal state is reconstructed by the filter $\dot{\hat\zeta}=F\hat\zeta+Gu+Ly$; because $F$ is Hurwitz, Lemma 3 makes the error dynamics $A-\Pi LC$ stable, and Lemma 4 replaces the unmeasurable error with a user-simulated auxiliary signal $\chi$ obeying $\dot\chi=F_0\chi$. This yields sampled batches $X,Z,\dot Z,U$ linked by the differential relation (37), so the LMI (29) is a pure data condition: its feasibility is equivalent to the full-rank condition (38), and its solution gives the gain $K=UQP^{-1}$. For MIMO plants with uniform observability index, Theorem 3 supplies explicit filter gains (78) built from a controllable pair $(\Lambda,\ell)$, and Algorithm 3 estimates $\nu$ by the rank loss of a data batch (86).

What would settle it

Choose a two-input, two-output continuous-time LTI plant with uniform observability index $\nu=2$, collect one noise-free trajectory, and run Algorithm 1 with $N$ samples satisfying the rank condition (38) and Algorithm 3 on the same data. If the computed gain $K$ ever leaves $F+LH+GK$ with an eigenvalue in the closed right half-plane, or if the index estimate differs from 2, then the paper's central claim is false.

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Extended reading notes

Core claim

The paper's central claim is that stabilization and output regulation in continuous time can be cast as a rank-conditioned linear matrix inequality built from filtered data. Given a canonical non-minimal realization—user-chosen Hurwitz filter dynamics $F,G,L$ with unknown output map $H$ satisfying $\Pi(F+LH)=A\Pi$, $\Pi G=B$, $H=C\Pi$—the paper proves (Theorem 1) that if the stacked data matrix of the sampled batches $X,Z,U$ has full rank, then any solution $P,Q$ of the LMI (29) yields $K=UQP^{-1}$ such that $F+LH+GK$ is Hurwitz, so the observer-like filter (31) globally exponentially stabilizes the plant. For regulation, an internal model $\dot\eta=\Phi\eta+\Gamma e$ is appended and the same LMI construction is applied to the augmented system; Theorem 2 states that the resulting controller (55) steers the regulated output $e$ to zero under the non-resonance condition. A data-driven algorithm (Algorithm 3) recovers the common observability index from the same trajectory, so the realization can be tuned from data. In short: a stabilizing output-feedback controller for a continuous-time MIMO LTI system can be computed directly from a single noise-free input-output trajectory without state or derivative measurements.

Load-bearing premise

Everything rests on every output having the same observability index $\nu$; if the indices differ, the tuning rules and the index-estimation algorithm are not known to work, and the paper leaves that case open.

Editorial extensions

If this is right

  • Any controllable and observable continuous-time LTI plant with a uniform observability index can be stabilized from one noise-free trajectory: the LMI (29) is feasible whenever the sampled batch satisfies the rank condition (38), and Theorem 1 guarantees the resulting closed loop is Hurwitz.
  • The deployed controller has an observer-plus-gain form (31), so its online implementation uses only the same filters and the computed gain; no model matrices or derivative measurements are needed after deployment.
  • For output regulation, given the exosystem frequencies and the non-resonance condition, the same LMI applied to the augmented system produces the internal-model regulator (55), and Theorem 2 guarantees the regulated output converges to zero.
  • The observability index can be read off the data: if the batch is full rank at $\hat\nu=\nu$, then increasing the filter dimension makes the batch lose rank (Theorem 5), so Algorithm 3 returns the correct uniform index.
  • The construction includes state-feedback ($\nu=1$) and SISO output-feedback ($\nu=n$) as special cases, so it unifies two previously separate derivative-free designs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's restriction to a uniform observability index is likely removable: following the discrete-time MIMO literature it cites, a filter bank with per-output dimensions $\nu_i$ and a similar rank test should extend the construction to generic MIMO systems, though the present proofs do not cover that case.
  • Because the LMI depends only on filtered data batches, replacing the exact batch equations with robust matrix relaxations for bounded noise is a natural route to measurement noise; the filter and internal-model structure would remain unchanged.
  • The auxiliary-dynamics trick of Lemma 4 turns a differential equation with an unknown perturbation into a purely data-driven equality, so the same device could reduce other continuous-time data-driven problems, such as optimal or constrained control, to LMIs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a data-driven output-feedback control framework for continuous-time LTI systems using non-minimal realizations whose state is obtained by filtering input and output signals, avoiding state and derivative measurements. For stabilization, the paper derives an LMI from sampled filtered data and proves that any feasible solution yields a stabilizing dynamic controller (Theorem 1). The framework is extended to output regulation by adding an internal model based on known exosystem frequencies (Theorem 2). A construction of canonical non-minimal realizations is given for systems with uniform observability index (Theorem 3), together with controllability of the resulting realization (Theorem 4) and an algorithm for estimating the observability index from data (Algorithm 3, Theorem 5). Numerical examples on a batch reactor and a surface vessel illustrate the approach; code is provided.

Significance. If the identified gaps are closed, this would be a valuable contribution to data-driven control of continuous-time systems: it provides a derivative-free, single-trajectory, output-feedback method for MIMO LTI systems, with an extension to output regulation and a filter-based implicit observer. Theorem 1's proof is detailed and internally consistent, and the LMI derivation is non-circular. The restriction to uniform observability index and the noise-free assumption are explicitly acknowledged, and the numerical implementation is reproducible via the provided repository. The main weaknesses are proof gaps in the output-regulation theorem and in the index-estimation certification.

major comments (3)
  1. [Section V, Theorem 2] The output-regulation theorem is stated without proof ("Its proof is omitted as it is based on Theorem 1 and the arguments above"), and the essential extension of Lemma 4 to the exosystem-augmented dynamics is also omitted ("The statement of such result and its proof are omitted for brevity"). These omissions are load-bearing: the data equation behind LMI (53), the rank condition (67), and the conclusion that the matrix in (68) is Hurwitz are not verifiable from the text. Please provide a complete proof in an appendix, including the explicit construction of D in col(w,ε)=Dχ, the derivation of the data equation from (62), and the verification of closed-loop regulation property (11) via (61).
  2. [Section VI-C and Section II-A] Theorem 1 is stated under Assumption 1 alone, but its initialization requires a canonical non-minimal realization, and the only general construction (Theorem 3) is proved under Assumption 3 (ν1=...=νp=ν). For generic MIMO systems the observability indices are as evenly balanced as possible, so uniform indices are possible only when p divides n; when p does not divide n, Assumption 3 cannot hold. Equations (116) decouple into identical ν-dimensional blocks precisely because of uniformity, and no alternative tuning is given for nonuniform indices. Thus the paper's MIMO stabilization and regulation claims are established only for the uniform-index class. Please state this restriction in Section II-A and in the abstract/contribution list, not only in Section VI and the conclusion.
  3. [Section VI-D, Algorithm 3 and Theorem 5] The index-estimation procedure is not fully certified. Theorem 5 proves rank deficiency for ν̂≥ν+1, but the full-rank condition at ν̂=ν is assumed, not proved: Algorithm 3 contains the comment "Assumes B full rank if ν̂=ν", and the text preceding Theorem 5 says "Suppose that, for ν̂=ν, B has full rank". No persistency-of-excitation or sampling condition is given under which this full-rank condition holds. Since this full-rank condition is necessary for the subsequent LMI feasibility condition (38), the claim that the observability index "can be directly estimated from the given input-output trajectory" needs either a proof under explicit excitation assumptions or a clear downgrade to a heuristic/assumption.
minor comments (4)
  1. [Section VI-C] The text introduces observability indices under the condition rank C=p, but Assumption 1 alone does not guarantee this when p>n. This implicit standing assumption should be stated explicitly in Section II.
  2. [Section IV, proof of Theorem 1] In part 1, the existence of M from the rank condition (38) is asserted by reference to [4] and [19]; a one-line argument that full row rank of [X;Z;U] makes the representation (40) possible would make the proof self-contained.
  3. [Section VI-D, proof of Theorem 5] The display (136) is typeset with blank entries and no explicit block dimensions; an explicit 4x4 block matrix with zero blocks and dimensions would substantially improve readability.
  4. [Introduction] The paper relies on [25] for Lemma 1 and the filter structure; although Lemma 1 is proved in the text, a brief remark on how the present results go beyond [25] (from SISO/state-feedback to MIMO and output regulation under the uniform-index assumption) would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the LMI and stabilization proofs are self-contained, with [25] used only as motivation and not as load-bearing evidence.

full rationale

The paper's central derivation chain is not circular. Lemma 1, Lemma 3, Lemma 4, and Theorem 1 are proved in the text from stated assumptions, and the LMI (29) is derived from the filtered data equation (37) rather than from the target closed-loop property. The gain K = U Q P^{-1} is not a fitted parameter renamed as a prediction; the proof shows directly that any solution of the LMI yields F + L H + G K Hurwitz using the identity (45). The construction of canonical non-minimal realizations in Theorem 3 is proved under Assumption 3 via the observer canonical form and Lemma 7, with no reliance on an unproved uniqueness theorem. The observability-index estimation in Algorithm 3 and Theorem 5 proves rank loss for overestimates; the full-rank condition at the true index is explicitly assumed in the algorithm comment, which is a completeness gap rather than circularity. The paper cites the authors' prior work [25] for motivation and for the filter structure, but the specific results needed here, including the generalization of [25, Lem. 3] as Lemma 1, are proved in the present manuscript. Other cited results, such as [4, Thm. 2], [19, Thm. 2], [21, Lem. 4.2], and [34, Ch. 3], are external standard facts. Thus no load-bearing step reduces to its own input by construction or to a self-citation chain.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. The non-minimal realization is a mathematical construction, not a physical invention. The free parameters are user-defined tuning gains, not fitted parameters, and the axioms are standard or explicit domain assumptions.

free parameters (4)
  • Filter matrices F, G, L (or Λ, ℓ in the tuned realization) = User-chosen; e.g., Λ=diag(-4,-8), ℓ=col(1,2) in Example 1
    These define the canonical non-minimal realization and are selected by the user, not fitted to data. The central claim holds for any valid choice, but the numerical behavior depends on them.
  • Tuning scalars ω_f, ω_s = Arbitrary non-zero; example uses Γ0=5, Γ0=1/10
    Used in the auxiliary dynamics (27) and internal model (57). They are user-defined and not fitted to data.
  • Tuning scalars λ_i, γ_i in Algorithm 3 = User-chosen; example uses λ_j=γ_j=j
    Used in the observability index estimation. They are hand-picked and affect the rank test, though the theoretical guarantee in Theorem 5 is independent of their values.
  • Sampling period τ_s and number of samples N = User-chosen; examples use N=50, τ_s=40 ms and N=80
    These affect the data batches (28) and the informativity condition (38). They are design choices, not fitted parameters.
assumptions (4)
  • domain assumption Assumption 1: (A,B) controllable and (C,A) observable
    Standard controllability/observability needed for the non-minimal realization existence and the LMI proofs.
  • domain assumption Assumption 2: non-resonance condition rank([A-sI B; Ce 0])=n+q for all s in σ(S)
    Necessary and sufficient for the output regulation problem to be solvable (by classical internal model theory).
  • domain assumption Assumption 3: uniform observability index ν1=...=νp=ν
    This is the load-bearing structural restriction. The construction of canonical non-minimal realizations and the observability index estimation rely on it. The paper does not handle general MIMO observability indices.
  • domain assumption Noise-free data
    The LMI formulation assumes exact measurements and no disturbances in the stabilization setting (except the exosystem in the regulation case). The paper explicitly states this is a simplifying assumption.

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Cite this review

Pith. "Pith review of Data-Driven Control of Continuous-Time LTI Systems via Non-Minimal Realizations." pith.science (2026). https://pith.science/paper/RTFAKGB7

@misc{pith2026250522505,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Control of Continuous-Time LTI Systems via Non-Minimal Realizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTFAKGB7}},
  note         = {Machine review of arXiv:2505.22505}
}
read the original abstract

This article proposes an approach to design output-feedback controllers for unknown continuous-time linear time-invariant systems using only input-output data from a single experiment. To address the lack of state and derivative measurements, we introduce non-minimal realizations whose states can be observed by filtering the available data. We first apply this concept to the disturbance-free case, formulating linear matrix inequalities (LMIs) from batches of sampled signals to design a dynamic, filter-based stabilizing controller. The framework is then extended to the problem of asymptotic tracking and disturbance rejection - in short, output regulation - by incorporating an internal model based on prior knowledge of the disturbance/reference frequencies. Finally, we discuss tuning strategies for a class of multi-input multi-output systems and illustrate the method via numerical examples.

Figures

Figures reproduced from arXiv: 2505.22505 by the authors.

Figure 1
Figure 1. Implementation of the controller (31) designed in Algorithm 1. Algorithm 1 Data-Driven Stabilization Initialization Dataset: (𝑢(𝑡), 𝑦(𝑡)), ∀𝑡 ∈ [0, 𝜏]. (25) Tuning: 𝐹, 𝐺, 𝐿 such that (12) is a canonical non-minimal realization of (1); (𝐹0, 𝐺0) as in (36); number of samples 𝑁 ∈ N, 𝑁 ≥ 1, 𝜏s B 𝜏/𝑁. Data Batches Construction Filter of the data: ¤ 𝜁ˆ(𝑡) = 𝐹𝜁ˆ(𝑡) + 𝐺𝑢(𝑡) + 𝐿𝑦(𝑡), ∀𝑡 ∈ [0, 𝜏]. (26) Initialization: 𝜁ˆ(0) =… view at source ↗
Figure 2
Figure 2. Implementation of the controller (55) designed in Algorithm 2. Algorithm 2 Data-Driven Output Regulation Initialization Dataset: (𝑢(𝑡), 𝑒(𝑡), 𝑦𝑟 (𝑡)), ∀𝑡 ∈ [0, 𝜏]. (48) Tuning: 𝐹, 𝐺, 𝐿 such that (12) is a canonical non-minimal realization of (1), with 𝐿 = [𝐿𝑒 𝐿r], 𝐿𝑒 ∈ R 𝜇×𝑞 , and 𝐿r ∈ R 𝜇× ( 𝑝−𝑞) ; (𝐹0, 𝐺0) as in (36); (𝑆0, Γ0) as in (57); number of samples 𝑁 ∈ N, 𝑁 ≥ 1, 𝜏s B 𝜏/𝑁. Data Batches Construction Filter o… view at source ↗
Figure 3
Figure 3. Simulation run for the batch reactor (control with integral action). First, we acquire a dataset of the form (𝑢(𝑡), 𝑦(𝑡)) by simulating the system for 𝜏 = 2 s. The applied input 𝑢 is the sum of 4 sinusoids in both entries, with 8 distinct frequencies. The initial condition 𝑥(0) is chosen randomly, with each entry extracted from the uniform distribution 𝑈(−1, 1). In the following, we describe a procedure that has bee… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Simulation run for the surface vessel example. B]. The matrices of system (4) are 𝐴=              −0.1 0.012 0.015 0 0 0.01 0.01 −0.0333 −0.05 0 0 −0.014 0.02 0.03 −0.18 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0              𝐵=        …

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Cited by 2 Pith papers

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.