Model Order Reduction from Data with Certification

Abolfazl Lavaei; Amy Nejati; Behrad Samari

REVIEW 2 major objections 5 minor 42 references

Model Order Reduction from Data with Certification

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that two recorded input-state trajectories of an unknown continuous-time linear system are enough to build a reduced-order model and a quadratic simulation function that formally bounds the difference between the reduced…

desk verdict New idea, flawed lemma: the data-driven simulation-function construction is attractive but the main theorem misses a rank condition on U0,T Q, so the certificate may be for the wrong system. read the letter →

arxiv 2502.01094 v1 pith:C75QW57Y submitted 2025-02-03 eess.SY cs.SY

Behrad Samari , Amy Nejati , Abolfazl Lavaei This is my paper · ORCID

classification eess.SYcs.SY MSC 93B1193C0593B3093D30

keywords modelorderreductiondata-drivencontrolsimulationfunctionsunknownlinearsystemscontrollersynthesisformalverificationreach-while-avoidtemporallogic

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 sets out to show that model order reduction with formal guarantees is possible when the underlying continuous-time linear system is completely unknown. The authors construct both a reduced-order model and a simulation function, a Lyapunov-like function that measures trajectory closeness, using only two recorded input-state trajectories: one driven by a general input and one by zero input. The main theorem certifies that, whenever three data-driven matrix conditions (3.8a)-(3.8c) hold, the reduced model's outputs are guaranteed to stay within an explicit bound of the unknown system's outputs. That makes the reduced model a sound substitute for controller synthesis: a controller enforcing safety, reach-while-avoid, or tracking on the small model transfers to the large unknown system with a quantified error. The authors verify the approach on five benchmark systems without ever knowing the system matrices.

What carries the argument

The central object is the quadratic simulation function $V(x,\hat{x}) = (x-\Theta\hat{x})^\top P(x-\Theta\hat{x})$, with $P \succ 0$ and $\Theta \in \mathbb{R}^{n \times \hat{n}}$. It works through three data-based identities: $I_n = X_{0,T}Q$ and $I_n = \bar{X}_{0,T}\bar{Q}$ express the drift matrix as $A = \bar{X}_{1,T}\bar{Q}$ and the pre-feedback matrix as $A+BF = X_{1,T}Q$, while $B$ is recovered as $(X_{1,T}Q-\bar{X}_{1,T}\bar{Q})(U_{0,T}Q)^\dagger$. The theorem's conditions then force the Lie derivative into the dissipativity inequality $\mathcal{L}V \le -\kappa V + \rho\|\hat{u}\|^2$: condition (3.8a) links the auxiliary matrix $H$ to $P^{-1}$, condition (3.8b) cancels the coupling terms between $x$ and $\hat{x}$, and condition (3.8c) makes the remaining quadratic term contractive. Because (3.8c) is equivalent to stabilizability of the pair $(A,B)$, the data-driven certificate inherits the classical necessary-and-sufficient condition of model-based design.

What would settle it

Pick a known stable linear system with two inputs and run the data-collection phase while applying control only through the first input, so the second column of $B$ is never excited; then follow Algorithm 1 and compare the certified bound from (2.5) with the actual sup-norm error $\|y(t)-\hat{y}(t)\|$ on a third trajectory driven through the second input. If the actual error exceeds the certified bound, the data-based representation (3.4) has not recovered the true $B$, and the simulation-function certificate fails.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the classical simulation-function framework for model order reduction does not require knowledge of $A$ and $B$; it requires only enough data to represent them. Lemma 3.2 shows that, under the rank condition that the stacked state matrices $X_{0,T}$ and $\bar{X}_{0,T}$ have full row rank, the closed-loop system under the interface map $u = U_{0,T}Q(x-\Theta\hat{x})+\Xi\hat{x}+\Psi\hat{u}$ has the data-based form (3.4), with $A$ recovered from the zero-input trajectory and $B$ recovered through the pseudoinverse expression $B = (X_{1,T}Q-\bar{X}_{1,T}\bar{Q})(U_{0,T}Q)^\dagger$. Theorem 3.4 then proves that the quadratic form $V(x,\hat{x}) = (x-\Theta\hat{x})^\top P(x-\Theta\hat{x})$ is a simulation function from the ROM to the unknown system whenever (3.8a)-(3.8c) hold, yielding the closeness guarantee $\|y(t)-\hat{y}(t)\| \le \frac{1}{\alpha}\beta(V(x,\hat{x}),t) + \frac{\rho}{\alpha\kappa}\|\hat{u}\|_\infty$. Because $\hat{C} = \Theta$ and the reduced-order model $\hat{A},\hat{B}$ is constructed entirely from this data, the guarantee applies to the actual unknown system rather than to an identified surrogate model.

Load-bearing premise

The load-bearing premise is that the two recorded trajectories reveal the full drift matrix $A$ (so $\bar{X}_{0,T}$ has full row rank) and that the control inputs used in the first trajectory excite every direction of $B$ that the interface map will use; without that, the data-based representation (3.4) can describe a different closed-loop system, and the certificate from Theorem 3.4 would not apply to the true unknown one.

Editorial extensions

If this is right

A controller that enforces a safety, reachability, or reach-while-avoid specification on the data-driven ROM, refined through the interface map $u = U_{0,T}Q(x-\Theta\hat{x})+\Xi\hat{x}+\Psi\hat{u}$, guarantees the same specification on the unknown system up to the quantified bound (2.5).
The construction needs only two recorded trajectories and no identification step: the rank conditions on $X_{0,T}$ and $\bar{X}_{0,T}$ are checkable from data, and the remaining matrices come from solving an LMI and the linear equation (3.8b).
Choosing the ROM input matrix $\hat{B} = I_{\hat{n}}$ (or a scaled identity) makes the ROM fully actuated, and choosing $\Psi$ according to (3.13) minimizes $\rho$, shrinking the guaranteed error bound.
Setting $x_0 = \Theta\hat{x}_0$ removes the transient term $\frac{1}{\alpha}\beta(V(x,\hat{x}),t)$ from (2.5), leaving the steady-state error $\frac{\rho}{\alpha\kappa}\|\hat{u}\|_\infty$.
For autonomous (input-free) systems the verification problem simplifies: no interface map is needed, and conditions (3.8b)-(3.8c) reduce to $\bar{X}_{1,T}\bar{Q}\Theta = \Theta\hat{A}$ and the corresponding LMI with $\bar{X}_{1,T}$.

Reading between the lines

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

Because condition (3.8c) is equivalent to stabilizability, the data-driven certificate inherits a diagnosable failure mode: for an unstabilizable system the LMI cannot be satisfied, so the method's pre-computation step already tells the user that no such certified ROM exists.
The zero-input trajectory requirement implicitly demands that every mode of the drift matrix $A$ shows up in the autonomous data; when $A$ is singular, condition (3.2b) cannot be met and, as the paper's own limitation section concedes, $B$ must be known instead, narrowing the 'fully unknown' setting in that case.
Following the paper's stated outlook, the same two-trajectory logic should extend to systems of the form $\dot{x} = AZ(x)+Bu$ by replacing $x$ with the nonlinear basis $Z(x)$, which would make the interface map nonlinear and would likely preserve the rank-and-LMI structure; testing this on a polynomial or trigonometric basis is a direct next step.
The bound in (2.5) grows linearly with $\|\hat{u}\|_\infty$, so the practical tightness of the certificate depends on restricting the ROM input magnitude; pairing the method with bounded-input symbolic controllers would make the guaranteed error a design parameter rather than a post-hoc number.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take on arXiv:2502.01094. The core idea is genuinely new: build a reduced-order model and a simulation function directly from two input-state trajectories, then use the ROM for LTL controller synthesis with a closeness certificate. The paper is clearly written, the LMI formulation is standard, and the benchmarks (motor, spacecraft, glucose, cart, 25-state) show practical intent. The authors also state their limitations honestly in Section 3.2, including the exact-derivative assumption and the full-rank requirement on A for the zero-input trajectory.

The problem is in Lemma 3.2, and it is load-bearing. From X1,T Q - Xbar1,T Qbar = B U0,T Q, the lemma concludes B equals M (U0,T Q)^\dagger. That only holds if U0,T Q has full row rank. If the collected input data doesn't excite every input direction, the reconstructed B is only correct on the row space of U0,T Q, and the interface term (Xi xhat + Psi uhat) can easily live outside that subspace. Then the Lie derivative in Theorem 3.4 is computed for a data-based system that is not the actual closed-loop system, and the "certified" closeness guarantee can fail for the true unknown system. The paper never states this rank condition, and the hypotheses on X0,T and Xbar0,T don't imply it. This is a fixable gap — persistently exciting inputs should make it hold — but as written, the main theorem is not established.

There is also a minor proof slip: Theorem 3.4 asserts Q = HP from X0,T HP = I and X0,T Q = I, which doesn't follow. The algorithm itself sets Q = HP, so that is repairable, but the equality should be stated as a construction, not a consequence. The benchmarks also use arbitrary initial conditions in the plots while the guaranteed bound assumes x0 = Theta xhat0; that's a presentation issue rather than a fatal one.

Net: the contribution is real and the machinery is mostly sound, but the central certificate currently lacks a necessary hypothesis. A serious referee should see it, because the repair is plausible and the direction is worth pushing. I would not use it as a citation in its current form, and I wouldn't sign off on acceptance. But I'd send it back for major revision rather than reject outright.

Referee Report

2 major / 5 minor

Summary. The paper proposes a data-driven approach to model order reduction for unknown continuous-time linear control systems. From two collected input-state trajectories, it constructs a reduced-order model (ROM) and a quadratic simulation function, and uses the latter to provide a guaranteed bound on the output closeness between the unknown system and the ROM. The main result is Theorem 3.4, which relies on Lemma 3.2 to obtain a data-based closed-loop representation and then formulates LMI conditions from which the simulation function is derived. The paper also gives an algorithm and five numerical benchmarks where the ROM is used for controller synthesis enforcing safety, tracking, or reach-while-avoid specifications.

Significance. If the main theorem were correct, this would be a useful contribution to data-driven control and formal synthesis: it bypasses system identification, uses only two trajectories, and provides a certificate suitable for controller synthesis over unknown linear systems. The paper also has practical strengths: it gives an explicit algorithm, validates the approach on several benchmarks, and correctly invokes the standard simulation-function closeness result in Theorem 2.5. However, the certification statement is not established as written, because Lemma 3.2 omits a necessary rank condition and the proof of Theorem 3.4 uses an unjustified equality Q = HP. These are load-bearing issues for the central claim that the simulation function certifies the actual unknown system.

major comments (2)

[Lemma 3.2, Eq. (3.4)] The derivation of the data-based closed-loop representation (3.4) is only valid if F = U0,T Q has full row rank, or if the interface inputs (Ξ\hat x + Ψ\hat u) always lie in the row space of F. From M := X1,T Q − \bar X1,T \bar Q = B U0,T Q, the paper concludes B = M(U0,T Q)†, which requires (U0,T Q)(U0,T Q)† = I_m. The hypotheses in Lemma 3.2 and Theorem 3.4 impose full-row-rank conditions only on X0,T and \bar X0,T (see Remark 3.3); no condition on U0,T Q is stated. If F is rank-deficient, for example because an input channel is identically zero during data collection, then B ≠ M F† on the complement of the row space of F, and the term B(Ξ\hat x + Ψ\hat u) in the true dynamics is replaced in (3.4) by M F†(Ξ\hat x + Ψ\hat u). The simulation function inequality is then certified for a different closed-loop system, not necessarily for the unknown ct-LCS. The authors should add the rank condition as an explicit assumption and explain how it can be checked from data, or alternatively restrict the interface map so that Ξ\hat x + Ψ\hat u lies in R(F⊤).
[Theorem 3.4, proof after Eq. (3.9)] The proof states that 'since X0,T HP = In from (3.8a) and X0,T Q = In from (3.2a), one can conclude that Q = HP'. This implication is false when T > n, because X0,T has a non-trivial right nullspace and two different right inverses Q and HP can both satisfy X0,T Q = X0,T HP = In. The subsequent equality X1,T Q P^{-1} = X1,T H, and therefore the use of LMI (3.8c), depends on Q = HP. While Algorithm 1 sets Q = HP in Step 4, the theorem statement does not include this condition; it should be added explicitly, for example by quantifying over Q = HP or by adding the constraint Q = HP, and the proof should be revised accordingly.

minor comments (5)

[Remark 3.5] The claim that (3.8c) is feasible if and only if (A,B) is stabilizable is not proved, and as stated it is questionable because (3.8c) is a data-dependent LMI whose interpretation in terms of (A,B) relies on the unstated identification Q = HP discussed above. The authors should either prove this equivalence or soften the remark.
[Section 3.2, Limitations] The statement that the zero-input trajectory condition 'implies that matrix A of the system should be full rank' is not sufficient; for \bar X0,T to have full row rank, the pair (A, x0) must be such that the zero-input trajectory spans R^n. The limitation should be rephrased accordingly.
[Eq. (3.13)] The proposed choice Ψ = Ψ1Ψ2 for minimizing ρ requires the matrix M^T P M to be invertible; this non-singularity condition is not stated.
[Algorithm 1, Step 5] Fixing \hat A to be Hurwitz before solving (3.8b) may make (3.8b) infeasible; the authors should either solve for \hat A and Ξ simultaneously or state the range condition from Lemma 3.7 explicitly.
[Section 4.4] The comparison with [Ion15] and [BBSC23] is not apples-to-apples because those methods target different ROM dimensions and different properties; a quantitative comparison at the same reduction order or on the same error metric would be more informative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the simulation-function certificate is derived from an LMI and a dissipation proof, not fitted to the output-closeness bound; self-citations are contextual and not load-bearing.

full rationale

We found no circular step. The central construction is not self-referential: Theorem 3.4 fixes a quadratic V and proves the dissipation inequality (2.3b) algebraically from LMI (3.8) via Cauchy-Schwarz and Young inequalities, and the closeness guarantee (2.5) is then imported from the external result [ZA17, Thm 3.3]. The data enter only as an identification of A+BF and B through Lemma 3.2; neither V nor the bound is fitted to observed trajectory errors. In the benchmarks the certified bound is compared against actual errors, rather than being tuned to match them, so the claimed closeness is not forced by construction. The self-citations in the introduction and related work ([LSAZ22, LSMZ17, LSZ19, LSZ20, ZLZC23]) are contextual and not load-bearing; the load-bearing cited results [ZA17] have no author overlap with the present paper. The main weaknesses are correctness/assumption gaps rather than circularity: Lemma 3.2's step B = (X1,T Q − bar{X}_{1,T} bar{Q})(U0,T Q)^dag requires U0,T Q to have full row rank, a condition never stated, and Theorem 3.4's assertion Q = HP does not follow from (3.2a) and (3.8a) unless Q is chosen so. These gaps could invalidate the certificate for the actual unknown system, but they do not make the derivation equivalent to its inputs.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. The free parameters are user-chosen design variables and data-collection settings. The main burden is the unstated input-richness condition for identifying B from data, plus the standard assumptions of exact derivative data and stabilizability.

free parameters (6)

\hat n = 1 or 2 in benchmarks
Desired reduced order, chosen by user.
\hat\kappa = 3, 5, 1.5, 2 in benchmarks
Design parameter in LMI (3.8c); user-selected to balance feasibility and error bound.
\varepsilon = 1 in benchmarks
Design parameter with 0 < epsilon < hat_kappa; set to 1.
\hat A = -1, -0.001, -0.002, -10^{-4} in benchmarks
ROM dynamics chosen arbitrarily as Hurwitz; not derived from data.
\hat B = 1 or 0.1 I in benchmarks
ROM input matrix chosen as identity or scaled identity per Remark 3.6.
Sampling time tau and input signal for first trajectory = not reported
Data-collection choices that affect X_{0,T}, X_{1,T}, and U_{0,T}; not specified in the paper, hampering reproduction.

assumptions (5)

domain assumption The unknown system is a continuous-time linear control system as in Definition 2.1.
The entire framework is built for ct-LCS; Section 2.2.
domain assumption Collected data is noise-free and derivatives are exactly known.
Section 3 and Remark 3.1 acknowledge differentiation error but main theorems assume exact X_{1,T} and \bar X_{1,T}.
ad hoc to paper X_{0,T} and \bar X_{0,T} are full row rank.
Required for (3.2) to hold; Remark 3.3 says readily validated but not guaranteed for every system.
domain assumption The pair (A,B) is stabilizable.
Needed for feasibility of (3.8c); mentioned in Remark 3.5.
ad hoc to paper The input data U_{0,T} Q has full row rank (or interface inputs lie in its row space).
Needed for Lemma 3.2 to identify B and for (3.4) to represent the true closed-loop; unstated but load-bearing.

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Pith. "Pith review of Model Order Reduction from Data with Certification." pith.science (2026). https://pith.science/paper/C75QW57Y

@misc{pith2026250201094,
  author       = {Pith},
  title        = {Pith review of: Model Order Reduction from Data with Certification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C75QW57Y}},
  note         = {Machine review of arXiv:2502.01094}
}

read the original abstract

Model order reduction (MOR) involves offering low-dimensional models that effectively approximate the behavior of complex high-order systems. Due to potential model complexities and computational costs, designing controllers for high-dimensional systems with complex behaviors can be challenging, rendering MOR a practical alternative to achieve results that closely resemble those of the original complex systems. To construct such effective reduced-order models (ROMs), existing literature generally necessitates precise knowledge of original systems, which is often unavailable in real-world scenarios. This paper introduces a data-driven scheme to construct ROMs of dynamical systems with unknown mathematical models. Our methodology leverages data and establishes similarity relations between output trajectories of unknown systems and their data-driven ROMs via the notion of simulation functions (SFs), capable of formally quantifying their closeness. To achieve this, under a rank condition readily fulfillable using data, we collect only two input-state trajectories from unknown systems to construct both ROMs and SFs, while offering correctness guarantees. We demonstrate that the proposed ROMs derived from data can be leveraged for controller synthesis endeavors while effectively ensuring high-level logic properties over unknown dynamical models. We showcase our data-driven findings across a range of benchmark scenarios involving various unknown physical systems, demonstrating the enforcement of diverse complex properties.

Figures

Figures reproduced from arXiv: 2502.01094 by the authors.

**Figure 1.** (a) Closed-loop trajectories of the original system, starting from 30 distinct initial conditions, together with (b) the closeness between trajectories of the motor system and those of its corresponding ROM, as well as the mean of errors. As illustrated, all trajectories of the original unknown motor remain within the safe region under the designed ROM controller. are as follows: A =         0 0 1 0 0 0 0 1 … view at source ↗

**Figure 2.** Trajectories should start from the initial set Xs and reach the target set Xt while avoiding debris in space. As illustrated in (a), trajectories of the spacecraft and its ROM are very close to one another, satisfying the spacecraft’s reach-while-avoid property. As shown in (b), the specification is properly fulfilled for 50 arbitrary trajectories of the spacecraft under the designed ROM controller. Algorithm 1, we … view at source ↗

**Figure 3.** As the reach-while-avoid specification, trajectories of x1 and x2 should start from the initial set Xs and reach the target set Xt without colliding with the obstacles . As seen, the specification is correctly fulfilled for 20 arbitrary trajectories of the original system under the designed ROM controller, offering the practicality of our framework for high-dimensional systems [PITH_FULL_IMAGE:figures/full_fig_p021… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The quantified error bound for this example is 0.8609 according to Theorem 2.5. As seen, all plotted errors are below the quantified upper bound 0.8609, demonstrating the formality of our results. 5. Conclusion In this work, we introduced a certified data-driven scheme…

**Figure 5.** Figure 5: The behavior of all states while the reach-while-avoid is solved for the first and second states. As illustrated, there is no anomaly in the behavior of states, and the desired property is clearly satisfied, i.e., x1 and x2 reached the target . two input-state trajecto…

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write newline

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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