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REVIEW 4 major objections 5 minor 1 cited by

Abstraction-based Control of Unknown Continuous-Space Models with Just Two Trajectories

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Two trajectories suffice to build a certified symbolic model and hybrid controller for unknown polynomial systems.

desk verdict First trajectory-based ASF construction for unknown polynomial systems, but the 'two trajectories' claim is inflated and the interface's admissibility w.r.t. the input set is unproven. read the letter →

arxiv 2412.03892 v1 pith:KCU4QHU5 submitted 2024-12-05 eess.SY cs.SY

Behrad Samari , Mahdieh Zaker , Abolfazl Lavaei This is my paper · ORCID
classification eess.SYcs.SY
keywords abstraction-basedcontrolsymbolicmodelsalternatingsimulationfunctionsdata-drivennonlinearpolynomialsystemssum-of-squaresoptimizationhybridinterfacepersistentexcitation
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 a formal, abstraction-based controller for an unknown nonlinear system can be built from only two input-state trajectories, one recorded from the original system and one from its discretized symbolic dynamics. The key ingredient is a data-driven alternating simulation function, a quadratic function that upper-bounds how far the unknown system's state can drift from its symbolic model under a designed hybrid interface map. When the collected data satisfy a rank condition and a coupled set of polynomial and linear-matrix-inequality constraints is feasible, the resulting simulation function yields an epsilon-approximate alternating simulation relation between the systems, so that a discrete controller designed on the symbolic model transfers to the unknown system with a quantifiable error. The paper demonstrates this pipeline on a two-state polynomial benchmark, enforcing both safety and reach-while-avoid specifications without ever identifying the system matrices.

What carries the argument

The load-bearing identity is the data-based representation of the one-step increment (Lemma 3.1): with the interface u = I Y1(x) P x - Ihat Y2(xhat) P xhat + uhat, and with G1 = Y1 P, G2 = Y2 P, the difference A M(x) + B u - (A M(xhat) + B uhat) is exactly O_plus G1 x - Ohat_plus G2 xhat, where O_plus and Ohat_plus are the one-step-shifted data matrices of the two collected trajectories. This identity removes the unknown matrices A and B from the analysis, replacing them by measured data and the monomial matrix M. Conditions (3.8a)-(3.8d) then ensure that the unknown A, B are consistently absorbed into the free functions Y1, Y2 and the constant Theta, while the linear matrix inequality (3.8e) guarantees that the quadratic function S(x, xhat) = (x - xhat)^T P (x - xhat) satisfies the dissipation inequality S(x+, xhat+) <= gamma S(x, xhat) + psi. The mechanism converts the abstract object 'alternating simulation function' into a concrete, data-conditioned semidefinite feasibility problem.

What would settle it

For a given unknown dt-IANSP, exhibit a two-trajectory dataset satisfying the stated rank condition (full row rank of the monomial data matrices and T >= M+1) for which the sum-of-squares feasibility problem (3.8a)-(3.8e) has no solution; such a counterexample would show the rank condition alone is not sufficient for the claimed ASF construction.

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Extended reading notes

Core claim

The central claim is that, for a discrete-time input-affine nonlinear system with polynomial dynamics whose matrices A and B are unknown, two finite input-state trajectories – one from the unknown system and one from the nominal (unquantized) symbolic dynamics – suffice to construct an alternating simulation function from the data-driven symbolic model to the unknown system, provided the data meet a rank condition reflecting persistent excitation. The construction is expressed as a sum-of-squares feasibility problem: find matrix-valued functions Y1 and Y2 and constant matrices Xi and Theta with Xi > 0 satisfying the coupled equations (3.8a)-(3.8d) and the linear matrix inequality (3.8e). When these are satisfied, the quadratic function S(x, xhat) = (x - xhat)^T P (x - xhat) with P = $Xi^{{-1}}$ is an alternating simulation function with $\alpha$ = lambda_min(P), rho = 0, and psi = (1 + 1/mu) ||$\sqrt$(P)||^2 $delta^{2}$, and the interface map u = I Y1(x) P x - Ihat Y2(xhat) P xhat + uhat transfers any discrete controller to the unknown system. Combined with the epsilon-approximate simulation relation theorem, this yields a certified closeness bound epsilon = $\sqrt$( psi / ( $\alpha$ (1 - gamma) eta1 ) ) in the rho = 0 case, which quantifies how faithfully the symbolic design is realized on the unknown system.

Load-bearing premise

The method works only if there exist matrix-valued functions Y1 and Y2 and a fixed matrix Theta that satisfy the coupled compatibility equations (3.8a)-(3.8d) for every state in the domain; the stated rank condition on the data does not by itself guarantee such a solution exists.

Editorial extensions

If this is right

  • A controller synthesized on a finite symbolic model can be refined to the unknown dt-IANSP with a certified epsilon closeness bound, so safety, reachability, and reach-while-avoid specifications transfer with formal guarantees.
  • Only two trajectories are needed, with T >= M+1 samples each, eliminating the i.i.d. sample burden and the large number of independent runs required by scenario-based abstraction methods.
  • The same ASF and interface map can be reused across different control objectives on the same system, as demonstrated by switching from safety to reach-while-avoid without re-solving the ASF problem.
  • As the state discretization parameter delta shrinks, the error bound psi scales as delta^2, so the symbolic model becomes arbitrarily accurate at the price of more discrete states.
  • The construction is algorithmic: once data are collected, the ASF is found by solving a sum-of-squares feasibility program with semidefinite constraints.

Reading between the lines

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

  • An immediate testable extension is to relax the constant-Theta condition (3.8c)-(3.8d) to allow Theta that depends on x and xhat; this would widen feasibility at the cost of a harder SOS program and would likely be necessary for higher-degree polynomial dynamics.
  • The two-trajectory construction is likely to carry over to other model classes (e.g., systems with rational or trigonometric drift) as long as a data-based representation of the one-step increment analogous to Lemma 3.1 can be established.
  • Because the interface map uses the measured data matrices, the framework implicitly assumes noise-free measurements; analysing the effect of measurement noise on the closeness bound is a natural next step not addressed in the paper.
  • A compositional extension to networks of unknown systems could be built by combining the data-driven ASFs of subsystems via small-gain conditions, following the pattern of model-based compositional symbolic control.
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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

4 major / 5 minor

Summary. This paper proposes a data-driven abstraction-based control framework for unknown discrete-time input-affine nonlinear polynomial systems (dt-IANSP). The authors claim that using only two input-state trajectories, together with a rank condition ensuring persistent excitation, one can construct a data-driven symbolic model, an alternating simulation function (ASF), and a hybrid interface controller. The main result, Theorem 3.4, gives SOS conditions under which a quadratic function S(x,xhat) = (x-xhat)^T P (x-xhat) is an ASF from the symbolic model to the unknown system, with a data-based interface map. The paper then applies Theorem 2.7 to obtain an epsilon-approximate alternating simulation relation and demonstrates the approach on a safety and a reach-while-avoid case study.

Significance. If the central claim held, the paper would be a significant advance over scenario-based data-driven abstraction methods that require large numbers of i.i.d. samples, since it would use only two finite-length trajectories. The data-based representation in Lemma 3.1 is elegant, and the reduction of ASF construction to an SOS program is concrete and aligns with existing tools. The derivation of Theorem 3.4 is internally coherent conditional on the stated conditions. However, the significance is contingent on closing a major gap: the interface map is never shown to take values in the admissible input set U, and the 'just two trajectories' framing is contradicted by the symbolic-model construction in Section 2.5, which requires querying the unknown system at every discrete state/input pair.

major comments (4)
  1. [Section 3, Theorem 3.4] The theorem proves inequality (2.5b) for the algebraic interface map u = I Y1(x) P x - Ihat Y2(xhat) P xhat + uhat, but it never proves that this u belongs to the admissible input set U. Definition 2.4 requires the existence of u in U for every (x,xhat,uhat) in X x Xhat x Uhat. When U is a proper subset of R^m, as it is in the case study (U=[-2.5,2.5]), the proposed u can leave U: the term -4.9773 x2 + 4.9773 xhat2 in (4.2) alone reaches 4.9773 for x2=-0.5 and xhat2=0.5, already exceeding the bound before adding the remaining terms. Consequently, Theorem 3.4 does not establish an ASF as defined, and the epsilon-bound and the correctness guarantees from Theorem 2.7 do not follow. A repair would require either an explicit SOS constraint enforcing u in U over X x Xhat x Uhat, or an explicit assumption that U = R^m.
  2. [Section 2.5] The proposed data-driven symbolic model is constructed by querying the unknown system at every discrete state/input pair (xhat,uhat) to compute AM(xhat)+B uhat and then apply the quantization map Pi. This requires on the order of |Xhat|*|Uhat| separate experiments. The two trajectories collected in Section 3 are used only for the ASF conditions; they are not sufficient to build the symbolic model. This directly contradicts the abstract and Problem 2.9's claim of 'just two trajectories,' and it overstates the data efficiency of the framework. The paper needs to either clarify that the symbolic-model construction is a separate data-collection phase or rescope the contribution to the ASF design given a preconstructed symbolic model.
  3. [Section 3, conditions (3.8a)-(3.8d)] The rank condition in Remark 3.2 ensures existence of some functions G1 and G2 satisfying the individual equalities M G1 = Upsilon(x) and Mhat G2 = Upsilon(xhat). However, Theorem 3.4 requires the much stronger coupled conditions: the same constant matrix Xi appears in both (3.8a) and (3.8b), the same constant Theta appears in both (3.8c) and (3.8d), and moreover Y1 and Y2 are related to G1 and G2 via P = Xi^{-1}. The paper provides no general feasibility test for these coupled equalities beyond attempting the SOS program, and the rank condition alone does not imply their satisfiability. This is a load-bearing assumption of the main theorem that is not characterized; the claim of a systematic data-driven design is therefore not fully supported.
  4. [Section 3, proof of Theorem 3.4] The inequality (3.13) applies the quantization bound (2.4) to the point AM(xhat)+B uhat, yielding ||Pi(AM(xhat)+B uhat) - (AM(xhat)+B uhat)|| <= delta. This is valid only if AM(xhat)+B uhat belongs to X, the domain of Pi. The paper does not state this assumption. In the case study, X is the safe set [-0.5,0.5]^2 and the open-loop dynamics can leave X during data collection, so the quantization map may be applied outside its domain. This is a technical gap in the correctness argument of Theorem 3.4 and in the construction of Section 2.5.
minor comments (5)
  1. [Throughout] The rendering of boldface notation is badly mangled: for example, O, I, and O+ appear as repeated characters such as 'OOO', 'III', and 'O+O+O+', and the hat versions appear as 'ˆOˆOˆO'. This severely harms readability and should be fixed.
  2. [Section 2.4, Definition 2.4] Remark 2.5 refers to 'u = uˆu(x, xhat, uhat)', which is a typo; it should read 'u = u(x, xhat, uhat)' or similar.
  3. [Section 3.5, Remark 3.6] The assertion that the same S is also an ASF from Sigma to Sigma-hat is stated with only a brief proof sketch. Given that the main theorem does not verify input admissibility (see Major Comment 1), the bisimulation claim requires more justification.
  4. [Section 4] After computing epsilon = 0.1831, the text states the error is always lower than the reported epsilon, but Figure 1b's axis labels are not described; the figure should include an explicit comparison to the epsilon line for clarity.
  5. [Section 1] The related-work discussion is thorough, but the comparison to scenario-based methods would be strengthened by quantifying the number of trajectories needed in prior work versus the two trajectories claimed here, given the additional queries required to build the symbolic model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ASF is a certified sufficient condition derived algebraically from data, not a fitted prediction.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The central theorem (Theorem 3.4) proposes a quadratic candidate S(x,xhat)=(x-xhat)^T P (x-xhat) and proves condition (2.5b) by the algebraic data-based identity of Lemma 3.1 (O+ G1 x - Ohat+ G2 xhat = A M(x)+B u - (A M(xhat)+B uhat)) combined with the SOS feasibility conditions (3.8a)-(3.8e). The resulting epsilon in Remark 2.8 is a closed-form consequence of the certified inequality, not a fit to the same trajectories dressed up as a prediction. The interface map is introduced as a design choice borrowed from [GDPT21], with the proof showing that if the SOS conditions hold the inequality holds; no step defines the ASF in terms of the target closeness or imports a uniqueness/forced-choice result. The self-citations in the introduction and related work ([SAZL24], [SNL24], [ALZ23], [Lav23]) are contextual and none is load-bearing for Theorem 3.4. A separate correctness concern, not a circularity, is that Theorem 3.4 proves the algebraic inequality for the proposed interface but does not explicitly verify u in the input constraint set U; this is a gap in the admissibility argument, but it does not make the derivation circular.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The method adds no new physical entities. It rests on the polynomial structure assumption, exact measurements, rank conditions, solvability of the SOS certificate, quantization bounds, and an implicit input-constraint check. The most fragile additions are the solvability of conditions (3.8a)-(3.8e) and the unverified u in U condition.

free parameters (6)
  • mu = 0.01 in case study
    Chosen for Young's inequality bound in Theorem 3.4; it directly enters the approximation term psi and hence the epsilon guarantee.
  • gamma = 0.99 in case study
    Chosen contraction rate in (3.8e); must be in (0,1) for the ASF condition to hold.
  • delta = 0.001 in case study
    State discretization parameter; enters psi through the quantization bound (2.4).
  • eta1 = 0.99 in case study
    Tuning parameter used in Remark 2.8 to compute the final closeness epsilon.
  • T = 9 in case study
    Trajectory horizon; must be large enough to satisfy the full-row-rank conditions on the data matrices.
  • beta2 = not specified
    An upper bound on the induced 2-norm of B is required in Remark 3.6 for the reverse ASF direction; it is not derived from data.
assumptions (7)
  • domain assumption The unknown system is a dt-IANSP (2.1) with known upper bound on monomial degree or known extended dictionary M(x).
    Definition 2.1 and Remark 2.2; without this polynomial structure, the data representation in Lemma 3.1 cannot be set up.
  • domain assumption States and inputs are measured exactly, without noise.
    Section 3 assumes all system states and inputs are measurable; no noise model or robustification is included.
  • domain assumption The data matrices M and Mhat have full row rank.
    Remark 3.2 uses this persistent-excitation condition to guarantee solutions of (3.8a) and (3.8b).
  • ad hoc to paper Conditions (3.8a)-(3.8e) are solvable for some Y1, Y2, Xi, Theta.
    Theorem 3.4 is conditional on this existence; the paper does not prove feasibility from the rank condition and gives no general test short of solving the SOS program.
  • domain assumption The quantization map Pi satisfies ||Pi(x)-x|| <= delta for the relevant next states, including states that may leave X.
    The bound (2.4) is stated for x in X only, but the proof in (3.13) applies it to AM(xhat)+B uhat, which could leave X.
  • ad hoc to paper The proposed interface map always takes values in the admissible input set U.
    ASF condition (2.5b) requires existence of u in U for every x, xhat, uhat; Theorem 3.4 does not verify this.
  • domain assumption For the reverse ASF, an upper bound beta2 on ||B|| is known.
    Remark 3.6 introduces beta2 to define psi for the reverse direction; B is otherwise unknown, so this is extra a priori knowledge.

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Pith. "Pith review of Abstraction-based Control of Unknown Continuous-Space Models with Just Two Trajectories." pith.science (2026). https://pith.science/paper/KCU4QHU5

@misc{pith2026241203892,
  author       = {Pith},
  title        = {Pith review of: Abstraction-based Control of Unknown Continuous-Space Models with Just Two Trajectories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCU4QHU5}},
  note         = {Machine review of arXiv:2412.03892}
}
read the original abstract

Finite abstractions (a.k.a. symbolic models) offer an effective scheme for approximating the complex continuous-space systems with simpler models in the discrete-space domain. A crucial aspect, however, is to establish a formal relation between the original system and its symbolic model, ensuring that a discrete controller designed for the symbolic model can be effectively implemented as a hybrid controller (using an interface map) for the original system. This task becomes even more challenging when the exact mathematical model of the continuous-space system is unknown. To address this, the existing literature mainly employs scenario-based data-driven methods, which require collecting a large amount of data from the original system. In this work, we propose a data-driven framework that utilizes only two input-state trajectories collected from unknown nonlinear polynomial systems to synthesize a hybrid controller, enabling the desired behavior on the unknown system through the controller derived from its symbolic model. To accomplish this, we employ the concept of alternating simulation functions (ASFs) to quantify the closeness between the state trajectories of the unknown system and its data-driven symbolic model. By satisfying a specific rank condition on the collected data, which intuitively ensures that the unknown system is persistently excited, we directly design an ASF and its corresponding hybrid controller using finite-length data without explicitly identifying the unknown system, while providing correctness guarantees. This is achieved through proposing a data-based sum-of-squares (SOS) optimization program, enabling a systematic approach to the design process. We illustrate the effectiveness of our data-driven approach through a case study.

Figures

Figures reproduced from arXiv: 2412.03892 by the authors.

Figure 1
Figure 1. The left-hand side subfigure illustrates the trajectories of x1 ( ), x2 ( ), xˆ1 ( ), and ˆx2 ( ). As shown, it is clear that the safety property is fulfilled. Moreover, the right-hand side subfigure depicts the norm of the error between the trajectories of the original system and those of its symbolic model. As illustrated, the error is always lower than the ϵ reported. synthesize a discrete controller for the data… view at source ↗
Figure 2
Figure 2. The trajectories must originate from the initial set , reach the target set , while avoiding the obstacle . As shown, the trajectories of the unknown system (4.1) ( ) and its data-driven symbolic model ( ) closely align, demonstrating compliance with the system’s reach-while-avoid property. 5. Conclusion In this work, we introduced a data-driven abstraction-based methodology that synthesizes a hybrid controller for … view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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