Pith. sign in

REVIEW 3 major objections 4 minor 27 references

Some Computational Tools for Solving a Selection of Problems in Control Theory

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Julia package brings certified exact solving to control-theory problems.

desk verdict A genuinely useful Julia toolbox for certified polynomial solving, but the paper's only complete identification example has inconsistent numbers and the 'guaranteed to be correct' language outruns what is actually certified. read the letter →

arxiv 2505.09191 v1 pith:VYOAVIUL submitted 2025-05-14 cs.SC cs.MScs.SYeess.SY

Alexander Demin , Christina Katsamaki , Fabrice Rouillier This is my paper · ORCID
classification cs.SCcs.MScs.SYeess.SY MSC 68W3093B40
keywords PACE.jlcertifiedcomputationsymboliceliminationrationalunivariaterepresentationdiscriminantvarietyparameteridentificationstructuralstabilityH-infinitynorm
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

The paper presents PACE.jl, a Julia package that wraps symbolic elimination algorithms—discriminant varieties, Rational Univariate Representation, Sturm–Habicht sequences, interval Newton refinement, and cylindrical algebraic decomposition—and applies them to three control-theory tasks. It claims that these tasks can be solved with guarantees of correctness: parameter estimation from ODE data, structural stability of multidimensional systems, and $H_\infty$-norm computation. The payoff is that outputs come as isolating intervals or rational descriptions of the solution set, so a user can trust the result without relying on heuristic numerical iteration. Demonstrations include a 16-state NF-kB identification model, a parametric two-dimensional stability test, and a transfer-matrix norm example.

What carries the argument

The engine is the certified solving pipeline: from a polynomial system with rational coefficients, compute a Rational Univariate Representation (RUR), a rational parametrization of the zero-dimensional solution set by a univariate polynomial; then isolate the real roots of that univariate polynomial in rational-bounded intervals and refine with multi-precision interval Newton steps. For parametric problems, discriminant varieties mark parameter values where the solution behavior can change, and the cells of maximal dimension of a cylindrical algebraic decomposition give regions where stability is uniform. The Sturm–Habicht sequence is the sign-counting device used for real-root counting and for the $H_\infty$ computation. These objects, not a single theorem, carry the certification.

What would settle it

Differentiate the rational interpolant $\hat{y}(t) = \frac{7.40t + 54.02}{1.52t^2 - 10.93t + 27.01}$ at $t=0$ and compare with the printed equations $1.000 = x_0^2 + x_0$, $0.608 = 2x_0x'_0 + x'_0$, $0.227 = 2x'_0{}^2 + 2x_0x''_0 + x''_0$. The interpolant gives $y(0) = 2.000$, $y'(0) \approx 1.083$, and $y''(0) \approx 0.652$, so the printed system cannot be the evaluation of the stated interpolant; reproducing the code would show which half of the example is in error.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that a bundle of classical exact-computation tools can be wrapped into a single Julia package and solve control problems with certifiable outputs. In parameter identification, the ODE is differentiated repeatedly and evaluated at an interpolant of the data, yielding a polynomial system whose Rational Univariate Representation isolates all candidate parameters and initial states, which can then be refined to arbitrary precision. In stability analysis of multidimensional systems, structural stability of a transfer-function denominator is reformulated as a real algebraic condition, and parametric stability regions are classified by computing a discriminant variety and cell decomposition. In $H_\infty$-norm computation, the norm is located as the maximal real $\gamma$-projection of solutions of a zero-dimensional polynomial system, obtained via Sturm–Habicht sign variations. The outputs in all three settings are intervals with rational bounds that are guaranteed to contain the true value.

Load-bearing premise

The identification pipeline rests on the assumption that the polynomial system obtained from the differentiated ODE and the estimated derivatives faithfully represents the true data; in the printed toy example the recorded derivative values do not match the stated rational interpolant, so that step is not actually certified as written.

Editorial extensions

If this is right

  • Certified parameter identification becomes a finite exact computation: all possible parameter vectors consistent with the data appear as isolated solutions of the polynomial system, and each can be refined to any requested precision.
  • Structural stability of a parametric multidimensional system can be certified region by region, with a sample point returned for every stable and unstable region, so no exhaustive simulation search is needed.
  • $H_\infty$ norms can be returned as rigorous intervals whose width shrinks as the requested precision grows, making them usable as certificates in robustness analysis.
  • The same pipeline handles both parameter-free and parameter-dependent systems, so a user can solve one parametric model and obtain all stability regions at once.

Reading between the lines

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

  • Because every candidate parameter set is enumerated as an isolated solution, identifiability itself is certified in practice: if the polynomial system has a unique real solution consistent with the domain and sign constraints, the parameters are identifiable under the stated reduction.
  • The derivative-estimation step is the natural place to push certification further: replacing the rational interpolant with interval-based derivative bounds, such as interval Taylor enclosures, would make the identification pipeline fully verified from data to parameters rather than verified only after the polynomial system is formed.
  • The same Sturm–Habicht machinery used for $H_\infty$ on transfer matrices should extend to parametric transfer functions once discriminant-variety methods for the norm are implemented, which is the paper's announced next direction.
  • A corrected toy example would make the identification demonstration reproducible, since the printed constants do not match the stated interpolant; this is a simple check anyone can run before relying on the tool.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 introduces PACE.jl, a Julia package that wraps and integrates certified symbolic elimination tools (rational univariate representation, discriminant varieties, Sturm-Habicht sequences, interval Newton methods, multi-precision interval arithmetic) and demonstrates their use on three control-theory problems: parameter identification, structural stability of multidimensional systems, and computation of the H-infinity norm. The main text explains the underlying algebraic objects, describes the package structure and its dependencies, and presents three worked examples with runnable notebooks. The authors claim that, by using certified algorithms, the displayed results are guaranteed to be correct.

Significance. If the claims are taken at face value, the paper is a useful software demonstration that brings a nontrivial stack of symbolic-certification algorithms (RUR, discriminant varieties, CAD, interval arithmetic) to bear on control problems in a reproducible Julia package. The strength of the paper is its packaging: it gives users access to certified zero-dimensional solving, parametric real-root counting, and exact interval output for H-infinity norm computation, with notebooks for all three applications. The significance is, however, tempered by two issues. First, the identification example in §4.1 is internally inconsistent as printed, with wrong numerical constants. Second, the identification pipeline interpolates data and differentiates the interpolant without any error bound, so the 'guaranteed to be correct' language in the abstract and conclusion is not justified for the identification application. The stability and H-infinity demonstrations appear sound and are the most compelling parts of the paper.

major comments (3)
  1. [§4.1, display after 'we evaluate the equations at t0 = 0.00'] The numerical constants on the left-hand side of the displayed polynomial system are inconsistent with the rational interpolant given immediately above. For \hat{y}(t) = (7.40t + 54.02)/(1.52t^2 - 10.93t + 27.01), one computes \hat{y}(0) = 54.02/27.01 = 2.000, \hat{y}'(0) \approx 1.083, and \hat{y}''(0) \approx 0.652. The paper instead prints 1.000, 0.608, and 0.227. As a consequence, x0 = 1.000 does not satisfy x0^2 + x0 = 1.000 (it gives 2.000), and x0 = -2.000 does not satisfy it either. The four printed solutions (±0.604, 1.000) and (±0.427, -2.000) would be consistent with the system if the left-hand side were 2.000, 1.083, and 0.652, so the error appears to be in the printed constants rather than in the solution values, but the demonstration is nevertheless internally contradictory as written and must be corrected.
  2. [§4.1, paragraph on the construction of the polynomial system] The reduction from measured data to a polynomial system is performed by interpolating the data with \hat{y}(t), differentiating \hat{y}(t) symbolically, and evaluating the resulting derivatives at t0. The certified solver then solves the polynomial system exactly, but that system is only an approximation to the true parameter-estimation problem unless the interpolation and differentiation step has a validated error bound or converges as the data density increases. No error estimate, convergence statement, or interval-enclosure of the derivatives is provided. Therefore the abstract and conclusion's claim that the identification results are 'guaranteed to be correct' is too strong; the guarantee applies to the solved algebraic system, not to the parameter estimates obtained from the interpolant. The authors should either supply an error analysis for the interpolation/differentiation step or qualify the guarantee language to state that certification covers only the polynomial system.
  3. [§4.2 and §4.3 (general assessment)] The stability and H-infinity examples appear internally consistent and are the strongest support for the package's claimed capabilities. The H-infinity example gives two intervals, [1.59375, 1.625] and [1.617919921875, 1.6180419921875], that are both plausible as enclosures of the same norm (the second at higher precision), but the text should briefly explain why the first, 'minimum precision necessary to isolate the roots,' is so much wider than the second. This is a minor clarity issue, not a technical error.
minor comments (4)
  1. [§2, paragraph on parametric systems] The sentence 'which is a polynomial in ∈ Q[U1,...,Ul]' contains a stray '∈' and should read 'which is a polynomial in Q[U1,...,Ul]'. Several similar typographical slips appear throughout, including 'a j,0,...a j,d1' where the index should be 'd_j'.
  2. [§2, definition of cells] The definition of the sets Poi and Ptj is dense and has minor notation inconsistencies (e.g., 'd1' used in multiple levels). A reference to the precise algorithm or a clearer display would help a reader verify the decomposition claim.
  3. [§4.1, Example 1] The table of ground-truth versus estimated parameters is not aligned in the extracted text, making it hard to read. Please format it with proper columns and report the error metric used to define 'relative error around 1%'.
  4. [References] The reference 'Bassik et al. (2023)' appears to be for a preprint (Robust parameter estimation for rational ordinary differential equations) but no DOI or arXiv identifier is given. Other entries have missing DOIs; adding them would improve reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: PACE.jl's certified-solver claims rest on independent published algorithms (RUR, discriminant varieties, Sturm–Habicht, interval Newton) rather than on the data or target outputs of the paper's control examples.

full rationale

The paper's central claim is that PACE.jl applies certified symbolic elimination to control problems. The load-bearing computational steps are citations to published, parameter-free algorithms with stated assumptions that do not include the control-theoretic conclusions: RUR (Rouillier 1999), discriminant varieties (Lazard and Rouillier 2007), Sturm–Habicht root counting (González-Vega et al.; Basu et al.), the Moebius reduction (Bouzidi et al. 2019), and interval Newton (Revol 2001). These are self-citations in part, but they are independent mathematical results rather than restatements of this paper's outputs. In Section 4.1, the identification system is constructed by differentiating the ODE and evaluating it at t0 using derivative estimates from an interpolant; the unknown parameters are then solved for from that system and selected by goodness of fit against the data, so the parameters are not inserted into the construction. No equation in the paper is defined in terms of the quantity it is supposed to predict, and no fitted value is renamed as a prediction. A separate defect, not a circularity, is that the printed constants in the toy example (1.000, 0.608, 0.227) do not match the displayed rational interpolant, whose values at 0 are about 2.000, 1.083, and 0.652; this undermines the worked example's reproducibility but does not make the derivation circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No invented entities. Free parameters are limited to the data-fitting interpolant in the toy example. Axioms are standard results from cited literature.

free parameters (1)
  • rational interpolant coefficients in toy identification = 7.40, 54.02, 1.52, -10.93, 27.01
    These coefficients are fit to the four data points to define \$\hat{y}(t)\$; the polynomial system and identified parameters depend on this fit.
assumptions (6)
  • standard math RUR provides a bijective, multiplicity-preserving representation of roots of zero-dimensional systems (Rouillier 1999)
    Invoked in Section 2 and used in Example 1 to encode solutions.
  • domain assumption Discriminant variety separates parameter space into cells where the zero set is an analytic covering (Lazard-Rouillier 2007)
    Used in Section 4.2 to conclude stability is constant on each cell.
  • domain assumption Decarlo et al. 1977 equivalence: structural stability iff univariate edge conditions and boundary hypersurface condition hold
    Reduces n-D stability to checking conditions (3) in Section 4.2.
  • standard math Sturm-Habicht sequences correctly count real roots and signs (González-Vega et al. 1998; Basu et al. 2006)
    Basis for the univariate stability tests and H-infinity norm computation.
  • domain assumption Moebius change of variables maps the unit circle condition to a real polynomial system (Bouzidi et al. 2019)
    Converts the boundary condition |z_i|=1 into R=I=0 in Section 4.2.
  • domain assumption Kanno-Smith 2006 characterization of H-infinity norm via determinant of Φ_γ(Iω)
    Used in Section 4.3 to reduce norm computation to real roots of n(ω,γ)=0.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Some Computational Tools for Solving a Selection of Problems in Control Theory." pith.science (2026). https://pith.science/paper/VYOAVIUL

@misc{pith2026250509191,
  author       = {Pith},
  title        = {Pith review of: Some Computational Tools for Solving a Selection of Problems in Control Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYOAVIUL}},
  note         = {Machine review of arXiv:2505.09191}
}
read the original abstract

This paper demonstrates how certified computational tools can be used to address various problems in control theory. In particular, we introduce PACE.jl, a Julia package that implements symbolic elimination techniques, including (among others) discriminant varieties and Rational Univariate Representation, while also supporting multi-precision interval computations. We showcase its applications to key control theory problems, including identification, stability analysis, and optimization, for both parameter-dependent and parameter-free systems.

Figures

Figures reproduced from arXiv: 2505.09191 by the authors.

Figure 1
Figure 1. Points sampled in each 2d-cell of the decomposition [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 17 canonical work pages

  1. [1]

    Aubry, P., Rouillier, F., and Safey El Din , M. (2002). Real solving for positive dimensional systems. Journal of Symbolic Computation, 34(6), 543--560. doi:10.1006/jsco.2002.0563

    work page arXiv 2002

  2. [2]

    Bassik, O., Berman, Y., Go, S., Hong, H., Ilmer, I., Ovchinnikov, A., Rackauckas, C., Soto, P., and Yap, C. (2023). Robust parameter estimation for rational ordinary differential equations

    work page 2023

  3. [3]

    Basu, S., Pollack, R., and Roy, M. (2006). Algorithms in Real Algebraic Geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag

    work page 2006

  4. [4]

    Bouzidi, Y., Quadrat, A., Rouillier, F., and Younes, G. (2021). Computation of the L _ -norm of Finite-Dimensional Linear Systems , 119–136. Springer International Publishing. doi:10.1007/978-3-030-81698-8_9

    work page doi:10.1007/978-3-030-81698-8_9 2021

  5. [5]

    and Rouillier, F

    Bouzidi, Y. and Rouillier, F. (2016). Certified Algorithms for proving the structural stability of two dimensional systems possibly with parameters . In Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems , Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems. Minneapolis...

    work page 2016

  6. [6]

    Bouzidi, Y.M., Quadrat, A., and Rouillier, F. (2019). Certified Non-conservative Tests for the Structural Stability of Discrete Multidimensional Systems . Multidimensional Systems and Signal Processing , 30(3), 31. doi:10.1007/s11045-018-0596-y

    work page doi:10.1007/s11045-018-0596-y 2019

  7. [7]

    Collins, G.E. (1967). Subresultants and reduced polynomial remainder sequences. J. ACM, 14(1), 128–142. doi:10.1145/321371.321381

    work page arXiv 1967

  8. [8]

    Collins, G.E. (1975). Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In H. Brakhage (ed.), Automata Theory and Formal Languages, 134--183. Springer Berlin Heidelberg, Berlin, Heidelberg

    work page 1975

Show all 27 references
  1. [9]

    Decarlo, R.A., Murray, J., and Saeks, R. (1977). Multivariable N yquist theory. International Journal of Control, 25(5), 657--675. doi:10.1080/00207177708922261

    1977 doi

  2. [10]

    and Gowda, S

    Demin, A. and Gowda, S. (2024). Groebner.jl: A package for gr\"obner bases computations in julia. ://arxiv.org/abs/2304.06935

    2024 arXiv

  3. [11]

    Demin, A., Rouillier, F., and Ruiz, J. (2024). Reading rational univariate representations on lexicographic groebner bases. ://arxiv.org/abs/2402.07141

    2024 arXiv

  4. [12]

    Doyle, J., A, F., and Tannenbaum, A. (2009). Feedback Control Theory. doi:10.1007/978-0-387-85460-1_1

    2009 doi

  5. [13]

    Gonz \'a lez-Vega, L., Recio, T., Lombardi, H., and Roy, M.F. (1998). Sturm---habicht sequences, determinants and real roots of univariate polynomials. In B.F. Caviness and J.R. Johnson (eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition, 300--316. Springer ...

    1998

  6. [14]

    Hong, H., Ovchinnikov, A., Pogudin, G., and Yap, C. (2019). Sian: software for structural identifiability analysis of ode models. Bioinformatics, 35(16), 2873--2874. doi:10.1093/bioinformatics/bty1069

    2019 doi

  7. [15]

    and Smith, M.C

    Kanno, M. and Smith, M.C. (2006). Validated numerical computation of the l_ -norm for linear dynamical systems. Journal of Symbolic Computation, 41(6), 697--707. doi:10.1016/j.jsc.2005.11.005

    2006 doi

  8. [16]

    Kobel, A., Rouillier, F., and Sagraloff, M. (2016). Computing Real Roots of Real Polynomials ... and now For Real! In ISSAC '16 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation , 7. Waterloo, Canada. doi:10.1145/2930889.2930937

    2016

  9. [17]

    and Rouillier, F

    Lazard, D. and Rouillier, F. (2007). Solving parametric polynomial systems . Journal of Symbolic Computation , 42(6), 636--667. doi:10.1016/j.jsc.2007.01.007

    2007 doi

  10. [18]

    and Hou, X

    Li, X. and Hou, X. (2021). A new approach based on the discriminant system of polynomial for robust stability and stabilization of two-dimensional systems. Journal of the Franklin Institute, 358(5), 2597--2622. doi:10.1016/j.jfranklin.2021.01.002

    2021 doi

  11. [19]

    Lipniacki, T., Paszek, P., Brasier, A.R., Luxon, B., and Kimmel, M. (2004). Mathematical model of nf-kb regulatory module. Journal of Theoretical Biology, 228(2), 195--215. doi:10.1016/j.jtbi.2004.01.001

    2004 doi

  12. [20]

    Quadrat, A., Rouillier, F., and Younes, G. (2024). l_ -norm computation for linear time-invariant systems depending on parameters. Maple Transactions, 4(1). doi:10.5206/mt.v4i1.17129

    2024 doi

  13. [21]

    Revol, N. (2001). Interval Newton Iteration in Multiple Precision for the Univariate Case . Research Report RR-4334, INRIA . ://inria.hal.science/inria-00072253

    2001

  14. [22]

    and Rouillier, F

    Revol, N. and Rouillier, F. (2005). Motivations for an arbitrary precision interval arithmetic and the MPFI library . Reliable Computing , 11(4), 275--290. doi:10.1007/s11155-005-6891-y

    2005 doi

  15. [23]

    Rouillier, F. (1999). Solving Zero-Dimensional Systems through the Rational Univariate Representation . Applicable Algebra in Engineering, Communication and Computing , 9(5), 433--461. doi:10.1007/s002000050114. Article dans revue scientifique avec comit \'e de lecture

    1999 doi

  16. [24]

    and Zimmermann, P

    Rouillier, F. and Zimmermann, P. (2004). Efficient isolation of polynomial's real roots . Journal of Computational and Applied Mathematics , 162(1), 33--50. doi:10.1016/j.cam.2003.08.015. Article dans revue scientifique avec comit \'e de lecture. internationale

    2004 doi

  17. [25]

    Zames, G. (1981). Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control, 26(2), 301--320. doi:10.1109/TAC.1981.1102603

    1981

  18. [26]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence afte...

  19. [27]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.