Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models and Piecewise Quadratic Lyapunov Functions

Artun Sel; Erdi Sayar; Mehmet Koruturk

REVIEW 3 major objections 4 minor 39 references

Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models and Piecewise Quadratic Lyapunov Functions

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that combining several coordinate-transformed Takagi-Sugeno models, each certified by a piecewise quadratic Lyapunov function, enlarges the proven region of attraction for nonlinear systems.

desk verdict The multi-transformation idea is plausible, but the paper's only numerical example has a fatal modeling error: the listed A matrices don't contain the actual transformed dynamics, so the enlarged ROA isn't certified. read the letter →

arxiv 2507.12718 v1 pith:VATMHLSF submitted 2025-07-17 math.DS cs.SYeess.SY

Artun Sel , Mehmet Koruturk , Erdi Sayar This is my paper · ORCID

classification math.DScs.SYeess.SY MSC 93D3093C1093D0590C22

keywords regionofattractionTakagi-SugenomodelscoordinatetransformationspiecewisequadraticLyapunovfunctionslinearmatrixinequalitiesnonlinearstabilityconvexoptimization

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 tries to establish that region-of-attraction (ROA) estimates for nonlinear systems can be enlarged by looking at the same dynamics in several coordinate systems instead of one. For each transformation $\bar{x} = T x$, the method builds a Takagi-Sugeno (TS) convex model of the transformed dynamics, certifies a ROA with a piecewise quadratic Lyapunov function $V(\bar{x}) = \max_{j} \bar{x}^{\top} P_j \bar{x}$, maps that region back to the original $x$-domain, and takes the union. A numerical example with the shear transformation $t_{11}=1$, $t_{12}=2$, $t_{21}=0$, $t_{22}=1$ is presented in which the back-mapped region extends beyond the ROA computed without transformation. The paper argues that this multi-transformation strategy reduces the conservatism of single-path TS analysis, making the choice of coordinates part of the search for a larger certified region.

What carries the argument

A TS convex model is a representation $\dot{x} = \sum_{i=1}^{r} h_i(z) A_i x$ in which the membership functions $h_i$ satisfy the convex sum property on a compact set. The paper combines this with the piecewise quadratic Lyapunov function $V(x) = \max_{j\in\{1,2\}} x^{\top} P_j x$ and the LMI feasibility conditions in (13), then applies a coordinate transformation $\bar{x} = T x$ to generate a second TS model. The optimization $\max k$ subject to $\{\bar{x} : V(\bar{x}) \le k\} \subseteq \bar{D}$ returns the certified sublevel set, and mapping back through $T^{-1}$ is what produces the enlarged region.

What would settle it

Evaluate the true transformed dynamics at $\bar{x}_2 = 0.55$: the $(2,2)$ entry of the state-dependent matrix equals $\bar{x}_2^2 - 1 = -0.6975$, whereas every listed $A_i$ has $(2,2)$ entry at most $-0.8336$; since a convex combination of numbers below $-0.8336$ cannot equal $-0.6975$, no membership functions satisfying the convex sum property can make the stated eight-vertex TS model reproduce the transformed dynamics on the claimed box.

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Extended reading notes

Core claim

The central claim is that coordinate transformations are more than a change of variables: they generate genuinely different TS representations whose ROA certificates complement one another. For the two-state example, applying $T$ with $t_{11}=1$, $t_{12}=2$, $t_{21}=0$, $t_{22}=1$ gives a transformed system on the box $[-0.55,0.55]^2$; the piecewise quadratic Lyapunov function with the printed $P_1, P_2$ yields the level set $V(\bar{x}) \le 1.54$. Mapping that set back to the original $x$-domain produces, per the paper, a region that extends beyond the ROA obtained by the untransformed TS method. The conclusion the authors draw is that systematic transformation selection should yield enlarged ROA estimates as a union of back-mapped level sets.

Load-bearing premise

The estimate collapses unless the transformed TS model is a true convex over-approximation of the transformed dynamics on the stated domain; the paper never gives the membership functions for the transformed system, and the printed $A_i$ matrices are numerically inconsistent with the dynamics on that domain.

Editorial extensions

If this is right

For polynomial-type autonomous systems, each added coordinate transformation yields a separate LMI feasibility problem, so the union of back-mapped sublevel sets is itself a valid inner approximation of the true ROA whenever the transformed TS model is valid.
ROA size becomes a searchable quantity: selecting transformation matrices and monomials can be automated, so the estimate is not fixed by the original coordinates.
Because piecewise quadratic Lyapunov functions are used, the method inherits lower conservatism than a single quadratic Lyapunov certificate for the same TS model.
The numerical example indicates that a single transformation can already extend the estimate beyond the conventional TS ROA, so the gap should widen as more transformations are united.

Reading between the lines

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

One extension left implicit is that the union over finitely many transformations could be iterated to generate a sequence of expanding inner approximations; convergence to the true ROA is not shown and would be a testable conjecture.
The method's success depends on the transformed TS model actually containing the transformed dynamics on the chosen domain; since the membership functions are never supplied, the reported eight-vertex model must be checked by evaluating the convex hull at sample states.
A practical extension would optimize the transformation matrix rather than hand-pick it, using LMI feasibility margin as the objective; the paper makes no claim about global optimality of the chosen $T$.

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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

I read the paper. The core idea — apply several coordinate transformations, build a TS model for each, compute a ROA, and take the union — is a reasonable extension of the standard TS pipeline, and the background sections on TS models and piecewise quadratic Lyapunov functions are correctly summarized. The transformation in Section IV is computed correctly, and the LMI framework is standard.

The problem is the demonstration. The paper promises N transformations but shows exactly one, with T and the domain Dbar hand-picked after the fact. The comparison in Fig. 2 is apples-to-oranges because the two ROAs are computed on different domains. But the fatal issue is that the eight A matrices listed for the transformed system are not a valid TS model for the actual dynamics on Dbar. The exact state-dependent matrix is A(xbar) = [[-xbar1+4*xbar2-2, 2*xbar2^2-4*xbar2],[0, xbar2^2-1]]. On Dbar = [-0.55,0.55]^2, a22 = xbar2^2-1 ranges from -1 to -0.6975, but all listed a22 are either -1.1664 or -0.8336; the true maximum is outside their convex hull. Similarly a11 reaches 0.75, while every listed a11 is at most -0.24. At xbar = (-0.55,0.55), the actual vector field is (-1.2898,-0.3836), but the convex hull of {A_i * xbar} gives a second component no larger than -0.4585. So the LMI certifies stability only for a different LDI, not for the transformed system. The membership functions are never given, so the convex-sum property cannot be checked. This is a load-bearing flaw: the claimed enlargement is an artifact, not a certified region.

To give credit where it is due, the theoretical idea would work if each transformed TS model were valid. The error is in the construction of the transformed model, not in the proof concept. But as it stands, the paper does not demonstrate its central claim.

This paper might be worth a quick look as a cautionary example of TS modeling pitfalls, but it does not deserve a full review in its current form. I would not cite it. If the authors can redo the example with correct vertex matrices and provide the membership functions, the idea could be worth revisiting.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes to enlarge Region of Attraction (ROA) estimates for nonlinear systems by applying coordinate transformations before Takagi-Sugeno (TS) modeling, computing piecewise-quadratic Lyapunov-function ROAs for each transformed representation, and then taking the union of the resulting sets. Section III walks through a standard TS sector-nonlinearity construction for a two-dimensional polynomial system and computes a sublevel-set ROA using a piecewise quadratic Lyapunov function. Section IV introduces a single coordinate transformation T = [[1,2],[0,1]], claims a transformed TS model on Dbar = [-0.55,0.55]^2 with eight vertex matrices, solves an LMI feasibility problem, and asserts that mapping the resulting invariant set back to the original x-domain yields an ROA larger than the Section III estimate. The central demonstration of the paper rests entirely on this transformed model and on the reported vertex matrices.

Significance. If the transformed TS model were valid and the union-of-transformations procedure were actually specified, the paper would illustrate a useful idea: coordinate changes can reduce conservatism in TS-based ROA estimation, and piecewise quadratic Lyapunov functions can be combined with such transformations. The Section III example is internally coherent and the LMI machinery is standard. However, the central numerical claim is not supported: the transformed vertex matrices are inconsistent with the actual transformed dynamics, the membership functions for the transformed model are never given, and the multi-transformation framework promised in the abstract is not implemented. These are load-bearing gaps, not presentation issues. The paper does not provide reproducible code or machine-checked verification of the LMI solutions, so the numerical results could not be independently confirmed.

major comments (3)

[Section IV, Eq. (27) and the list of A_i matrices] The transformed TS model is not a valid convex representation of system (27) on Dbar. For (27), the exact state-dependent system matrix is A(xbar) = [[-xbar1+4*xbar2-2, 2*xbar2^2-4*xbar2],[0, xbar2^2-1]]. On Dbar = [-0.55,0.55]^2, the (2,2) entry ranges over [-1,-0.6975], but every listed A_i has a22 in {-1.1664,-0.8336}, so the interval (-0.8336,-0.6975] is not contained in their convex hull. Likewise, the (1,1) entry reaches 0.75 at xbar = (-0.55,0.55), while every listed a11 is at most -0.24. A concrete check at xbar = (-0.55,0.55) gives f(xbar) = (-1.2898,-0.3836), but every convex combination of the products A_i*xbar has first component at least -0.7165, so no membership weights can reproduce the dynamics. Therefore the LMI solution in (13) certifies stability only for a different differential inclusion, not for (27), and the red boundary in Fig. 2b is not a certified ROA.
[Section IV, membership functions] The membership functions for the transformed TS model are never stated. The text only lists the premise variables z1 = xbar1, z2 = xbar2, z3 = xbar2^2 and then jumps to the eight vertex matrices. To apply the sector-nonlinearity construction one must specify the sector bounds and the associated weighting functions on Dbar and verify the convex-sum property (15). Without this information, even a consistent set of vertex matrices could not be checked for exact representation of (27). This is a completeness failure in the central construction, not a minor omission.
[Abstract and Section I] The abstract promises a systematic framework using transformation matrices T1, T2, ..., TN and a final ROA obtained as the union of all computed regions. The body provides only one hand-chosen matrix T and one hand-chosen domain Dbar; there is no algorithm for generating the T_i, no criterion for choosing the transformed domain, and no union operation is actually carried out. The concluding remark about 'iterative application' and 'systematic automation of monomial function selection' is not supported by any procedure or additional example. As a result, the paper's claimed contribution is not demonstrated even apart from the numerical inconsistencies in Section IV.

minor comments (4)

[Section II-B2, Definition 5] Definition 5 is informal: 'the corresponding region around the equilibrium point is called ROA.' The ROA should be defined as the set of initial conditions from which trajectories converge to the equilibrium, for example {x0 : lim_{t to infinity} phi(t;x0) = 0}.
[Section II-B6, Eq. (13)] In the two LMI constraints, the same symbol lambda_i appears in both blocks; please clarify whether the multipliers may differ between the two constraints and whether they are required to be nonnegative.
[Fig. 2 and caption] The caption of Fig. 2 does not clearly distinguish the red-shaded region from the mapped boundary, and the axes of panel (a) are not labeled consistently with the definition of Dbar in Eq. (28).
[Throughout] There are numerous typos and spacing artifacts, including 'V A' in the author affiliation line, 'F or' at the start of Theorem 1, and 'conser vatism' in Section I. A careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ROA estimates are computed from an LMI feasibility problem, not fitted to the claimed enlargement; the main concerns are soundness and selection, not circularity.

full rationale

The derivation chain is self-contained for the purposes of circularity: the transformed dynamics (27) are obtained from the stated transformation T, the TS vertex matrices are inputs to the LMI feasibility problem (13), and the values k=0.054 and k=1.54 are solutions of the optimization problem (26), not parameters fitted to reproduce the claimed enlarged ROA. The final result is a mapped level set of a Lyapunov function computed in the transformed coordinates, so no prediction is statistically forced by a fitted input. The choice of T and the domain Dbar is a heuristic selection issue, and a separate soundness concern exists because the reported A_i may not contain the actual transformed dynamics on Dbar, but that would be a modeling error rather than a circular argument. The self-citations in the introduction, such as [18]–[20] and [39], are background or algorithmic references and are not load-bearing for the core ROA enlargement claim. No circular step can be exhibited by reducing any equation of the paper to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central argument rests on standard LMI and Lyapunov theory, on the exactness of the TS convex representation over chosen compact domains, and on the validity of coordinate changes and unions of invariant sets. The TS membership functions for the transformed system are never given, the vertex matrices are not verified, and the transformation and domain are chosen ad hoc, so the example carries several unstated assumptions.

free parameters (2)

Coordinate transformation matrix T = T = [[1, 2], [0, 1]]
The transformation is chosen by hand for the example; the paper provides no systematic procedure for selecting T, and the claimed ROA enlargement depends on this choice.
Transformed domain bounds = xbar1 and xbar2 in the interval from -0.55 to 0.55
The domain bounding the TS model is chosen by hand; the apparent ROA improvement over Section III may be an artifact of using a different domain rather than the transformation itself.

assumptions (4)

standard math Lyapunov stability theorem and LMI feasibility for piecewise quadratic Lyapunov functions (equation 13) are sufficient for ROA estimation.
Used without derivation in Sections III and IV; the condition is a known sufficient condition from the max-min Lyapunov literature but is stated here without proof or explicit nonnegativity constraints on the lambda terms.
domain assumption The constructed TS models exactly represent the nonlinear dynamics on the chosen compact domains, with membership functions satisfying the convex sum property.
Invoked in Remark 2 and used for both D and the transformed domain; for the transformed system the membership functions are never given and the vertex matrices appear inconsistent with the stated domain.
standard math An invertible linear coordinate transformation maps ROA subsets bijectively, and the union of valid ROA subsets is itself a valid ROA subset.
Relied on to justify the union and mapping in Section IV; true for diffeomorphisms but never stated explicitly.
ad hoc to paper The transformation T equal to the matrix with rows [1,2] and [0,1], and the transformed domain, are chosen by hand as the method's inputs.
The paper gives no algorithm for selecting these; the claimed enlargement may depend on them.

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Pith. "Pith review of Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models and Piecewise Quadratic Lyapunov Functions." pith.science (2026). https://pith.science/paper/VATMHLSF

@misc{pith2026250712718,
  author       = {Pith},
  title        = {Pith review of: Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models and Piecewise Quadratic Lyapunov Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VATMHLSF}},
  note         = {Machine review of arXiv:2507.12718}
}

abstract

This paper presents a novel approach for computing enlarged Region of Attractions (ROA) for nonlinear dynamical systems through the integration of multiple coordinate transformations and piecewise quadratic Lyapunov functions within the Takagi-Sugeno (TS) modeling framework. While existing methods typically follow a single-path approach of original system $\rightarrow$ TS model $\rightarrow$ ROA computation, the proposed methodology systematically applies a sequence of coordinate transformations to generate multiple system representations, each yielding distinct ROA estimations. Specifically, the approach transforms the original nonlinear system using transformation matrices $T_1, T_2, \ldots, T_N$ to obtain $N$ different coordinate representations, constructs corresponding TS models for each transformed system, and computes individual ROAs using piecewise quadratic Lyapunov functions. The final ROA estimate is obtained as the union of all computed regions, leveraging the flexibility inherent in piecewise quadratic Lyapunov functions compared to traditional quadratic approaches. The enhanced methodology demonstrates significant improvements in ROA size estimation compared to conventional single-transformation techniques, as evidenced through comparative analysis with existing TS-based stability methods.

Figures

Figures reproduced from arXiv: 2507.12718 by the authors.

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