Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum

Guillaume Ducard; Lorenzo Fici

REVIEW 2 major objections 6 minor 11 references

Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum

T0 review · 2 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Backward-Reaching the Region of Attraction

desk verdict Solid, practical RoA enlargement for a TWIP — the predecessor expansion is sound, the hardware validation is thin but genuine. read the letter →

arxiv 2607.07231 v1 pith:42AL7JSD submitted 2026-07-08 eess.SY cs.SY

Lorenzo Fici , Guillaume Ducard This is my paper · ORCID

classification eess.SYcs.SY

keywords certifiedapproximationguaranteesinvariantlyapunov-basednonlinearpredecessor-basedregion

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 targets a basic tension in nonlinear control: Lyapunov-based methods can certify that a set of initial conditions will converge to a stable equilibrium, but the certified set is typically far smaller than the set of states from which the system actually recovers. The authors propose a two-stage procedure to close this gap. First, they compute a conventional Lyapunov-based positively invariant set for a two-wheeled inverted pendulum stabilized by a saturated LQR controller. Second, they iteratively compute backward reachable sets (predecessors) of this initial set under the exact nonlinear closed-loop dynamics. Because the initial set is invariant, any state that maps into it after N steps is also guaranteed to converge to the equilibrium. The key technical move is to represent these predecessor sets implicitly through a scalar membership function rather than constructing their boundaries explicitly, which would be analytically intractable for nonlinear dynamics. The result is a substantially enlarged certified inner approximation of the region of attraction that, on simulation and hardware experiments, closely tracks the empirically observed boundary between recovering and non-recovering initial conditions.

What carries the argument

The central mechanism is the recursive predecessor operator. Given an initial invariant set V0 defined by a Lyapunov sublevel condition v0(x) <= 1, the N-step predecessor set VN consists of all states x for which the N-step nonlinear dynamics map f^N(x) lands back inside V0. Membership in VN is tested by evaluating the implicit function vN(x) = v0(f^N(x)) and checking whether it is at most 1. Because V0 is positively invariant, the nested sequence V0 subset V1 subset ... subset VN is guaranteed to lie within the true region of attraction, and the union over all N converges to it.

What would settle it

Find a state x for which the implicit membership test vN(x) <= 1 holds (certified as inside the RoA) but the nonlinear closed-loop trajectory starting from x does not converge to the upright equilibrium. This would break the chain of invariance inheritance from V0 to VN.

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Extended reading notes

Core claim

By composing a Lyapunov sublevel-set membership test with N iterations of the nonlinear closed-loop dynamics, the paper defines an implicit function whose sublevel set is a certified subset of the region of attraction. This implicit predecessor representation avoids explicit geometric construction of the expanded set while preserving the formal convergence guarantee inherited from the initial Lyapunov-certified invariant set. Applied to a saturated-LQR-stabilized inverted pendulum, the certified region grows from a small conservative neighborhood to a boundary that matches Monte Carlo and hardware observations.

Load-bearing premise

The entire predecessor expansion rests on the initial Lyapunov-based set being truly positively invariant under the exact nonlinear closed-loop dynamics. This invariance depends on a local bound on the nonlinear remainder terms being correct and on the saturation model faithfully capturing the actuator limits. If that bound is wrong or the saturation model is inaccurate, the initial set is not actually invariant, and every predecessor set built on top of it loses its formalgu

Editorial extensions

If this is right

The implicit predecessor representation can be applied to other underactuated or saturated nonlinear systems where Lyapunov certificates exist but are overly conservative, provided the closed-loop dynamics can be iterated numerically.
The robust positive invariance analysis, which yields a disturbance-dependent sixth-degree polynomial condition on the local radius, offers a template for certifying robustness margins of Lyapunov sets under actuator disturbances.
Hardware experiments show the implicit membership test correctly predicts recovery versus failure on a physical robot, suggesting the method could serve as a real-time safety monitor: evaluate vN(x) for the current disturbed state and flag whether the system remains within the certified recovery region.
The gap between the certified predecessor-expanded boundary and the true empirical boundary visible in Monte Carlo samples quantifies how much additional conservatism remains, guiding future work on tighter initial Lyapunov functions or alternative set representations.

Reading between the lines

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

If the implicit membership function vN(x) can be evaluated in real time, it could function as a runtime safety certificate for robotic systems: a supervisor could check whether the current state lies inside the certified region and trigger fallback behaviors when it does not.
The predecessor expansion is limited by the prediction horizon N and computational cost of iterating nonlinear dynamics; for high-dimensional systems, the implicit membership test may become expensive, suggesting the method is most practical for moderate-dimensional systems like the 4-state pendulum studied here.
The robust positive invariance analysis reveals a disturbance-dominated inner radius below which convergence cannot be certified, implying that for sufficiently large disturbances the equilibrium itself is practically destabilized even if the nominal controller is stable.

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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

This is a competent, clearly written paper that combines a standard quadratic Lyapunov invariant set with an implicit predecessor expansion to enlarge the certified region of attraction for a two-wheeled inverted pendulum under saturated LQR. The core idea is not new — predecessor sets and implicit representations both appear in prior work (Balint et al., Serry and Liu) — but the specific combination, applied to a real nonlinear robotic system with actuator saturation baked into the closed-loop dynamics, is a useful and honest contribution. The math is internally consistent. The Lyapunov decrease condition is properly derived from a local Taylor bound, the invariant set is correctly chosen as the largest sublevel set within the validity ball, and the predecessor recursion logically guarantees convergence for any state whose N-step forward image lands in V0. The authors are transparent that the implicit set has no closed-form boundary and that the plotted contour is numerical approximation only for visualization — the membership test itself is exact. That scoping is clean. The Monte Carlo validation (5000 samples) looks good but is not independent confirmation: it uses the same model f used to compute the predecessor sets, so agreement is expected. The two hardware experiments are the real out-of-model evidence, and they do work — one state inside the certified region recovers, one outside fails. But two data points is thin. The RPI analysis is the weakest part. It is scoped only to V0, not the expanded region, and the authors acknowledge the resulting certificates are conservative relative to practice. The sixth-degree polynomial condition is derived but not really stress-tested — no numerical examples of disturbance bounds are given, and the gap between certified and actual robustness is left unquantified. The free parameters (LQR weights, rho, N=1500) are reasonable but somewhat ad hoc; no sensitivity analysis is provided. These are minor-to-moderate concerns. The central claim — that you can take a cheap quadratic Lyapunov set and enlarge it substantially via implicit predecessor expansion while keeping formal guarantees — holds up. This is a paper for control engineers working on certified stability of underactuated robots, or anyone wanting a practical recipe for less conservative RoA approximation without resorting to expensive Lyapunov function search. It deserves a serious referee who can check the algebra carefully and push on the RPI section.

Referee Report

2 major / 6 minor

Summary. The paper proposes combining a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to enlarge the certified inner approximation of the region of attraction (RoA) for a nonlinear two-wheeled inverted pendulum (TWIP) stabilized by a saturated LQR. The Lyapunov-certified set V0 is computed via a local Taylor bound on the nonlinear remainder, and the predecessor expansion VN = {x : v0(f^N(x)) <= 1} is represented implicitly, enabling exact membership tests without explicit set construction. A robust positive invariance (RPI) analysis of V0 under bounded additive input disturbances is also provided. The approach is validated via Monte Carlo simulations (5000 samples) and two hardware experiments. The mathematical framework is internally consistent, the predecessor expansion logic is sound, and the formal guarantees are correctly scoped. The paper is a solid contribution to the practical certification of nonlinear control systems.

Significance. The paper addresses a well-known limitation of Lyapunov-based RoA estimation: conservatism. By combining a simple quadratic Lyapunov function with predecessor-based implicit expansion, the authors obtain a significantly larger certified region while preserving formal guarantees. The implicit membership representation is a practical strength, as it avoids the need to explicitly construct predecessor sets, which is analytically intractable for nonlinear dynamics. The inclusion of RPI analysis and hardware validation adds practical value. The methodology is demonstrated on a representative underactuated nonlinear system (TWIP), making the results relevant to the broader robotics and control community.

major comments (2)

Section 4.1 (Monte Carlo Validation): The Monte Carlo simulation uses the same nonlinear closed-loop model f(x) that defines the predecessor expansion. Therefore, agreement between the simulated stable/unstable boundary and the predecessor contour VN(x) = 1 is expected by construction and does not constitute independent validation of the certified region. The paper should explicitly acknowledge this: the Monte Carlo results confirm that the numerical contour extraction is consistent with the model, but they do not independently verify the correctness of the Lyapunov decrease condition or the invariance of V0. The two hardware experiments (Section 4.2) provide genuine out-of-model validation, but only two data points are insufficient to claim that the approximation 'matches the empirical closed-loop behavior' broadly. This claim should be tempered accordingly.
Section 3.3, Eq. (20): The RPI analysis introduces the disturbance model x_{k+1} = A_cl x_k + g(x_k) + B_w w_k + h(x_k, w_k), but the terms B_w and h(x_k, w_k) are not explicitly defined. The paper states that B_w w_k is the 'linear contribution of the disturbance' and h collects 'higher-order nonlinear interaction terms,' but it is unclear how these terms arise from the additive input disturbance model. If the disturbance enters at the control input (as stated), the linear contribution should be B_d w_k (or B_d times the disturbance gain), and the nonlinear interaction terms should be specified. Without this, the sixth-degree polynomial p(rho, w_bar) in Eq. (21) cannot be independently verified. Please provide the explicit derivation of B_w and h(x_k, w_k), and if possible, include the expression for p(rho, w_bar) or its coefficients.

minor comments (6)

Section 3.1, Eq. (7): The notation |P A_cl| is used but not defined. It appears to denote a matrix norm, but the specific norm (e.g., induced 2-norm, Frobenius norm) should be specified, as the bound on gamma depends on it.
Section 2.1, Eq. (1): The dead-zone compensation term d_b u_0(u_c) is described verbally but the functional form of u_0(u_c) is not given. A brief explicit definition would aid reproducibility.
Figure 2 and Figure 3: The axis labels use notation like '3 (rad)' and '_xw (m/s)', which appear to be rendering artifacts (likely theta and x_dot_w). Please correct the axis labels for readability.
Section 3.2, Eq. (14): The claim that the union of predecessor sets converges to the RoA is attributed to backward reachable set theory (Serry and Liu, 2025). A brief justification or reference to the specific theorem would strengthen this statement.
Section 4.2: The two hardware experiments use velocity references of -0.5 m/s and -1.0 m/s. It would be useful to clarify whether the RoA analysis is performed around the upright equilibrium (zero velocity) or around the tracking equilibrium, and how the reference shift is handled in the predecessor function evaluation.
The reference 'Fici et al., 2026' (arXiv:2604.04455) is cited for the Lyapunov construction method. If this is a companion or prior work by the same authors, the relationship should be clearly stated to distinguish the novel contribution of the present paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the predecessor expansion is a standard backward-reachability construction, the Lyapunov set is certified via an independent Taylor-bound argument, and the one self-citation is non-load-bearing.

full rationale

The paper's central derivation is self-contained. The Lyapunov-based invariant set V0 is certified using a standard discrete-time Lyapunov equation and a local Taylor bound on the nonlinear remainder g(x), following Theorem 4.7 in Khalil (2002) — an external, standard reference. The predecessor expansion VN = {x : v0(f^N(x)) ≤ 1} is a standard backward-reachability construction: any state whose N-step forward image lands in the invariant set V0 is guaranteed to converge, since V0 is positively invariant. This is not circular — it is a correct logical implication from a certified invariant set, not a definition of the RoA in terms of itself. The Monte Carlo validation uses the same model f, so its agreement with the predecessor boundary is expected (as the paper itself acknowledges by calling it 'empirical reference'), but this is not circularity — it is consistency checking, and the two hardware experiments provide genuine out-of-model validation. The one self-citation (Fici et al., 2026) is for the Lyapunov construction procedure, which is itself based on the standard Khalil theorem and is not load-bearing for the predecessor expansion or the RPI analysis. No step in the derivation chain reduces to its own inputs by construction, and no 'prediction' is a renamed fit. The formal guarantees are correctly scoped: the Lyapunov decrease condition is independently verified, and the predecessor expansion preserves (rather than assumes) convergence guarantees. Score: 0 — no significant circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper does not invent new entities. It uses standard control theory constructs (Lyapunov functions, predecessor sets) and applies them to a physical system.

free parameters (3)

LQR weights Q and R = See Eq. 4
Selected following Bryson's rule and adjusted to prioritize selected states. These are design choices, not fitted to data.
Local validity radius rho = 2.724e-2
Computed from the local bound on the nonlinear remainder, not fitted to experimental data.
Prediction horizon N = 1500
Chosen to correspond to a 15s simulation horizon. This is a computational parameter.

assumptions (3)

standard math The nonlinear closed-loop dynamics can be rewritten as x_{k+1} = A_cl x_k + g(x_k) where g(x_k) is locally bounded by |g(x_k)| < gamma |x_k|.
This is a standard application of Taylor's theorem for nonlinear systems linearized around an equilibrium.
domain assumption The initial Lyapunov-based set V0 is positively invariant under the nominal closed-loop dynamics.
This is proven in the paper based on the Lyapunov decrease condition, but it relies on the accuracy of the nonlinear model and the local bound gamma.
domain assumption The DC motor dead-zone is adequately compensated by a constant offset u0.
This is a modeling assumption that simplifies the control input. If the dead-zone is not well-compensated, the closed-loop dynamics f(x) used for predecessor expansion will not match reality.

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Cite this review

Pith. "Pith review of Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum." pith.science (2026). https://pith.science/paper/42AL7JSD

@misc{pith2026260707231,
  author       = {Pith},
  title        = {Pith review of: Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42AL7JSD}},
  note         = {Machine review of arXiv:2607.07231}
}

read the original abstract

Estimating the region of attraction (RoA) of nonlinear systems is fundamental for assessing closed-loop stability and ensuring safe operation. While Lyapunov-based approaches provide certified stability guarantees, they often yield conservative inner approximations of the RoA. This paper combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to compute a significantly less conservative inner approximation of the RoA while preserving formal stability guarantees. In addition, the robust positive invariance of the initial certified Lyapunov-based invariant set is analyzed under bounded additive input disturbances, providing formal robustness guarantees. The proposed methodology is demonstrated on a nonlinear two-wheeled inverted pendulum stabilized by a saturated linear quadratic regulator. The resulting RoA approximation is compared with the initial Lyapunov-certified invariant set and validated through Monte Carlo simulations and hardware experiments, showing a substantially enlarged certified operating region that matches the empirical closed-loop behavior. These results demonstrate the practical applicability of combining certified Lyapunov analysis with predecessor-based set propagation for RoA approximation and robustness assessment of nonlinear systems.

Figures

Figures reproduced from arXiv: 2607.07231 by the authors.

**Figure 1.** Schematic of the TWIP showing the coordinate frames and physical parameters. The corresponding model parameters are listed in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Certified Lyapunov-based invariant set. a prediction horizon of N = 1500 sampling instants, corresponding to a simulation horizon of 15 s. Since the predecessor-expanded set is defined implicitly through the condition VN(x) ≤ 1, its boundary does not admit an explicit analytical representation. Therefore, the contour shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Implicit predecessor-expanded RoA. The contour obtained is an approximation. position and the angular velocity to xw = 0, θ˙ = 0 for visualization purposes. Each initial condition was propagated through the nonlinear closed-loop dynamics under the saturated LQR controller over a simulation horizon of 15 s, matching the time horizon used to compute the predecessor expansion. Each trajectory was classified according … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

M., and Grigis, A

Balint, S., Kaslik, E., Balint, A. M., and Grigis, A. (2006). Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems. Advances in Difference Equations, 2006:1–16

work page 2006
[2]

Fici, L., Wahby, D., Detailleur, A., Barreau, M., and Ducard, G. (2026). Region of Attrac- tion Estimation for Linear Quadratic Regu- lator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendu- lum. arXiv:2604.04455

work page Pith review arXiv 2026
[3]

Gross, L., Maywald, L., Kumar, S., Kirchner, F., and Lüth, C. (2022). Analytic Estimation of Region of Attraction of an LQR Controller for Torque Limited Simple Pendulum. pages 2695–2701. arXiv:2211.15437 [cs]

work page Pith review arXiv 2022
[4]

Han, D., Rizaldi, A., El-Guindy, A., and Althoff, M. (2016). On enlarging backward reach- able sets via Zonotopic set membership. In 2016 IEEE International Symposium on In- telligent Control (ISIC), pages 1–8, Buenos Aires. IEEE

work page 2016
[5]

Horibe, T., Zhou, B., Hara, S., and Tsubakino, D. (2018). Quantitative measure for non- linear unstable systems based on the region of attraction and its application to designing parameter optimization – inverted pendulum example. Advanced Robotics, 32(8):399–410. Joao P. Hespanha (2010). Linear System Theory. Lecture notes, University of California

work page 2018
[6]

Khalil, H. K. (2002). Nonlinear systems. Pren- tice Hall, Upper Saddle River, N.J, 3rd ed edition

work page 2002
[7]

Underapproximating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Representations of Backward Reachable Sets

Serry, M. and Liu, J. (2025). Underapprox- imating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Rep- resentations of Backward Reachable Sets. arXiv:2409.10657 [math.OC]

work page Pith review arXiv 2025
[8]

and Collado, J

Tellez, J. and Collado, J. (2013). Estimate of the region of attraction of a Kapitza pendu- lum subject to an harmonic excitation and analysis of the pendulum behavior for a non harmonic excitation. In 2013 10th Interna- tional Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), pages 87–91, Mexico City, Mexico. IEEE

work page 2013

Show all 11 references

[9]

and Anderson, J

Valmorbida, G. and Anderson, J. (2017). Region of attraction estimation using invariant sets and rational Lyapunov functions. Automat- ica, 75:37–45

2017
[10]

and Vidyasagar, M

Vannelli, A. and Vidyasagar, M. (1985). Maximal Lyapunov Functions and Domains of Attrac- tion for Autonomous Nonlinear Systems. Au- tomatica

1985
[11]

and Mohamad Nor, N

Zhang, H. and Mohamad Nor, N. (2025). Control Strategies for Two-Wheeled Self-Balancing Robotic Systems: A Comprehensive Review. Robotics, 14(8):101

2025

Pith tools

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

[1] [1]

M., and Grigis, A

Balint, S., Kaslik, E., Balint, A. M., and Grigis, A. (2006). Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems. Advances in Difference Equations, 2006:1–16

work page 2006

[2] [2]

Fici, L., Wahby, D., Detailleur, A., Barreau, M., and Ducard, G. (2026). Region of Attrac- tion Estimation for Linear Quadratic Regu- lator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendu- lum. arXiv:2604.04455

work page Pith review arXiv 2026

[3] [3]

Gross, L., Maywald, L., Kumar, S., Kirchner, F., and Lüth, C. (2022). Analytic Estimation of Region of Attraction of an LQR Controller for Torque Limited Simple Pendulum. pages 2695–2701. arXiv:2211.15437 [cs]

work page Pith review arXiv 2022

[4] [4]

Han, D., Rizaldi, A., El-Guindy, A., and Althoff, M. (2016). On enlarging backward reach- able sets via Zonotopic set membership. In 2016 IEEE International Symposium on In- telligent Control (ISIC), pages 1–8, Buenos Aires. IEEE

work page 2016

[5] [5]

Horibe, T., Zhou, B., Hara, S., and Tsubakino, D. (2018). Quantitative measure for non- linear unstable systems based on the region of attraction and its application to designing parameter optimization – inverted pendulum example. Advanced Robotics, 32(8):399–410. Joao P. Hespanha (2010). Linear System Theory. Lecture notes, University of California

work page 2018

[6] [6]

Khalil, H. K. (2002). Nonlinear systems. Pren- tice Hall, Upper Saddle River, N.J, 3rd ed edition

work page 2002

[7] [7]

Underapproximating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Representations of Backward Reachable Sets

Serry, M. and Liu, J. (2025). Underapprox- imating Safe Domains of Attraction for Discrete-Time Systems Using Implicit Rep- resentations of Backward Reachable Sets. arXiv:2409.10657 [math.OC]

work page Pith review arXiv 2025

[8] [8]

and Collado, J

Tellez, J. and Collado, J. (2013). Estimate of the region of attraction of a Kapitza pendu- lum subject to an harmonic excitation and analysis of the pendulum behavior for a non harmonic excitation. In 2013 10th Interna- tional Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), pages 87–91, Mexico City, Mexico. IEEE

work page 2013