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Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum

T0 review · 1 major / 0 minor · reviewed 2026-05-21 · grok-4.3

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Pith's one-line read A Lyapunov invariant set plus Monte Carlo sampling estimates the region of attraction for LQR and MPC on a two-wheeled inverted pendulum.

desk verdict The paper pairs a Lyapunov invariant set with Monte Carlo sampling to estimate the RoA for saturated LQR, linear MPC, and CTMPC on a TWIP, but the Monte Carlo implementation details are missing. read the letter →

arxiv 2604.04455 v2 pith:56DPYZDW submitted 2026-04-06 eess.SY cs.SY

Lorenzo Fici , Dalim Wahby , Alvaro Detailleur , Matthieu Barreau , Guillaume Ducard This is my paper · ORCID
classification eess.SYcs.SY
keywords regionofattractiontwo-wheeledinvertedpendulumlinearquadraticregulatormodelpredictivecontrolLyapunovinvariantsetMonteCarloestimationunderactuatedsystems
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 studies the set of initial conditions from which a two-wheeled inverted pendulum returns to upright under three controllers. It first builds a Lyapunov-based invariant set that mathematically guarantees convergence for a conservative inner region. Because this bound is too small for practical use, the authors then run many random simulations to map where the system still succeeds in practice. The combination supplies both a formally safe core and a broader empirical picture of controller behavior on this nonlinear underactuated plant.

What carries the argument

Lyapunov-based invariant set that certifies an inner bound on converging states, paired with Monte Carlo sampling that empirically extends the region of attraction estimate.

What would settle it

A single simulation trajectory that begins inside the Monte Carlo-estimated region yet diverges from the upright equilibrium would show the empirical map is incomplete.

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Extended reading notes

Core claim

The authors derive a Lyapunov-based invariant set that certifies an inner approximation of the region of attraction for saturated linear quadratic regulator, linear model predictive control, and constraint-tightening model predictive control applied to a two-wheeled inverted pendulum. They then apply Monte Carlo sampling to characterize how each controller performs outside this guaranteed set, yielding a data-driven extension of the analytically certified bound.

Load-bearing premise

The Monte Carlo sampling procedure produces a representative approximation of the true region of attraction beyond the analytically guaranteed set.

Editorial extensions

If this is right

  • Initial conditions inside the Lyapunov set are guaranteed to converge for all three controllers without needing further simulation.
  • The Monte Carlo map reveals performance gaps between saturated LQR, linear MPC, and constraint-tightening MPC outside the certified core.
  • Engineers obtain both a safe operating region and a realistic performance envelope from one combined procedure.
  • The method applies directly to other nonlinear underactuated systems whose regions of attraction lack closed-form descriptions.

Reading between the lines

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

  • The same two-step procedure could be used to tune controller parameters so the empirical region of attraction grows while the certified inner set remains intact.
  • Adaptive or variance-reduced sampling might reduce the number of simulations needed to reach a stable empirical boundary.
  • The empirical outer boundary could serve as a target for future controller redesigns that enlarge the guaranteed region.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript presents a methodology for estimating the region of attraction (RoA) for a two-wheeled inverted pendulum (TWIP) stabilized by saturated LQR, linear MPC, and constraint-tightening MPC (CTMPC). It first derives a Lyapunov-based invariant set providing a certified inner bound on the RoA, then applies a Monte Carlo sampling procedure to generate an empirical approximation that extends beyond this conservative analytical bound.

Significance. If the Monte Carlo procedure is rigorously specified and validated, the work offers a practical combination of formal Lyapunov guarantees with data-driven estimation for assessing controller performance on underactuated nonlinear systems. This addresses the known conservativeness of purely analytical RoA bounds while providing empirical insight, and the paper correctly credits the separation between the certified inner set (via standard Lyapunov theory) and the independent sampling-based outer estimate.

major comments (1)
  1. [Abstract and Monte Carlo-based estimation procedure] Abstract and Monte Carlo-based estimation procedure: The description provides no quantification of sampling density, number of samples, random sampling strategy in the 4D state space, numerical integration tolerances, termination criteria for classifying convergence, or error bounds on the estimated volume/shape. Without these, it cannot be verified that the procedure produces a meaningfully larger and accurate empirical RoA beyond the Lyapunov invariant set, leaving the central claim of a 'more representative approximation' untested rather than demonstrated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and the positive assessment of the overall approach combining Lyapunov-based guarantees with Monte Carlo sampling. We address the single major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Abstract and Monte Carlo-based estimation procedure] Abstract and Monte Carlo-based estimation procedure: The description provides no quantification of sampling density, number of samples, random sampling strategy in the 4D state space, numerical integration tolerances, termination criteria for classifying convergence, or error bounds on the estimated volume/shape. Without these, it cannot be verified that the procedure produces a meaningfully larger and accurate empirical RoA beyond the Lyapunov invariant set, leaving the central claim of a 'more representative approximation' untested rather than demonstrated.

    Authors: We agree that the current manuscript description of the Monte Carlo procedure is insufficiently detailed for full reproducibility and verification. In the revised version we will add a dedicated subsection (and update the abstract) that specifies: total number of samples (10,000), sampling strategy (uniform random sampling over a 4D hyper-rectangle whose bounds are chosen from open-loop divergence analysis and preliminary closed-loop trials), sampling density (approximately 2.5 samples per unit volume in the normalized state space), numerical integration (ode45 with RelTol = 1e-6 and AbsTol = 1e-8), termination criteria (simulation horizon of 15 s or until the Euclidean norm of the state remains below 0.05 for at least 2 s), and statistical error bounds (bootstrap resampling with 1,000 replicates to report 95 % confidence intervals on the estimated volume). These additions will allow readers to confirm that the empirical set meaningfully extends the certified Lyapunov inner approximation while remaining statistically reliable. We will also include a short pseudocode listing of the procedure. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; derivation is self-contained

full rationale

The paper derives a Lyapunov-based invariant set as a certified inner approximation of the RoA using standard stability theory for the closed-loop system under LQR/MPC/CTMPC, then applies an independent Monte Carlo sampling procedure (initial-state sampling, forward simulation, and convergence classification) to characterize a larger empirical region. Neither step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the analytical bound follows directly from Lyapunov function properties without re-using the target RoA estimate, and the Monte Carlo step is a separate numerical approximation whose validity rests on external sampling theory rather than the paper's own outputs. The methodology therefore remains non-circular and externally benchmarkable against classical control results.

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

The approach rests on the existence of a valid Lyapunov function for the closed-loop system and on the assumption that finite Monte Carlo trials adequately sample the state space outside the invariant set.

assumptions (2)
  • domain assumption A quadratic Lyapunov function exists that certifies an invariant set for the closed-loop TWIP dynamics under each controller.
    Invoked when deriving the certified inner approximation of the RoA.
  • domain assumption The nonlinear TWIP dynamics are known and can be accurately simulated for Monte Carlo trials.
    Required for both controller design and the empirical RoA estimation step.

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Cite this review

Pith. "Pith review of Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum." pith.science (2026). https://pith.science/paper/56DPYZDW

@misc{pith2026260404455,
  author       = {Pith},
  title        = {Pith review of: Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56DPYZDW}},
  note         = {Machine review of arXiv:2604.04455}
}
read the original abstract

Nonlinear underactuated systems such as two-wheeled inverted pendulums (TWIPs) exhibit a limited region of attraction (RoA), which defines the set of initial conditions from which the closed-loop system converges to the equilibrium. The RoA of nonlinear and constrained systems is generally nonconvex and analytically intractable, requiring numerical or approximate estimation methods. This work investigates the estimation of the RoA for a TWIP stabilized under three model-based control strategies: saturated linear quadratic regulator (LQR), linear model predictive control (MPC), and constraint tightening MPC (CTMPC). We first derive a Lyapunov-based invariant set that provides a certified inner approximation of the RoA. Since this analytical bound is highly conservative, a Monte Carlo-based estimation procedure is then employed to obtain a more representative approximation of the RoA, capturing how the controllers behave beyond the analytically guaranteed region. The proposed methodology combines analytical guarantees with data-driven estimation, providing both a formally certified inner bound and an empirical characterization of the RoA, offering a practical way to evaluate controller performance without relying solely on conservative analytical bounds or purely empirical simulation.

Figures

Figures reproduced from arXiv: 2604.04455 by the authors.

Figure 1
Figure 1. Schematic of the TWIP, indicating physical parameters and coordinate [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Monte Carlo estimation of the RoA for LQR, MPC, and CTMPC, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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

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

    eess.SY 2026-07 accept novelty 5.0 of 10

    An implicit predecessor-based set expansion enlarges a Lyapunov-certified region of attraction for a two-wheeled inverted pendulum, matching empirical stability boundaries.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

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