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arxiv: 2604.04455 · v2 · pith:56DPYZDWnew · submitted 2026-04-06 · 📡 eess.SY · cs.SY

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

Lorenzo Fici , Dalim Wahby , Alvaro Detailleur , Matthieu Barreau , Guillaume Ducard This is my paper

Pith reviewed 2026-05-21 10:10 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords region of attractiontwo-wheeled inverted pendulumlinear quadratic regulatormodel predictive controlLyapunov invariant setMonte Carlo estimationunderactuated systems
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The pith

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

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

Figures

Figures reproduced from arXiv: 2604.04455 by Alvaro Detailleur, Dalim Wahby, Guillaume Ducard, Lorenzo Fici, Matthieu Barreau.

Figure 1
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. 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 ↗
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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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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Reference graph

Works this paper leans on

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