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

summary

Video file (mp4)

The gist

The gist The work investigates the estimation of the Region of Attraction (RoA) for a two-wheeled inverted pendulum stabilized under three model-based control strategies: saturated LQR, linear MPC,

In short

The work estimates the Region of Attraction (RoA) for a two-wheeled inverted pendulum using three control strategies: saturated LQR, linear MPC, and constraint tightening MPC. It combines an analytical inner bound derived from Lyapunov functions with Monte Carlo simulations to empirically characterize the RoA. This provides both a formally certified safe region and an empirical measure of controller performance.

Key concepts

Region of Attraction (RoA)
This is the set of all possible starting points (initial conditions) for the system from which it is guaranteed to converge to its stable equilibrium point when using the specific control law. For nonlinear systems like the inverted pendulum, this region is often complex and hard to calculate analytically.
Lyapunov-based Invariant Set
This is a mathematical set derived from a Lyapunov function that guarantees stability. It acts as an inner approximation of the true RoA, meaning if the system stays within this specific set, it will definitely converge to equilibrium under the control law being tested.
Monte Carlo Estimation
This is a numerical technique where many random initial states are sampled and simulated to see how often they reach a certain condition. In this paper, it is used to empirically estimate the RoA by testing thousands of starting points against the analytically derived invariant set.

Terminology used across episodes

This episode discusses

The paper

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

Lorenzo Fici, Dalim Wahby, Alvaro Detailleur, Matthieu Barreau, Guillaume Ducard

Universite C´ote d'Azur CNRS I3S Sophia Antipolis France · ETH Zurich IDSC Zurich Switzerland KTH Royal Institute of Technology

DOI: 10.1109/ICCAD69956.2026.11643097

Transcript

Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Hi, I'm Rosa.

Dev: And I'm Dev, here with Taro.

Rosa: We're looking at the paper: "Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum".

Dev: Here's the gist of the paper. The gist The work investigates the estimation of the Region of Attraction (RoA) for a two-wheeled inverted pendulum stabilized under three model-based control strategies: saturated LQR, linear MPC, and constraint tightening MPC.

Rosa: The paper proposes a methodology that combines an analytical inner approximation with data-driven Monte Carlo estimation to characterize the RoA of the system <ref:2604.04455#pg2>.

Dev: This approach provides both a formally certified inner bound and an empirical characterization of the RoA, offering a practical way to evaluate controller performance <ref:2604.04455#pg4>.

Rosa: The analysis begins with deriving a Lyapunov-based invariant set that serves as an inner approximation of the RoA under the LQR control law <ref:2604.04455#pg2>.

Paper summary: Rosa: We've talked about how this paper investigates the Region of Attraction for systems like two-wheeled inverted pendulums, focusing on how to find that region when analytical methods are usually too conservative.

Dev: The thesis is that they propose combining a Lyapunov-based inner approximation with Monte Carlo sampling to get both a formally certified core and an empirical characterization of the surrounding region.

Taro: They are testing this against three control strategies: saturated LQR, linear MPC, and constraint tightening MPC to see how they compare in terms of their RoA estimation.

Rosa: The paper claims the main contribution is giving us a mathematically-certified core of stable initial conditions derived from the nonlinear closed-loop dynamics under the LQR control law <ref:2604.04455#pg3>.

Dev: And they also provide that Monte Carlo simulation to empirically characterize the Region of Attraction for those three controllers by identifying initial conditions that reach this invariant set without violating input constraints >

Taro: So, it’s about combining the analytical guarantee with a numerical estimation method, which gives you both a certified inner bound and an empirical characterization >

Rosa: The paper shows that this combination works across different control strategies because the same condition holds for linear MPC and CTMPC when certain cost functions are used <ref:2604.04455#pg3>.

Dev: It also verified that both controllers match the LQR law inside this terminal set when no constraints are active, and they satisfy their respective state, input, and terminal constraints within this region >

Taro: The analysis involves deriving a scaling factor alpha star to ensure that the ellipsoid defined by the invariant set remains fully contained within these constraint sets for both controllers >

Rosa: It’s about proving that this analytical bound is valid for comparing the LQR, linear MPC, and CTMPC on this TWIP system <ref:2604.04455#pg3>.

Dev: The results show that when they run their Monte Carlo simulation using the analytically derived invariant set as a stopping condition, all three controllers yielded the same RoA estimates of fifty-eight point four eight percent of the sampled states entering the set >

Conclusion: Rosa: So, wrapping up this discussion on "Region of Attraction Estimation for Linear Quadratic Regulator, Linear and Robust Model Predictive Control on a Two-Wheeled Inverted Pendulum."

Dev: The paper focuses on how to find that Region of Attraction when analytical methods are usually too conservative for these kinds of nonlinear systems.

Taro: The authors used the combination of a Lyapunov-based inner approximation and Monte Carlo sampling to provide both a formally certified core and an empirical characterization of the surrounding region.

Rosa: The key finding is that this approach works for saturated LQR, linear MPC, and CTMPC because they all seem to stabilize essentially the same set of initial conditions on this TWIP >

Dev: This suggests that their true Regions of Attraction are probably very similar based on the simulation results showing identical estimates across all three methods >

Taro: This framework gives us a general but model-dependent way to do RoA estimation for nonlinear systems where analytical methods get too conservative and data-driven approaches often lack formal guarantees >

Rosa: So, this paper provides a general but model-dependent framework for RoA estimation <ref:2604.04455#pg3>.

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