Reachability Guarantees for Cart-Pole Swing-Up and Stabilization

2026-06-26

AI Summary

This work addresses the lack of end-to-end reachability guarantees in existing methods for the full swing-up and stabilization process of the inverted pendulum system. The authors propose a switching control strategy that integrates an energy-based Lyapunov function with a linear quadratic regulator (LQR). During the swing-up phase, energy control is employed, and upon reaching a neighborhood of the upright equilibrium, the controller seamlessly switches to LQR. Formal verification of this transition is achieved through a strictly sign-definite Lyapunov derivative and LaSalle’s invariance principle. An augmented Lyapunov function is constructed to ensure the cart’s steady-state velocity converges to zero, thereby establishing almost global asymptotic stability. Numerical simulations confirm reliable convergence from a compact set of initial conditions to the upright equilibrium, along with robust stabilization performance.

Abstract

The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up law is derived from an energy-error Lyapunov function; canceling the autonomous conservative term yields a strictly sign-definite Lyapunov derivative, and convergence follows from LaSalle's invariance principle. We also propose an augmented Lyapunov function to regulate the steady-state cart velocity to zero, for which we establish almost-global convergence. For the controller handoff, a switching region is designed to lie strictly within the LQR region of attraction, formally certifying the swing-up-to-stabilization transition. Numerical simulations corroborate the theoretical analysis.

Problem

• cart-pole
• swing-up
• stabilization
• reachability
• underactuated systems

Innovation

• reachability analysis
• switched control
• Lyapunov-based stabilization
• cart-pole swing-up
• almost-global convergence