Formal reachability guarantees for cart-pole swing-up and stabilization

The cart-pole swing-up is a classic benchmark in nonlinear control, but rigorous end-to-end guarantees linking the global swing-up maneuver to local stabilization have been missing. In a new paper, Mohamed Khalid M Jaffar fills this gap with a formal reachability analysis for a switched controller combining an energy-based swing-up law and an LQR stabilizer. The swing-up law is derived from an energy-error Lyapunov function; by canceling the conservative term, the Lyapunov derivative becomes strictly sign-definite, and convergence follows from LaSalle’s invariance principle. The author also proposes an augmented Lyapunov function that drives the cart’s steady-state velocity to zero, achieving almost-global convergence.

Crucially, the handoff between the two controllers is designed so that the switching region lies entirely within the LQR's region of attraction. This certifies that once the swing-up phase brings the pole sufficiently close to upright, the LQR controller can always take over and stabilize the system. Numerical simulations corroborate the theoretical guarantees. The work provides a blueprint for formally verifying safety and convergence in other underactuated robotics systems, offering a path toward certifiable control in real-world applications like drones and walking robots.