Global asymptotic stabilization of a pendulum using a single Lyapunov proportional bang-bang control strategy

Abstract

The existence of a Lyapunov function is known to ensure either local or global stability of a system’s equilibrium state. Inspired by the control-Lyapunov method, we here construct a Lyapunov candidate function by analyzing a pendulum system’s total energy and then applying appropriate control actions such that the conditions of a Lyapunov function are met. More specifically, our controller evaluates the Lyapunov function’s time derivative at each time step, and applies control torque such as to ensure that the Lyapunov function decreases for each step toward the goal upright state. Unlike the control-Lyapunov method, which aims to select control input as to minimize the Lyapunov function’s time derivative, our method provides up front the satisfactory conditions that yield a globally stable controller by using a rigorously designed proportional bang-bang control strategy. We show how to derive the controllers evaluation function, and how the controller is implemented in code. We further demonstrate the effectiveness of our method through numerical simulations. The result of our approach is a globally stable upright pendulum using a single-controller strategy.