Robust and Exponential Stabilization of a Cart–Pendulum System via Geometric PID Control

Abstract

This paper addresses the robust stabilization problem of a cart–pole system. The controlled dynamics of this interconnected system are deduced by following the analytic framework of Lagrangian mechanics, and the residual terms are formulated as a bias depending on the angle and angular velocity. A geometric definition of Proportional–Integral–Derivative (PID) control algorithm is proposed, and a Lyapunov function is explicitly constructed through two stages of variable change. Local exponential stability of the stable equilibrium is proved, and a criterion for parameter tuning is provided by ensuring an exponential decrease in the Lyapunov function. Enlarging the control parameters to infinity allows for the extension of attraction region almost to the half circle. The effectiveness of geometric PID controller and the local exponential stability of the resulting close system are verified by simulating a numerical example.