This paper addresses the position control problem for a Lagrangian pendulum. Using a strict Lyapunov function, a rigorous analysis is presented to prove that the closed-loop system equilibrium point composed of the pendulum dynamics and a classical linear PID control is globally asymptotically stable. Motivated by these results, the theoretical proposal is extended to analyze a novel hyperbolic PID-type control scheme; reformulating the Lyapunov function, global asymptotic stability of the equilibrium point for the corresponding closed-loop equation is demonstrated. The proposed hyperbolic scheme is a rational function with bounded control action composed of a suitable combination of hyperbolic sine and cosine functions. The hyperbolic structure is used in the proportional, integral, and derivative terms of the control algorithm to drive the position error and joint velocity to zero. Experimental results of both a linear PID and a novel hyperbolic PID-type controller on a direct-drive pendulum are presented to illustrate the effectiveness and performance of the proposed control algorithm.
Rico et al. (Wed,) studied this question.