Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $ L_\infty $-gain analysis

Abstract

<abstract><p>This paper considers a model-free control approach to Euler-Lagrange equations and proposes a new quantitative performance measure with its Lyapunov-based computation method. More precisely, this paper aims to solve a trajectory tracking problem for uncertain Euler-Lagrange equations by using a model-free controller with a proportional-integral-derivative (PID) control form. The $ L_\infty $-gain is evaluated for the closed-loop systems obtained through the feedback connection between the Euler-Lagrange equation and the model-free controller. To this end, the input-to-state stability (ISS) for the closed-loop systems is first established by deriving an appropriate Lyapunov function. The study further extends these arguments to develop a computational approach to determine the $ L_\infty $-gain. Finally, the theoretical validity and effectiveness of the proposed quantitative performance measure are demonstrated through a simulation of a $ 2 $-degree-of-freedom ($ 2 $-DOF) robot manipulator, which is one of the most representative examples of Euler-Lagrange equations.</p></abstract>