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

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 $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-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.