A new optimal control approach to uncertain Euler-Lagrange equations: $H_{\mathrm{\infty }}$ disturbance estimator and generalized $H_2$ tracking controller

This paper proposed a new optimal control method for uncertain Euler-Lagrange systems, focusing on estimating model uncertainties and improving tracking performance. More precisely, a linearization of the nonlinear equation was achieved through the inverse dynamic control (IDC) and an $H_{\mathrm{\infty }}$ optimal estimator was designed to address model uncertainties arising in this process. Subsequently, a generalized $H_2$ optimal tracking controller was obtained to minimize the effect of the estimation error on the tracking error in terms of the induced norm from $L_2$ to $L_{\mathrm{\infty }}$. Necessary and sufficient conditions for the existences of these two optimal estimator and controller were characterized through the linear matrix inequality (LMI) approach, and their synthesis procedures can be operated in an independent fashion. To put it another way, this developed approach allowed us to minimize not only the modeling error between the real Euler-Lagrange equations and their nominal models occurring from the IDC approach but also the maximum magnitude of the tracking error by solving some LMIs. Finally, the effectiveness of both the $H_{\mathrm{\infty }}$ optimal disturbance estimator and the generalized $H_2$ tracking controller were demonstrated through some comparative simulation and experiment results of a robot manipulator, which was one of the most representative examples of Euler-Lagrange equations.