On Adaptive Continuous Finite‐Time Regulators for Euler‐Lagrange Systems

ABSTRACT We propose an adaptive set‐point controller for n‐degrees of freedom Euler‐Lagrange systems, particularly for robot manipulators. The controller is reminiscent of a simple proportional‐derivative controller with adaptive gravity compensation but ensures finite‐time (FT) stability. To that end, the parameter estimation law employs a Dynamic Regressor Extension and Mixing (DREM) estimator and both the proportional‐derivative terms and the estimation law are modified using nonlinearities of the type used in FT stabilization of second‐order systems (called Hölder's ontinuous terms). The value of our contribution lies in the fact that, to the best of the authors' knowledge, FT regulation (set‐point control) for Euler–Lagrange systems in presence of uncertain potential energy ensuring FT stability is a long‐standing open problem. Moreover, an interesting byproduct of our main results is a novel asymptotic adaptive control scheme which employs an FT‐convergent adaptation law and improves the system's robustness, notably with respect to certain unmodeled dynamics, such as friction, as well as to measurement noise. These properties are illustrated through extensive simulation results.