ABSTRACT This article addresses the global finite‐time regulation problem for Euler–Lagrange systems using an energy‐shaping plus damping injection approach. This methodology ensures that the trajectories of the closed‐loop system converge to any desired position with zero velocity in finite time, regardless of the initial conditions. As it is typical of the energy‐shaping approach, the control structure is based on the gradients of artificial potential and dissipative energy functions. In this work, new exponent conditions are proposed for the lower bounds of the desired potential energy and energy dissipative functions. Notably, these conditions eliminate the need for the energy functions to satisfy specific homogeneity constraints. By removing this requirement, the method enables the design of a broader class of finite‐time (state‐feedback and output‐feedback) controllers and allows for controllers that exhibit an unconventional form of exponential convergence—distinct from the classical definition. A byproduct of our proposal is the development of several novel finite‐time and unconventional exponential controllers that do not satisfy homogeneity constraints and compensate for the gravity force via a feed‐forward term. Such controllers have not been reported in the literature. We further include experimental results on a 6‐degrees of freedom robot to illustrate their performance.
Cruz‐Zavala et al. (Thu,) studied this question.