Hysteresis nonlinearity degrades the positioning accuracy of a piezostage and requires a suppression for precision micro-/nanopositioning applications. This paper proposes two new approaches to modeling and compensating the rate-dependent hysteresis of a piezostage driven by piezoelectric stack actuators. By formulating the hysteresis modeling as an online nonlinear regression problem, online least squares support vector machine (SVM) (LS-SVM) and online relevance vector machine (RVM) models are proposed to capture the hysteretic behavior. Both hysteresis models are capable of updating continually with subsequent samples. After a comparative study on modeling performances, an inverse model-based feedforward combined with proportional–integral–derivative feedback control is presented to alleviate the hysteresis effect. Experimental results show that the LS-SVM model-based control scheme is over 86% more accurate than the RVM model-based one in the motion tracking task, whereas the latter is 14 times faster than the former in terms of updating time. Moreover, both LS-SVM and RVM model-based control schemes can suppress the rate-dependent hysteresis to a negligible level, which validates the feasibility and effectiveness of the proposed approaches.