Although adaptive control design with function approximators, for example, neural networks (NNs) and fuzzy logic systems, has been studied for various nonlinear systems, the classical adaptive laws derived based on the gradient descent algorithm with σ -modification or e -modification cannot guarantee the parameter estimation convergence. These nonconvergent learning methods may lead to sluggish response in the control system and make the parameter tuning complex. The aim of this paper is to propose a new learning strategy driven by the estimation error to design the alternative adaptive laws for adaptive control of nonlinear servo systems. The parameter estimation error is extracted and used as a new leakage term in the adaptive laws. By using this new learning method, the convergence of both the estimated parameters and the tracking error can be achieved simultaneously. The proposed learning algorithm is further tailored to retain finite-time convergence. To handle unknown nonlinearities in the servomechanisms, an augmented NN with a new friction model is used, where both the NN weights and some friction model coefficients are estimated online via the proposed algorithms. Comparisons with the σ -modification algorithm are addressed in terms of convergence property and robustness. Simulations and practical experiments are given to show the superior performance of the suggested adaptive algorithms.