In this paper, we investigate a nonlinear extended state observer (ESO) constructed from piece-wise smooth functions consisted of linear and fractional power functions. This structure of ESO was first proposed in the 1990's and has been widely used in active disturbance rejection control for engineering controls. Its convergence, however, has remained an open problem up to this day. The main objective of this paper is to provide a convergence theory with explicit error estimation. The performances of this type ESO are studied by numerical simulation and compared with linear ESO. The numerical results show that the ESO proposed in this paper enjoys the advantages of smaller peaking value and better measurement noise tolerance. (C) 2017 Elsevier Ltd. All rights reserved.