The class of nonlinear dynamical systems whose dynamics depends linearly on the unknown parameter vector is considered. After reviewing a standard observer-based identifier for estimating the unknown parameters, we propose a family of new identifiers which exploit the a priori known parameter dependencies. Then we establish that, under mild assumptions on the dynamical system, 1) the proposed identifiers are stable, 2) the weighted norm of state-parameter errors using the proposed identifiers are less than the corresponding errors using the standard identifier for a length of time after t = 0. The main contribution of this paper is that it introduces a family of observer-based identifiers which exhibit better transient performance than the standard identifier.