System identification and damage detection for structural dynamic systems have received more and more attention in recent years. One of the time domain methodologies, the extended Kalman filter (EKF), has been widely applied in identifying the states and parameters of dynamic systems simultaneously. In EKF algorithm, the original dynamic systems have been transformed into nonlinear state-space system models. Therefore, the observability problem of the nonlinear state-space systems is required to be investigated before the application of EKF algorithm. In this paper, the definitions and the rank criterion of nonlinear observability for continuous-time systems are presented, which are only discussed in a few areas involved nonlinear systems previously. The analysis on the nonlinear observability of SISO and SIMO state-space structure-systems show that the rank criterion presented can provide sufficient conditions for the observability of the nonlinear systems to be identified. In practice, this criterion gives out a theoretical guideline for the selection of appropriate sensor types and locations before putting up the system identification, which is quite important but neglected in most of the EKF-identification literature thus far.