In this paper, a stable and robust filter is proposed for structural identification. This filter resolves the instability problems of the traditional extended Kalman filter (EKF). Instead of ad hoc assignment of the noise covariance matrices in the EKF, the proposed stable robust extended Kalman filter (SREKF) provides real-time updating of the noise parameters. This resolves the well-known instability problem of the EKF due to improper assignment of the noise covariance matrices. Furthermore, the proposed SREKF is capable of removing abnormal data points in a real-time manner. As a result, the parametric identification results will be more reliable and have fewer fluctuations. The proposed approach will be applied to structural damage detection of degrading linear and nonlinear structures in comparison with the plain EKF, utilizing highly contaminated response measurements. It turns out that the estimation error of the state vector and the structural parameters is lower than the EKF by one and two orders of magnitude, respectively.