This paper presents identification of second order plus dead time (SOPDT) integrating and critically damped systems based on relay feedback testing. Relay with hysteresis is applied to the unknown system to get the sustained oscillations also called as limit cycle. The limit cycle parameters are utilized in mathematical expressions which are derived using state space technique so that exact process model parameters are estimated. As the relay with hysteresis helps in generating sustained oscillations and also reduces effect of measurement noise which is an important issue in system identification. Different types of processes in the form of transfer function models are considered to show the efficacy of the proposed method and results are compared with available methods in the literature with and without noise effect.