Abstract: a continuous-time rational approximation (CRA) for the foundation frequency response is a starting point for constructing various high-order lumped-parameter models (LPMs) in the foundation vibration analysis. The stability and identification of CRA determine directly the stability and accuracy of its resulting LPMs and a soil-structure interaction system. In this paper, the necessary and sufficient stability conditions for the CRA and its LPMs are proposed based on linear-system stability theory and the input-output case of the LPMs. A parameter identification method including the proposed stability constraints is further developed by using the penalty func-tion method and the hybrid genetic-simplex optimization algorithm. A stable and accurate CRA is thus obtained by this method and is then realized as Wu-Lee’s and Wolf’s LPMs. The proposed stability and identification methods are verified by analyzing several typical foundation vibration problems and the comparision with Wu-Lee’ and Wolf’s results.