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　function　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　comparison　with　Wu-Lee＇　and　Wolf＇s　results.