Theanalysisanddesignoftheextendedstateobserver(ESO)involvesacontinuousnon-smoothstructure,thusthestudyoftheESOdynamicrequiresmathematicaltoolsofthenonlinearsystemsanalysis.ThispaperestablishesthesufficientconditionsforabsolutestabilityoftheESO.Basedonthisstudy,amethodologytoestimateseveralnonlinearfunctionsindynamicssystemsisproposed.