The　estimation　for　state　variables　of　systems　with　non-linear　or　unknown　signals　has　great　significance.　In　comparison　with　traditional　NNs　algorithm,　the　ELM　algorithm　has　much　shorter　training　time.　Adopting　kernel　function　and　on-line　update　method,　the　OS-KELM　algorithm　can　obtain　better　performance　than　ELM　algorithm.　In　this　paper,　an　on-line　state　estimation　method　has　been　proposed　based　on　OS-KELM　algorithm　and　Lyapunov　stability　analysis.　Compared　to　estimators　based　on　NNs　algorithm　or　ELM　algorithm,　the　proposed　method　takes　less　computing　resource　and　faster　computing　speed.　Simulations　show　that,　the　estimators　designed　by　this　method　can　approximate　state　variables　of　systems　with　non-linear　or　unknown　signals　in　satisfying　accuracy.　At　the　same　time,　the　approximations　keep　stability　and　robustness.　©　2022,　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.