Lyapunov-based adaptive state estimation for a class of continuous-time nonlinear stochastic systems

This paper is concerned with an adaptive state estimation problem for a class of continuous-time nonlinear stochastic systems with unknown constant parameters. These nonlinear systems have a linear-in-parameter (affine) structure and the nonlinearity is assumed to be bounded in a Lipschitz-like manner. Using stochastic counterparts of Lyapunov stability theory, we present adaptive state and parameter estimators with ultimately exponentially bounded estimator errors in the sense of mean square. Sufficient conditions are given in terms of the solvability of LMIs. In addition, we introduce a suboptimal design approach. By a martingale method, we demonstrate that this suboptimal design procedure also minimizes an upper bound of estimation error in almost sure sense.