Stability of the SDDRE based Estimator for Stochastic Nonlinear System

Bounded-Input-Bounded-State and Bounded-Input-Bounded-Output stability of the State Dependent Differential Riccati Equation (SDDRE) based estimator of nonlinear stochastic time-varying systems is proved by three approaches. Each proof highlights different aspect of the SDDRE based estimator stability. The State Dependent Coefficient (SDC) form representation of nonlinear stochastic system is used. In the first approach theory of linear time-varying stochastic systems is used. In second approach it is shown that the state of the SDDRE based estimator is always bounded by defining a Lyapunov function and usage the Barkana's invariance principle. The exponential stability of the state transition matrix of the estimator and computation of bounds on the state, under specific conditions, is proved by the third approach. It is shown that uniform complete observability and controllability, and respective boundedness conditions along the estimator's trajectories is sufficient for the results above.