Stochastic stability of singularly perturbed nonlinear systems

The stability of a nonlinear stochastic dynamic system with singular perturbations is considered. Based on the notion of stochastic input-to-state stability and using time scale decomposition, a result of the "total stability" type is obtained, i.e. if the fast subsystem and the slow subsystem are both input-to-state stable with respect to disturbances, then this property continues to hold for the full-order system as long as the singular perturbation parameter is sufficiently small and a stochastic small gain condition is satisfied. The result is general in that it holds for a broad class of disturbances, and resembles similar results for deterministic systems