Stochastic stability of fault tolerant control systems in the presence of noise

The stochastic stability of fault tolerant control systems (FTCS) in the presence of noise is studied using the Lyapunov function approach. We consider FTCS in the presence of white Gaussian noise. The noise is of three types, state-dependent noise, control-dependent noise, and purely additive noise. It is shown that the exponential stability in the mean square is sufficient for almost sure asymptotic stability. Specifically, a testable necessary and sufficient condition for the exponential stability in the mean square is derived. A numerical example is presented to validate the theoretical analysis.