Robustness of discrete-time systems for unstructured stochastic perturbations

Several stochastic stability robustness measures are presented for nominally exponentially stable linear discrete-time systems with unstructured perturbations having second-moment bounds. Dependence of these measures on the stability degree of the nominal system and other parameters used in the procedure is illustrated. By using the time evolution of the second moment of the system state and stochastic Lyapunov stability results (positive super-martingale convergence theorems), the ability of nominally exponentially stable systems to maintain stability in the presence of unstructured stochastic (linear and nonlinear) perturbations is demonstrated. Quantitative results are given to determine the maximum modeling uncertainty which can be tolerated in design. Upper bounds on the second moments of stochastic perturbations to maintain the mean-square and almost sure stability of these systems in the presence of unstructured perturbations are obtained