Robustness of feedback-stabilized systems in the presence of non-linear and random perturbations

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Abstract
The general robustness problem is discussed for a controlled discrete-time system subject to a Lur'e type feedback non-linearity together with random multiplicative noises effective on state, input and the non-linearity. A sufficient condition is given utilizing the discrete stochastic version of the second method of Lyapunov to guarantee that any stabilizing controller designed for the deterministic linear part of the system renders the closed-loop system with perturbations, asymptotically stable with probability one. Then, specializations to systems given in controllability and controller canonical forms are made. Independent noise signals are assumed to be effective on individual parameters of these canonical forms whose inputs are designed to assign the closed-loop poles to zero. In these cases, stronger results are obtained in terms of system parameters and known characteristics of perturbations.