A probabilistic approach to robust disturbance attenuation for LTI systems with deterministic and stochastic noises

We consider a robust disturbance attenuation problem for a set of LTI SISO discrete-time systems with two types of exogenous inputs, namely a deterministic disturbance input and a stochastic noise. The performance is measured by a probability with respect to the stochastic noise of which the worst case 2-norm of the output against disturbance input in a class of deterministic signals with bounded 2-norm is less than a specified level. We first provide a matrix inequality characterization of the probability based on the Toeplitz formulae of the system and derive a lower bound of the probability. We then show that a guaranteed performance level satisfying a specified worst case probability condition against a set of bounded perturbations can be computed by solving an LMI convex optimization problem. We also show that the corresponding synthesis problem can be also reduced to an LMI optimization problem.