Robust stabilization of linear systems in the presence of Gaussian perturbation of parameters

Stabilization of linear systems in state space in the presence of parametric perturbations is considered. The perturbed system is represented by a matrix differential equation with the elements of the matrices given by Gaussian processes with known mean and covariance. Using methods from stochastic control theory, certain pole-placement-like results are derived which hold in the mean square sense. In the absence of any perturbation, these results reduce to the well-known results of pole placement for deterministic linear systems. Minimizing the real part of the largest eigenvalue of the expected closed-loop matrix, we obtain the optimal feedback gain that stabilizes the system at the fastest possible rate. The question of existence of a guaranteed stabilizing feedback is also investigated. As a consequence of the main result we obtain a method of designing fault-tolerant systems that will survive in the events of catastrophic controller failure. An extension of the Luenberger observer for uncertain systems is also presented.