Probabilistic design of discrete-time guaranteed cost regulators

A guaranteed cost regulator design is presented for uncertain linear discrete-time systems, where its objective is to find a state feedback such that a quadratic performance index is less than a given value for any uncertainty. The design is recast as a feasibility problem described by a parametrized linear matrix inequality (LMI). A notion of probabilistic solution that satisfies the LMI for almost all uncertainties is introduced and an efficient randomized algorithm based on the gradient method with random samples of the uncertainty is developed. The proposed algorithm always stops in a finite number of iterations. Then, it can either find a probabilistic solution with a given probabilistic confidence or detect the infeasibility of the problem in the sense that the feasibility set is too small to contain a ball with a given radius.