www.elsevier.com/locate/sysconle Probabilistic robust design with linear quadratic regulators

In this paper, we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators (LQR). We consider systems a2ected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism. The approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters. In particular, we propose an iterative algorithm for designing a controller which is based upon subgradient iterations. At each step of the sequence, we 6rst generate a random sample and then we perform a subgradient step for a convex constraint de6ned by the LQR problem. The main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a 6nite number of steps. In addition, at a 6xed step, we compute a lower bound of the probability that a quadratically stabilizing controller is found. c © 2001 Elsevier Science B.V. All rights reserved