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. We consider systems affected 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 first generate a random sample and then we make a subgradient step for a convex constraint defined 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 finite number of steps. In addition, at a fixed step, we compute a lower bound of the probability that a quadratically stabilizing controller is found.