Probabilistic robust design with linear quadratic regulators

In this paper, we study robust design of uncertain systems in a probabilist ic setting by means of linear quadratic regulators (LQR). We consider syste ms affected by random bounded nonlinear uncertainty so that classical optim ization methods based on linear matrix inequalities cannot be used without conservatism. The approach followed here is a blend of randomization techni ques for the uncertainty together with convex optimization for the controll er parameters. In particular, we propose an iterative algorithm for designi ng a controller which is based upon subgradient iterations. At each step of the sequence, we first generate a random sample and then we perform a subg radient step for a convex constraint defined by the LQR problem. The main r esult of the paper is to prove that this iterative algorithm provides a con troller which quadratically stabilizes the uncertain system with probabilit y 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 controll er is found. (C) 2001 Elsevier Science B.V. All rights reserved.