Infinite-horizon performance bounds for constrained stochastic systems

We present a new method to bound the performance of causal controllers for uncertain linear systems with mixed state and input constraints. The performance is measured by the expected value of a discounted linear quadratic cost function over an infinite horizon. Our method computes a lower bound on the lowest achievable cost of any causal control policy. We compare our lower performance bound with the best performance achievable using the restricted class of disturbance affine control policies, both of which can be computed by solving convex conic optimization problems that are closely connected. The feasible sets of both convex programs have a natural relationship with respect to the maximal robust control invariant (RCI) set of the control problem. We present two numerical examples to illustrate the utility of our method.