Performance bounds for min-max uncertain constrained systems

We present a method to bound the performance of causal controllers for uncertain linear systems with mixed state and input constraints. The performance is measured by the worst-case value (over all possible disturbance inputs) of a convex piecewise linear cost function over a finite horizon. Our method computes a causal affine control policy that satisfies the constraints robustly, while simultaneously providing a lower bound on the achievable performance of this, or any other causal control policy constructed by any other method. The controller and performance guarantee can be found by solving a number of convex conic optimization problems which are closely connected.