  In this paper, we introduce a significant extension, called scenario with
certificates (SwC), of the so-called scenario approach for uncertain
optimization problems. This extension is motivated by the observation that in
many control problems only some of the optimization variables are used in the
design phase, while the other variables play the role of certificates. Examples
are all those control problems that can be reformulated in terms of linear
matrix inequalities involving parameter-dependent Lyapunov functions. These
control problems include static anti-windup compensator design for uncertain
linear systems with input saturation, where the goal is the minimization of the
nonlinear gain from an exogenous input to a performance output. The main
contribution of this paper is to show that randomization is a useful tool,
specifically for anti-windup design, to make the overall approach less
conservative compared to its robust counterpart. In particular, we demonstrate
that the scenario with certificates reformulation is appealing because it
provides a way to implicitly design the parameter-dependent Lyapunov functions.
Finally, to further reduce the computational cost of this one-shot approach, we
present a sequential randomized algorithm for iteratively solving this problem.
