  In this paper, we address robust static anti-windup compensator design and
performance analysis for saturated linear closed loops in the presence of
nonlinear probabilistic parameter uncertainties via randomized techniques. The
proposed static anti-windup analysis and robust performance synthesis
correspond to several optimization goals, ranging from minimization of the
nonlinear input/output gain to maximization of the stability region or
maximization of the domain of attraction. We also introduce a novel paradigm
accounting for uncertainties in the energy of the disturbance inputs. Due to
the special structure of linear static anti-windup design, wherein the design
variables are decoupled from the Lyapunov certificates, we introduce a
significant extension, called scenario with certificates (SwC), of the
so-called scenario approach for uncertain optimization problems. This extension
is of independent interest for similar robust synthesis problems involving
parameter-dependent Lyapunov functions. We demonstrate that the scenario with
certificates robust design formulation is appealing because it provides a way
to implicitly design the parameter-dependent Lyapunov functions and to remove
restrictive assumptions about convexity with respect to the uncertain
parameters. Subsequently, to reduce the computational cost, we present a
sequential randomized algorithm for iteratively solving this problem. The
obtained results are illustrated by numerical examples.
