  This article considers the $\mathcal{H}_\infty$ static output-feedback
control for linear time-invariant uncertain systems with polynomial dependence
on probabilistic time-invariant parametric uncertainties. By applying
polynomial chaos theory, the control synthesis problem is solved using a
high-dimensional expanded system which characterizes stochastic state
uncertainty propagation. A closed-loop polynomial chaos transformation is
proposed to derive the closed-loop expanded system. The approach explicitly
accounts for the closed-loop dynamics and preserves the $\mathcal{L}_2$-induced
gain, which results in smaller transformation errors compared to existing
polynomial chaos transformations. The effect of using finite-degree polynomial
chaos expansions is first captured by a norm-bounded linear differential
inclusion, and then addressed by formulating a robust polynomial chaos based
control synthesis problem. This proposed approach avoids the use of high-degree
polynomial chaos expansions to alleviate the destabilizing effect of truncation
errors, which significantly reduces computational complexity. In addition, some
analysis is given for the condition under which the robustly stabilized
expanded system implies the robust stability of the original system. A
numerical example illustrates the effectiveness of the proposed approach.
