A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

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. A numerical example illustrates the effectiveness of the proposed approach.