A Polynomial Chaos Approach to Robust $\mathcal{H}_\infty$ Static Output-Feedback Control with Bounded Truncation Error

TL;DR

This paper introduces a polynomial chaos-based method for robust $\\mathcal{H}_\\infty$ static output-feedback control of uncertain systems, reducing computational complexity and truncation errors while ensuring stability.

Contribution

It develops a novel polynomial chaos approach that explicitly accounts for closed-loop dynamics and truncation errors, improving robustness and computational efficiency.

Findings

Reduces transformation errors compared to existing methods.

Avoids high-degree polynomial chaos expansions, lowering complexity.

Demonstrates effectiveness through a numerical example.

Abstract

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.