Robust and Gain-Scheduling ${\mathcal{H}}_2$ Control Techniques for LFT Uncertain and Parameter-Dependent Systems

This paper addresses the robust ${\mathcal{H}}_2$ synthesis problem for linear fractional transformation (LFT) systems subject to structured uncertainty (parameter) and white-noise disturbances. By introducing an intermediate matrix variable, we derive convex synthesis conditions in terms of linear matrix inequalities (LMIs) that enable both robust and gain-scheduled controller design for parameter-dependent systems. The proposed framework preserves the classical white-noise and impulse-response interpretation of the ${\mathcal{H}}_2$ criterion while providing certified robustness guarantees, thereby extending optimal ${\mathcal{H}}_2$ control beyond the linear time-invariant setting. Numerical and application examples demonstrate that the resulting robust ${\mathcal{H}}_2$ controllers achieve significantly reduced conservatism and improved disturbance rejection compared with conventional robust ${\mathcal{H}}_{\mathrm{\infty }}$-based designs.