Robust ${\mathcal{H}}_{\mathrm{\infty }}$ Observer Design via Finsler's Lemma and IQCs

This paper develops a Finsler-based LMI for robust ${\mathcal{H}}_{\mathrm{\infty }}$ observer design with integral quadratic constraints (IQCs) and block-structured uncertainty. By introducing a slack variable that relaxes the coupling between the Lyapunov matrix, the observer gain, and the IQC multiplier, the formulation addresses two limitations of the standard block-diagonal approach: the LMI requirement $\mathrm{H}\mathrm{e}(PA)\prec 0$ (which fails for marginally stable dynamics), and a multiplier–Lyapunov trade-off that causes infeasibility for wide uncertainty ranges. For marginally stable dynamics, artificial damping in the design model balances certified versus actual performance. The framework is demonstrated on quaternion attitude estimation with angular velocity uncertainty and mass-spring-damper state estimation with uncertain physical parameters.