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

TL;DR

This paper introduces a Finsler-based LMI approach for robust $_ ext{infty}$ observer design that effectively handles marginally stable systems and wide uncertainties using IQCs and slack variables.

Contribution

It develops a novel Finsler-LMI formulation with slack variables to relax coupling constraints, improving robustness and feasibility in $_ ext{infty}$ observer design for uncertain systems.

Findings

Addresses limitations of standard block-diagonal approaches.

Balances certified and actual performance for marginally stable dynamics.

Demonstrates effectiveness on quaternion and mass-spring-damper systems.

Abstract

This paper develops a Finsler-based LMI for robust $\mathcal{H}_\infty$ 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 $\mathrm{He}(PA) \prec 0$ (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.