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Robust $\Hinf$ Observer Design via Finsler's Lemma and IQCs

Raktim Bhattacharya, Felix Biertümpfel

Abstract

This paper develops a Finsler-based LMI for robust $\Hinf$ 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 $\He{PA} \prec 0$ (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.

Robust $\Hinf$ Observer Design via Finsler's Lemma and IQCs

Abstract

This paper develops a Finsler-based LMI for robust 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 (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.

Paper Structure

This paper contains 27 sections, 3 theorems, 29 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classical $\mathcal{H}_\infty$ Observer Design
  3. Robust Observer Problem Setup
  4. Linear Fractional Transformation (LFT)
  5. Plant Dynamics
  6. Design Objective
  7. Integral Quadratic Constraints (IQCs)
  8. Multiplier for Norm-Bounded Uncertainty
  9. LMI Derivation of Robust $\mathcal{H}_\infty$ Observer
  10. Dissipation Inequality with IQC Supply
  11. The Bilinear Matrix Inequality
  12. General Resolution via Finsler's Lemma
  13. Gain Recovery and Certificate Verification
  14. Block-Diagonal Specialization
  15. Artificial Damping for Marginally Stable Dynamics
  16. ...and 12 more sections

Key Result

Lemma 1

Let $Q = Q^\top \in \mathbb{R}^{m \times m}$ and $\mathcal{B} \in \mathbb{R}^{k \times m}$. The following are equivalent: $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Upper LFT: $\Delta$ satisfies $p = \Delta q$; $w$ is the exogenous input, $z$ the performance output, $y$ the measurement.
  • Figure 2: Quaternion Monte Carlo: $\|e(t)\|_2$ over 50 random $\omega$ realizations ($\alpha = 0.15$, $\delta_\omega = 0.20$). Shaded: 5th--95th percentile band. Thick line: median.
  • Figure 3: Quaternion certificate validation: $\mathcal{H}_\infty$ norm distribution over 200 random $\omega$ realizations (shifted system) with certified $\gamma$ (red line).
  • Figure 4: MCK Monte Carlo: $\|e(t)\|_2$ over 50 random parameter realizations. Shaded: 5th--95th percentile band. Thick line: median.
  • Figure 5: MCK certificate validation: $\mathcal{H}_\infty$ norm distribution over 200 random parameter realizations (histogram) with certified $\gamma$ (red line). Both robust certificates are valid. The nominal design's worst-case $\mathcal{H}_\infty$ norm ($0.67$) exceeds its design $\gamma$ ($0.50$), confirming the need for robust synthesis.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3: Why the augmented state is needed
  • Lemma 1: Finsler finsler1937
  • Theorem 1: IQC-based $\mathcal{H}_\infty$ Observer via Finsler's Lemma
  • proof
  • Remark 4: Connection to the IQC theorem
  • Corollary 1: Block-diagonal specialization
  • Remark 5: Multiplier--damping interaction
  • Remark 6: Multiplier--Lyapunov trade-off