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Filtering Homogeneous Observer for MIMO System

Xubin Ping, Konstantin Zimenko, Andrey Polyakov, Denis Efimov

TL;DR

This work develops a filtering homogeneous observer for linear MIMO systems with an output prefilter to enhance noise robustness. Gains are computed via an LMI that is feasible whenever the plant is observable, and the design supports both prescribed-time convergence (ν<0) and standard asymptotic convergence (ν≥0) within a unified homogeneous framework. A high-order sliding mode–style extension is also provided, along with an ISS robustness analysis against measurement noise and unmodeled perturbations, and a numerical example demonstrates superior noise rejection compared to a Luenberger observer. The approach yields constructive, scalable tuning and broad applicability to practical MIMO state estimation tasks.

Abstract

Homogeneous observer for linear multi-input multi-output (MIMO) system is designed. A prefilter of the output is utilized in order to improve robustness of the observer with respect to measurement noises. The use of such a prefilter also simplifies tuning, since the observer gains in this case are parameterized by a linear matrix inequality (LMI) being always feasible for observable system. In particular case, the observer is shown to be applicable in the presence of the state and the output bounded perturbations. Theoretical results are supported by numerical simulations.

Filtering Homogeneous Observer for MIMO System

TL;DR

This work develops a filtering homogeneous observer for linear MIMO systems with an output prefilter to enhance noise robustness. Gains are computed via an LMI that is feasible whenever the plant is observable, and the design supports both prescribed-time convergence (ν<0) and standard asymptotic convergence (ν≥0) within a unified homogeneous framework. A high-order sliding mode–style extension is also provided, along with an ISS robustness analysis against measurement noise and unmodeled perturbations, and a numerical example demonstrates superior noise rejection compared to a Luenberger observer. The approach yields constructive, scalable tuning and broad applicability to practical MIMO state estimation tasks.

Abstract

Homogeneous observer for linear multi-input multi-output (MIMO) system is designed. A prefilter of the output is utilized in order to improve robustness of the observer with respect to measurement noises. The use of such a prefilter also simplifies tuning, since the observer gains in this case are parameterized by a linear matrix inequality (LMI) being always feasible for observable system. In particular case, the observer is shown to be applicable in the presence of the state and the output bounded perturbations. Theoretical results are supported by numerical simulations.
Paper Structure (16 sections, 7 theorems, 65 equations, 3 figures)

This paper contains 16 sections, 7 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Generalized homogeneity
  4. Linear Dilation
  5. Canonical homogeneous norm
  6. Homogeneous Functions and Vector fields
  7. Filtering homogeneous observer
  8. Basic ideas of the observer design and analysis
  9. Main results
  10. Robustness analysis
  11. Numerical example
  12. Conclusions
  13. Appendix
  14. The proof of Theorem \ref{['thm:hom_conv2stab']}
  15. The proof of Theorem \ref{['thm:ch14_ext_hom_observer']}
  16. ...and 1 more sections

Key Result

Corollary 1

A linear continuous dilation in $\mathbb{R}^n$ is strictly monotone with respect to the weighted Euclidean norm $\|x\|=\sqrt{x^{\top} Px}$ with $0\prec P\in \mathbb{R}^{n\times n}$ if and only if $PG_{\mathbf{d}}+G_{\mathbf{d}}^{\top}P\succ 0, P\succ 0.$

Figures (3)

  • Figure 1: Estimation errors for homogeneous (left) and linear (right) observers in the nominal case
  • Figure 2: Estimation errors for homogeneous (left) and linear (right) observers with the additive perturbation $0.1B\sin(5t)$
  • Figure 3: Estimation errors for homogeneous (left) and linear (right) observers with the additive perturbation $0.1B\sin(5t)$ and with the noisy measurements of the magnitude $0.001$

Theorems & Definitions (10)

  • Definition 1
  • Corollary 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Corollary 3