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Homogeneous Distributed Observers for Quasilinear Systems

Min Li, Andrey Polyakov, Siyuan Wang, Gang Zheng

TL;DR

This work addresses cooperative state estimation for nonlinear quasilinear plants with Hölder nonlinearities by developing distributed observers that leverage generalized homogeneity. The authors construct globally and locally homogeneous observer designs, with LMIs providing gain tuning to achieve finite-time and fixed-time convergence, respectively, while ensuring ISS with respect to bounded perturbations. The methods accommodate nonlinearities via Hölder conditions and include robustness analysis under $q_x$ and $q_y$, validated by numerical simulations showing faster convergence and smaller steady-state errors than linear counterparts. The approach advances distributed estimation by delivering non-asymptotic performance guarantees and practical resilience in networked cyber-physical systems.

Abstract

The problem of finite/fixed-time cooperative state estimation is considered for a class of quasilinear systems with nonlinearities satisfying a Hölder condition. A strongly connected nonlinear distributed observer is designed under the assumption of global observability. By proper parameter tuning with linear matrix inequalities, the observer error equation possesses finite/fixed-time stability in the perturbation-free case and input-to-state stability with respect to bounded perturbations. Numerical simulations are performed to validate this design.

Homogeneous Distributed Observers for Quasilinear Systems

TL;DR

This work addresses cooperative state estimation for nonlinear quasilinear plants with Hölder nonlinearities by developing distributed observers that leverage generalized homogeneity. The authors construct globally and locally homogeneous observer designs, with LMIs providing gain tuning to achieve finite-time and fixed-time convergence, respectively, while ensuring ISS with respect to bounded perturbations. The methods accommodate nonlinearities via Hölder conditions and include robustness analysis under and , validated by numerical simulations showing faster convergence and smaller steady-state errors than linear counterparts. The approach advances distributed estimation by delivering non-asymptotic performance guarantees and practical resilience in networked cyber-physical systems.

Abstract

The problem of finite/fixed-time cooperative state estimation is considered for a class of quasilinear systems with nonlinearities satisfying a Hölder condition. A strongly connected nonlinear distributed observer is designed under the assumption of global observability. By proper parameter tuning with linear matrix inequalities, the observer error equation possesses finite/fixed-time stability in the perturbation-free case and input-to-state stability with respect to bounded perturbations. Numerical simulations are performed to validate this design.
Paper Structure (14 sections, 2 theorems, 104 equations, 5 figures)

This paper contains 14 sections, 2 theorems, 104 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Generalized Homogeneity
  5. Problem Statement
  6. Basic Idea of the Observer Design
  7. Homogeneous Observer Design
  8. Globally Homogeneous Distributed Observer
  9. Locally Homogeneous Distributed Observer
  10. Robustness Analysis
  11. Simulation Results
  12. On the Robust Finite-time Distributed Observer
  13. On the Robust Fixed-time Distributed Observer
  14. Conclusion

Key Result

Proposition 1

Let conditions of Theorem thm_fin_homo hold. The error equation eq:error_eq_fin is ISS with respect to the bounded perturbation $q\!\!=\!\!(q^\top_x,q^\top_y)^\top\!\!\!\!\!\in\!\!L^\infty(\mathbb{R}, \mathbb{R}^{n+p})$.

Figures (5)

  • Figure 1: The communication graph of the distributed observer.
  • Figure 2: The trajectory of $|e|$ and $|e_l|$, with $q\!\!=\!\!\mathbf{0}$, by employing the finite-time distributed observer and the linear distributed observer, respectively.
  • Figure 3: The trajectory of $|e|$ and $|e_l|$, with $q\!\!\neq\!\!\mathbf{0}$, by employing the finite-time distributed observer and the linear distributed observer, respectively.
  • Figure 4: The trajectory of $|e|$ and $|e_l|$ by employing the fixed-time distributed observer and the linear distributed observer, respectively; with $q\!\!=\!\!\mathbf{0}$ and $m\!\!\in\!\!\{-1,0,1,2,3\}$.
  • Figure 5: The trajectory of $|e|$ and $|e_l|$, with $q\!\!\neq\!\!\mathbf{0}$, by employing the fixed-time distributed observer and the linear distributed observer, respectively.

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Proposition 2