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Distributed Prescribed-Time Observer for Nonlinear Systems in Block-Triangular Form

Vincent de Heij, M. Umar B. Niazi, Karl H. Johansson, Saeed Ahmed

TL;DR

The paper tackles distributed state estimation for nonlinear systems in block-triangular observable canonical form, proposing a network of observers that cooperatively reconstructs the full state within a user-specified time $T$ independent of initial conditions. It combines a time-varying gain design, a coordinate transformation, and LMIs to guarantee EF-GUAS of the distributed estimation error over a strongly connected graph, with convergence guaranteed exactly at $t_0+T$. A Lyapunov-based analysis yields explicit error bounds and a constructive gain synthesis procedure, demonstrated through a four-agent numerical example where the prescribed-time convergence is achieved and the impact of the parameter $m$ on speed is illustrated. The work provides the first rigorous framework for distributed prescribed-time observers in this nonlinear block-triangular setting and discusses practical implementation considerations and avenues for future robustness and generalization.

Abstract

This paper proposes a distributed prescribed-time observer for nonlinear systems representable in a block-triangular observable canonical form. Using a weighted average of neighbor estimates exchanged over a strongly connected digraph, each observer estimates the system state despite the limited observability of local sensor measurements. The proposed design guarantees that distributed state estimation errors converge to zero at a user-specified convergence time, irrespective of observers' initial conditions. To achieve this prescribed-time convergence, distributed observers implement time-varying local output injection gains that monotonically increase and approach infinity at the prescribed time. The theoretical convergence is rigorously proven and validated through numerical simulations, where some implementation issues due to increasing gains have also been clarified.

Distributed Prescribed-Time Observer for Nonlinear Systems in Block-Triangular Form

TL;DR

The paper tackles distributed state estimation for nonlinear systems in block-triangular observable canonical form, proposing a network of observers that cooperatively reconstructs the full state within a user-specified time independent of initial conditions. It combines a time-varying gain design, a coordinate transformation, and LMIs to guarantee EF-GUAS of the distributed estimation error over a strongly connected graph, with convergence guaranteed exactly at . A Lyapunov-based analysis yields explicit error bounds and a constructive gain synthesis procedure, demonstrated through a four-agent numerical example where the prescribed-time convergence is achieved and the impact of the parameter on speed is illustrated. The work provides the first rigorous framework for distributed prescribed-time observers in this nonlinear block-triangular setting and discusses practical implementation considerations and avenues for future robustness and generalization.

Abstract

This paper proposes a distributed prescribed-time observer for nonlinear systems representable in a block-triangular observable canonical form. Using a weighted average of neighbor estimates exchanged over a strongly connected digraph, each observer estimates the system state despite the limited observability of local sensor measurements. The proposed design guarantees that distributed state estimation errors converge to zero at a user-specified convergence time, irrespective of observers' initial conditions. To achieve this prescribed-time convergence, distributed observers implement time-varying local output injection gains that monotonically increase and approach infinity at the prescribed time. The theoretical convergence is rigorously proven and validated through numerical simulations, where some implementation issues due to increasing gains have also been clarified.

Paper Structure

This paper contains 17 sections, 2 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. System Definition
  4. Design Problem
  5. Coordinate Transformation and Time-Varying Gains
  6. Prescribed-Time Convergence of Estimation Error
  7. Main Result
  8. Numerical Simulations
  9. Conclusion
  10. Proof of Lemma \ref{['lem:key-ineq']}
  11. Analysis of Mathematical Inconsistencies in Adil2024
  12. Incorrect Application of Lipschitz Property
  13. Circular Reasoning and Inconsistent Parameter Requirements
  14. Incomplete Proof of Prescribed-Time Stability
  15. Missing Design Criterion
  16. ...and 2 more sections

Key Result

Theorem 1

Given an initial time $t_0\in\mathbb R_{\geq 0}$, the prescribed-time $T\in\mathbb R_{>0}$, and $\mu(t;t_0,T)$ as in eq:mu, suppose there exist symmetric positive definite matrices $P_1,\dots,P_N\in\mathbb R^{n\times n}$, row vectors $Q_1,\dots,Q_N\in\mathbb R^{1\times n}$, scalars $\epsilon_1,\epsi where $P=\operatorname{diag}(P_1,\dots,P_N)$, $Q=\operatorname{diag}(Q_1,\dots,Q_N)$, $R$ is given

Figures (4)

  • Figure 1: Framework of distributed state estimation.
  • Figure 2: Directed graph for the example.
  • Figure 3: From top left to bottom right: the observed system, the errors of local observers, the estimates $z^{1}_{12}(t)$ for various $m$.
  • Figure 4: Graph representation of the system structure considered in Chang2021.

Theorems & Definitions (8)

  • Definition 1: FGUAS Holloway19
  • Definition 2: E-FGUAS
  • Theorem 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:newthm']}
  • Remark 1
  • Remark 2: Implementation