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Leader-following Consensus over Jointly Connected Switching Networks is Achievable for Exponentially Unstable Linear Systems

Yuhan Chen, Tao Liu, Jie Huang

TL;DR

This work resolves a long-standing limitation in leader-following consensus by proving exponential convergence for general linear agents over jointly connected switching networks even when $A$ is exponentially unstable. The authors derive a bound on the allowable instability using a projection-based analysis of the leader-follower matrix and construct a distributed stabilizing gain $K$ via a weighted controllability Gramian, with explicit lower bounds on nonzero eigenvalues of the switching matrix. A dual formulation yields an exponentially convergent output-based distributed observer for the leader, with parallel Gramian-based design for $L$. Numerical examples corroborate the theory, showing convergence of follower states to the leader and convergence of the dual observer under the same connectivity and stability constraints. Overall, the paper substantially extends the scope of solvable leader-following problems in time-varying networks and enhances distributed observer design for unstable leaders.

Abstract

The leader-following consensus problem for general linear multi-agent systems over jointly connected switching networks has been a challenging problem and the solvability of the problem has been limited to the class of linear multi-agent systems whose system matrix is marginally stable. This condition is restrictive since it even excludes the most commonly used double-integrator system. This paper presents a breakthrough by demonstrating that leader-following exponential consensus is achievable for general linear multi-agent systems over jointly connected switching networks, even when the system matrix is exponentially unstable. The degree of instability can be explicitly characterized by two key quantities that arise from the jointly connected condition on a switching graph. By exploiting duality, we further show that the output-based distributed observer design problem for a general leader system is solvable over jointly connected switching networks, even when the system matrix is exponentially unstable. This is also in sharp contrast to the existing distributed observers, which rely on the assumption that the leader system is marginally stable.

Leader-following Consensus over Jointly Connected Switching Networks is Achievable for Exponentially Unstable Linear Systems

TL;DR

This work resolves a long-standing limitation in leader-following consensus by proving exponential convergence for general linear agents over jointly connected switching networks even when is exponentially unstable. The authors derive a bound on the allowable instability using a projection-based analysis of the leader-follower matrix and construct a distributed stabilizing gain via a weighted controllability Gramian, with explicit lower bounds on nonzero eigenvalues of the switching matrix. A dual formulation yields an exponentially convergent output-based distributed observer for the leader, with parallel Gramian-based design for . Numerical examples corroborate the theory, showing convergence of follower states to the leader and convergence of the dual observer under the same connectivity and stability constraints. Overall, the paper substantially extends the scope of solvable leader-following problems in time-varying networks and enhances distributed observer design for unstable leaders.

Abstract

The leader-following consensus problem for general linear multi-agent systems over jointly connected switching networks has been a challenging problem and the solvability of the problem has been limited to the class of linear multi-agent systems whose system matrix is marginally stable. This condition is restrictive since it even excludes the most commonly used double-integrator system. This paper presents a breakthrough by demonstrating that leader-following exponential consensus is achievable for general linear multi-agent systems over jointly connected switching networks, even when the system matrix is exponentially unstable. The degree of instability can be explicitly characterized by two key quantities that arise from the jointly connected condition on a switching graph. By exploiting duality, we further show that the output-based distributed observer design problem for a general leader system is solvable over jointly connected switching networks, even when the system matrix is exponentially unstable. This is also in sharp contrast to the existing distributed observers, which rely on the assumption that the leader system is marginally stable.
Paper Structure (15 sections, 11 theorems, 112 equations, 7 figures)

This paper contains 15 sections, 11 theorems, 112 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Statement of Main Result
  4. Proof of Main Result
  5. Solution of Linear Switched System \ref{['eq-der-L0-otimes-x']}
  6. Design of Feedback Gain Matrix $K$
  7. Proof of Theorem \ref{['Theorem']}
  8. A Dual Problem
  9. Output-based Distributed Observer
  10. Dual System
  11. Solvability of Problem \ref{['Problem-observer']}
  12. Numerical Examples
  13. Conclusions
  14. Notation on Graph
  15. Proof of Lemma \ref{['lemma-ratio']}

Key Result

Lemma 1

Under Assumptions Ass-undirected and Ass-connected, there exists a number $0 < \delta < 1$ such that

Figures (7)

  • Figure 1: Switching graph $\bar{\mathcal{G}}_{\sigma(t)}$.
  • Figure 2: Profile on the first component of the states of the leader and followers.
  • Figure 3: Profile on the second component of the states of the leader and followers.
  • Figure 4: Profile on the third component of the states of the leader and followers.
  • Figure 5: Profile on the first component of the states of the leader and observers.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Theorem 1
  • Remark 4
  • Remark 5
  • Lemma 2
  • Remark 6
  • Lemma 3
  • ...and 12 more