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High Performance Distributed Control for Large-Scale Linear Systems: A Cover-Based Distributed Observer Approach

Haotian Xu, Shuai Liu, Ling Shi

TL;DR

It is proved that the distributed control law based on the cover-based distributed observer can also arbitrarily approximate the control performance of the centralized control law, and the dimension of the local observer is greatly reduced compared with the traditional method.

Abstract

In recent years, the distributed-observer-based distributed control law has shown powerful ability to arbitrarily approximate the centralized control performance. However, the traditional distributed observer requires each local observer to reconstruct the state information of the whole system, which is unrealistic for large-scale scenarios. To fill this gap, This paper presents a coverage solution algorithm for large-scale systems that accounts for both physical and communication network characteristics, which can significantly reduce the dimension of local observers. Then, the cover-based distributed observer for large-scale systems is proposed to overcome the problem that the system dynamics are difficult to estimate due to the coupling between cover sets. Furthermore, the two-layer Lyapunov analysis method is adopted and the dynamic transformation lemma of compact errors is proved, which solves the problem of analyzing stability of the error dynamic of the cover-based distributed observer. Finally, it is proved that the distributed control law based on the cover-based distributed observer can also arbitrarily approximate the control performance of the centralized control law, and the dimension of the local observer is greatly reduced compared with the traditional method. The simulation results show the validity of the developed theories.

High Performance Distributed Control for Large-Scale Linear Systems: A Cover-Based Distributed Observer Approach

TL;DR

It is proved that the distributed control law based on the cover-based distributed observer can also arbitrarily approximate the control performance of the centralized control law, and the dimension of the local observer is greatly reduced compared with the traditional method.

Abstract

In recent years, the distributed-observer-based distributed control law has shown powerful ability to arbitrarily approximate the centralized control performance. However, the traditional distributed observer requires each local observer to reconstruct the state information of the whole system, which is unrealistic for large-scale scenarios. To fill this gap, This paper presents a coverage solution algorithm for large-scale systems that accounts for both physical and communication network characteristics, which can significantly reduce the dimension of local observers. Then, the cover-based distributed observer for large-scale systems is proposed to overcome the problem that the system dynamics are difficult to estimate due to the coupling between cover sets. Furthermore, the two-layer Lyapunov analysis method is adopted and the dynamic transformation lemma of compact errors is proved, which solves the problem of analyzing stability of the error dynamic of the cover-based distributed observer. Finally, it is proved that the distributed control law based on the cover-based distributed observer can also arbitrarily approximate the control performance of the centralized control law, and the dimension of the local observer is greatly reduced compared with the traditional method. The simulation results show the validity of the developed theories.
Paper Structure (17 sections, 8 theorems, 54 equations, 10 figures, 1 table, 2 algorithms)

This paper contains 17 sections, 8 theorems, 54 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Overall objectives
  4. An illustrative example for the target problem
  5. Design of cover-based distributed observer
  6. Cover sets selecting
  7. Cover-based distributed observer
  8. Distributed control law for large-scale linear systems
  9. Distributed control law
  10. Performance of cover-based distributed observer
  11. Closed-loop performance under the distributed control law
  12. The ability of performance recovery
  13. Simulation
  14. System formulation
  15. Effectiveness of coverage solving method
  16. ...and 2 more sections

Key Result

Lemma 1

Consider an undirected connected graph $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$. Let $\mathcal{S}=diag\{1,0,\ldots,0\}\in\mathbb{R}^{N\times N}$ and then all eigenvalues of matrix $\mathcal{H}=\mathcal{L}+\mathcal{S}$ locate at the open right half of plane, where $\mathcal{L}$ is the La

Figures (10)

  • Figure 1: Architecture diagram of distributed observer.
  • Figure 2: Illustrations of coverage issue in simple network systems.
  • Figure 3: Illustrations of covering in simple network systems.
  • Figure 4: Usage example of Algorithm \ref{['alg1']}: initial cover set. The part marked in red is the node with $\ell_i\neq 0$, which needs to consider whether to merge covers sets.
  • Figure 5: Usage example of Algorithm \ref{['alg1']}: Final coverage.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Lemma 1: Hong2008Distributed
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Remark 5
  • ...and 5 more