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Optimal Network Topology of Multi-Agent Systems subject to Computation and Communication Latency (with proofs)

Luca Ballotta, Mihailo R. Jovanović, Luca Schenato

TL;DR

It is proved that such a control design can be solved efficiently for circular formations and compute near-optimal control gains in closed form and it is shown that the centralized control is in general a poor design choice when adding communication links to the network increases the latency.

Abstract

We study minimum-variance feedback-control design for a networked control system with retarded dynamics, where inter-agent communication is subject to latency. We prove that such a design can be solved efficiently for circular formations and compute near-optimal control gains in closed form. We show that the centralized control architecture is in general suboptimal when the communication increase with the number of links, and propose a control-driven optimization of the network topology.

Optimal Network Topology of Multi-Agent Systems subject to Computation and Communication Latency (with proofs)

TL;DR

It is proved that such a control design can be solved efficiently for circular formations and compute near-optimal control gains in closed form and it is shown that the centralized control is in general a poor design choice when adding communication links to the network increases the latency.

Abstract

We study minimum-variance feedback-control design for a networked control system with retarded dynamics, where inter-agent communication is subject to latency. We prove that such a design can be solved efficiently for circular formations and compute near-optimal control gains in closed form. We show that the centralized control architecture is in general suboptimal when the communication increase with the number of links, and propose a control-driven optimization of the network topology.

Paper Structure

This paper contains 16 sections, 6 theorems, 52 equations, 9 figures.

Table of Contents

  1. Introduction
  2. contribution and paper outline
  3. setup
  4. system model
  5. decoupling the error dynamics
  6. minimum-variance control of scalar delay systems
  7. Optimization of feedback gains
  8. Single parameter
  9. multiple parameters
  10. discussion: the role of latency in decentralized control
  11. Conclusions
  12. Proof of \ref{['lemma:convexVariance']}
  13. Proof of \ref{['prop:optimumGain']}
  14. Proof of \ref{['thm:AlphaSuboptClosedForm']}
  15. Proof of \ref{['thm:suboptKClosedForm']}
  16. ...and 1 more sections

Key Result

Theorem 1

Eq. eq:retarded-diff-eq admits a steady-state solution $x_{\textit{ss}}(t)$ if and only if If it exists, $x_{\textit{ss}}(t)$ is unique and it is a zero-mean Gaussian process with variance where $x_{d}(t)$ is the so-called fundamental solution of the deterministic equation corresponding to eq:retarded-diff-eq: If eq:retarded-eq-steady-state-condition holds, $x_{d}(t)$ and $\dot{x}_d(t)$ are exp

Figures (9)

  • Figure 1: Variance $\sigma^2_{\textit{ss}}$ as a function of the gain $\lambda$ ($\tau = 1$).
  • Figure 2: Minimum variance $\sigma^{2,*}_{\textit{ss}}$ as a function of $\tau$.
  • Figure 3: Optimal gain $\lambda^{*}$ as a function of the delay $\tau$.
  • Figure 4: Cost functions in \ref{['eq:problem-recast']} and in \ref{['eq:problem-recast-eig-squares']} about the minimum.
  • Figure 5: Delay rate $f(n) = n$.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1: KuchlerLangevinEqs
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 2: Control regularization
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 5 more