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An Algorithm for Distributed Computation of Reachable Sets for Multi-Agent Systems

Omanshu Thapliyal, Shanelle Clarke, Inseok Hwang

TL;DR

This paper utilizes the ideas of polytopic reachable set approximation and generalize them to a multi-agent systems (MAS) setup, and forms the resulting sub-problems in a fully distributed manner and provides convergence guarantees for the associated computations.

Abstract

In this paper, we consider the problem of distributed reachable set computation for multi-agent systems (MASs) interacting over an undirected, stationary graph. A full state-feedback control input for such MASs depends no only on the current agent's state, but also of its neighbors. However, in most MAS applications, the dynamics are obscured by individual agents. This makes reachable set computation, in a fully distributed manner, a challenging problem. We utilize the ideas of polytopic reachable set approximation and generalize it to a MAS setup. We formulate the resulting sub-problems in a fully distributed manner and provide convergence guarantees for the associated computations. The proposed algorithm's convergence is proved for two cases: static MAS graphs, and time-varying graphs under certain restrictions.

An Algorithm for Distributed Computation of Reachable Sets for Multi-Agent Systems

TL;DR

This paper utilizes the ideas of polytopic reachable set approximation and generalize them to a multi-agent systems (MAS) setup, and forms the resulting sub-problems in a fully distributed manner and provides convergence guarantees for the associated computations.

Abstract

In this paper, we consider the problem of distributed reachable set computation for multi-agent systems (MASs) interacting over an undirected, stationary graph. A full state-feedback control input for such MASs depends no only on the current agent's state, but also of its neighbors. However, in most MAS applications, the dynamics are obscured by individual agents. This makes reachable set computation, in a fully distributed manner, a challenging problem. We utilize the ideas of polytopic reachable set approximation and generalize it to a MAS setup. We formulate the resulting sub-problems in a fully distributed manner and provide convergence guarantees for the associated computations. The proposed algorithm's convergence is proved for two cases: static MAS graphs, and time-varying graphs under certain restrictions.
Paper Structure (9 sections, 4 theorems, 19 equations, 4 figures, 1 algorithm)

This paper contains 9 sections, 4 theorems, 19 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Distributed Reachable Set Computation
  4. Centralized Reachable Set Computation
  5. Distributed Reachable Set Algorithm
  6. Convergence Discussion
  7. Convergence of Algorithm \ref{['algo:algo1']} for static graphs
  8. Convergence of Algorithm \ref{['algo:algo1']} for Time-Varying Graphs
  9. Conclusions

Key Result

Lemma 1

The admissible exogenous input set $\mathcal{W}$ is convex and bounded.

Figures (4)

  • Figure 1: Dynamical coupling of reachable sets in an MAS
  • Figure 2: Polytopic Approximation of Reachable Set: Evolution of a selected Hyperplane
  • Figure 3: (a) Distributed, and (b) centralized structures of the polytope
  • Figure 4: Distribution of system matrices information across the agents

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 1: Polytopic Reachability varaiya2000reach
  • proof
  • Remark 1: Tightness of Approximation in (\ref{['eq:outer-inner-approx']})
  • Lemma 2
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2
  • ...and 2 more