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Minimum time consensus for damped second order agents using Gröbner basis

Akansha Rautela, Deepak U. Patil, Ameer Mulla, Indra Narayan Kar

TL;DR

The paper addresses minimum-time consensus for a network of identical damped second-order LTI agents under input bounds and a fixed fuel budget. It develops a set-based approach by characterizing attainable sets $\mathcal{A}_i^\beta(t_f,\mathbf{x}_{0,i})$ and their boundaries via Gröbner-basis analysis, then leverages Helly's theorem to reduce the $N$-agent problem to tractable triplet computations. The main contributions are a constructive procedure to compute the minimum consensus time and corresponding consensus point, a boundary-based algebraic description of attainable sets, and a region-of-consensus criterion that links initial spread to fuel constraints. The results enable distributed computation of optimal consensus in connected networks and offer a path toward extending the framework to higher-order systems with damping and fuel considerations.

Abstract

A problem of achieving minimum time consensus for a set of $N$ second-order LTI system agents with bounded inputs and fuel constraints is considered. Unlike our other works, here the damping effect in agent dynamics is included. First, the attainable set for each agent with fuel budget constraints is characterized, and its boundary equations are derived. Then, using the convexity property, the minimum time at which attainable sets of all agents have a non-empty intersection is computed. By applying Helly's theorem, the computation reduces to finding the minimum time to consensus and the corresponding consensus point for each of the triplets separately.

Minimum time consensus for damped second order agents using Gröbner basis

TL;DR

The paper addresses minimum-time consensus for a network of identical damped second-order LTI agents under input bounds and a fixed fuel budget. It develops a set-based approach by characterizing attainable sets and their boundaries via Gröbner-basis analysis, then leverages Helly's theorem to reduce the -agent problem to tractable triplet computations. The main contributions are a constructive procedure to compute the minimum consensus time and corresponding consensus point, a boundary-based algebraic description of attainable sets, and a region-of-consensus criterion that links initial spread to fuel constraints. The results enable distributed computation of optimal consensus in connected networks and offer a path toward extending the framework to higher-order systems with damping and fuel considerations.

Abstract

A problem of achieving minimum time consensus for a set of second-order LTI system agents with bounded inputs and fuel constraints is considered. Unlike our other works, here the damping effect in agent dynamics is included. First, the attainable set for each agent with fuel budget constraints is characterized, and its boundary equations are derived. Then, using the convexity property, the minimum time at which attainable sets of all agents have a non-empty intersection is computed. By applying Helly's theorem, the computation reduces to finding the minimum time to consensus and the corresponding consensus point for each of the triplets separately.

Paper Structure

This paper contains 13 sections, 8 theorems, 32 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Problem Formulation
  4. Preliminaries
  5. Attainable Set Characterization
  6. Region of Consensus
  7. Minimum Time Consensus
  8. Minimum time consensus for a triplet
  9. $\partial{\mathcal{A}}_{i}^{\beta,s_1} \cap \partial{\mathcal{A}}_{j}^{\beta,s_3} \cap \partial{\mathcal{A}}_{k}^{\beta,s_1}$
  10. Scenario 7: $\partial{\mathcal{A}}_{i}^{\beta_2,s_3} \cap \partial{\mathcal{A}}_{j}^{\beta_2,s_2} \cap \partial{\mathcal{A}}_{k}^{\beta_2,s_1}$
  11. Minimum time consensus for N agents
  12. Example
  13. Conclusion

Key Result

Lemma 1

athans2013optimalHjek1979L1optimizationIL For the system given by n-agents, control ${u}_i(t)$ that steers state-trajectory $\mathbf{x}_i(t)$ of agent $a_i$ from an initial condition $\mathbf{x}_{0,i} \in \mathbb{R}^2$ to the origin with minimum fuel i.e., $F(u_i(t),t_f)$ and in finite time $t_f$ is where $\gamma_l \in \{+1,-1\}$ for $l=1,2$.

Figures (4)

  • Figure 1: Attainable set
  • Figure 2: Region of Consensus
  • Figure 3: Intersection of Attainable sets
  • Figure 4: State-trajectories, control profile and attainable set intersection for $N=6$ agents

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 6 more