Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete-Time Stress Matrix-Based Formation Control of General Linear Multi-Agent Systems

Okechi Onuoha, Suleiman Kurawa, Zezhi Tang, Yi Dong

TL;DR

The paper addresses distributed affine formation control for discrete-time linear multi-agent systems with both stationary and dynamic leaders using a stress-matrix-based approach. It develops two protocols for single-integrator dynamics (stationary and dynamic leaders) and extends to general linear systems by selecting a stabilizing gain via a discrete-time algebraic Riccati equation, under the assumption of a universally rigid framework and appropriate rank conditions on the stress matrix $\Omega$. Sufficient conditions, including $T<2$ and $\text{rank}(\Omega)=n-d-1$, guarantee convergence of followers to the affine image $\mathcal{A}(r)$ of a nominal formation, as demonstrated by simulations. The work broadens affine formation control to discrete-time settings with time-varying leader velocities and general linear dynamics, enabling robust maneuvers such as scaling, translation, and rotation in practical multi-agent deployments.

Abstract

This paper considers the distributed leader-follower stress-matrix-based affine formation control problem of discrete-time linear multi-agent systems with static and dynamic leaders. In leader-follower multi-agent formation control, the aim is to drive a set of agents comprising leaders and followers to form any desired geometric pattern and simultaneously execute any required manoeuvre by controlling only a few agents denoted as leaders. Existing works in literature are mostly limited to the cases where the agents' inter-agent communications are either in the continuous-time settings or the sampled-data cases where the leaders are constrained to constant (or zero) velocities or accelerations. Here, we relax these constraints and study the discrete-time cases where the leaders can have stationary or time-varying velocities. We propose control laws in the study of different situations and provide some sufficient conditions to guarantee the overall system stability. Simulation study is used to demonstrate the efficacy of our proposed control laws.

Discrete-Time Stress Matrix-Based Formation Control of General Linear Multi-Agent Systems

TL;DR

The paper addresses distributed affine formation control for discrete-time linear multi-agent systems with both stationary and dynamic leaders using a stress-matrix-based approach. It develops two protocols for single-integrator dynamics (stationary and dynamic leaders) and extends to general linear systems by selecting a stabilizing gain via a discrete-time algebraic Riccati equation, under the assumption of a universally rigid framework and appropriate rank conditions on the stress matrix . Sufficient conditions, including and , guarantee convergence of followers to the affine image of a nominal formation, as demonstrated by simulations. The work broadens affine formation control to discrete-time settings with time-varying leader velocities and general linear dynamics, enabling robust maneuvers such as scaling, translation, and rotation in practical multi-agent deployments.

Abstract

This paper considers the distributed leader-follower stress-matrix-based affine formation control problem of discrete-time linear multi-agent systems with static and dynamic leaders. In leader-follower multi-agent formation control, the aim is to drive a set of agents comprising leaders and followers to form any desired geometric pattern and simultaneously execute any required manoeuvre by controlling only a few agents denoted as leaders. Existing works in literature are mostly limited to the cases where the agents' inter-agent communications are either in the continuous-time settings or the sampled-data cases where the leaders are constrained to constant (or zero) velocities or accelerations. Here, we relax these constraints and study the discrete-time cases where the leaders can have stationary or time-varying velocities. We propose control laws in the study of different situations and provide some sufficient conditions to guarantee the overall system stability. Simulation study is used to demonstrate the efficacy of our proposed control laws.
Paper Structure (15 sections, 6 theorems, 47 equations, 2 figures)

This paper contains 15 sections, 6 theorems, 47 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Basic Graph Notations
  4. Affine Span
  5. Stress Matrix
  6. Stress Matrix Design
  7. Leader Selection and Affine Localizability
  8. Problem Formulation
  9. Preamble
  10. Network of Integrator Cases
  11. Stationary Leaders
  12. Dynamic Leaders
  13. Simulation Study
  14. General Linear Systems
  15. Conclusion

Key Result

Lemma 1

a2a3a4: Given a framework ($\mathcal{G},p$) with an undirected communication graph, a generic configuration in $\mathbb{R} ^d$ and number of nodes $n \geq d+2$. The framework is universally rigid only if its communication graph is $(d+1)$-connected and has a stress matrix, $\Omega$ that has a rank o

Figures (2)

  • Figure S1: Framework depicting 5 agents communication together with their $2$-dimensional reference positions me_usme_icca. The agents communications are depicted with straight lines.
  • Figure S2: Illustration of required formation control accomplishment. Here, $\mu_{\min}$ is computed as $-1.49$. Control law \ref{['control law 1 equation']} is used for the simulation and $T=1$ is chosen. Leader agents $1-3$ were initialized to $P_1(1, 0), ~ P_2(0,1)~ \hbox{and} ~ P_3(0,-1)$ respectively. Similarly, the follower agents $4$ and $5$ were initialized to $P_4(-4,3) ~ \hbox{and} ~ P_5(-3,-2)$ respectively. The simulation was first allowed to run for 3 mins and then the data for the new positions of the agents were obtained. As expected, the followers agents correctly tracked their targets, i.e., agent 4 moved to positions (-1,0) and agent 5 moved to position (-2,0) as the position of the leaders remained unchanged. Clearly, this is the required reference configuration as shown in Figure \ref{['Sample Formation 5 Agents']}. After this is accomplished, the position of the leaders were then changed and the followers correctly tracked their their new corresponding positions after the movement of the leaders.

Theorems & Definitions (11)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Proof 1
  • Theorem 1
  • Proof 2
  • Theorem 2
  • Proof 3
  • Theorem 3
  • ...and 1 more