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Pattern formation using an intrinsic optimal control approach

Tianhao Li, Yibei Li, Zhixin Liu, Xiaoming Hu

TL;DR

This work addresses pattern formation for a Laplacian multi-agent system where only a subset of agents are actuated as leaders. It formulates an intrinsic infinite-horizon linear-quadratic problem with an integrator to drive the system toward a desired spatial pattern $\mathcal{SP}(\alpha)$, rather than a fixed point, by selecting $Q = D+\mathcal{A}^{(1)}(\alpha)-\mathcal{A}^{(2)}(\alpha)$ and solving a Riccati-based controller. It shows the existence of an optimal control and proves convergence to the pattern under mild assumptions, then extends the design to a distributed setting using distributed observers so each leader can implement the controller with only local information. Numerical simulations on grid graphs illustrate both centralized and distributed strategies achieving the target pattern, highlighting the practical viability of the approach.

Abstract

This paper investigates a pattern formation control problem for a multi-agent system modeled with given interaction topology, in which $m$ of the $n$ agents are chosen as leaders and consequently a control signal is added to each of the leaders. These agents interact with each other by Laplacian dynamics on a graph. The pattern formation control problem is formulated as an intrinsic infinite time-horizon linear quadratic optimal control problem, namely, no error information is incorporated in the objective function. Under mild conditions, we show the existence of the optimal control strategy and the convergence to the desired pattern formation. Based on the optimal control strategy, we propose a distributed control strategy to achieve the given pattern. Finally, numerical simulation is given to illustrate theoretical results.

Pattern formation using an intrinsic optimal control approach

TL;DR

This work addresses pattern formation for a Laplacian multi-agent system where only a subset of agents are actuated as leaders. It formulates an intrinsic infinite-horizon linear-quadratic problem with an integrator to drive the system toward a desired spatial pattern , rather than a fixed point, by selecting and solving a Riccati-based controller. It shows the existence of an optimal control and proves convergence to the pattern under mild assumptions, then extends the design to a distributed setting using distributed observers so each leader can implement the controller with only local information. Numerical simulations on grid graphs illustrate both centralized and distributed strategies achieving the target pattern, highlighting the practical viability of the approach.

Abstract

This paper investigates a pattern formation control problem for a multi-agent system modeled with given interaction topology, in which of the agents are chosen as leaders and consequently a control signal is added to each of the leaders. These agents interact with each other by Laplacian dynamics on a graph. The pattern formation control problem is formulated as an intrinsic infinite time-horizon linear quadratic optimal control problem, namely, no error information is incorporated in the objective function. Under mild conditions, we show the existence of the optimal control strategy and the convergence to the desired pattern formation. Based on the optimal control strategy, we propose a distributed control strategy to achieve the given pattern. Finally, numerical simulation is given to illustrate theoretical results.
Paper Structure (6 sections, 7 theorems, 68 equations, 4 figures)

This paper contains 6 sections, 7 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The optimal control strategy
  4. Distributed control strategy
  5. Numerical examples
  6. Concluding remarks

Key Result

Lemma 1

Under Assumption assumption1, we have $\mathcal{SP}(\alpha)=\mathcal{S}(\alpha)$.

Figures (4)

  • Figure 1: The figures of patterns in $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$.
  • Figure 2: The figures of graph and the target pattern.
  • Figure 3: The state of the system (\ref{['eq1']}) controlled by centralized control strategy.
  • Figure 4: The state of the system (\ref{['eq1']}) controlled by distributed control strategy.

Theorems & Definitions (11)

  • Definition 1
  • Example 1
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Remark 2
  • Theorem 2
  • ...and 1 more