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Optimal intrinsic formation using exogenous systems

Yueyue Xu, Panpan Zhou, Lin Wang, Xiaoming Hu

TL;DR

The paper tackles intrinsic formation control for multi-agent systems by embedding an exogenous system into an infinite-horizon linear-quadratic framework, enabling convergence to formations without explicit formation-error terms in the cost. It derives conditions for nonzero steady states, characterizes the steady-state and maximal steady-state spaces, and then solves the inverse problem by designing the input matrix and exogenous system (with minimal dimension) so that a desired formation is achieved within the maximal space. It also provides a method to set the initial exogenous state to realize formation scaling, and validates the theory through illustrative 3D four-agent simulations that show secure, non-shortest-path convergence to line, square, and tetrahedral formations. Collectively, the approach extends intrinsic control to essentially arbitrary formations in any dimension while offering security benefits and design flexibility for large-scale multi-agent networks.

Abstract

This paper investigates the intrinsic formation problem of a multi-agent system using an exogenous system. The problem is formulated as an intrinsic infinite time-horizon linear quadratic optimal control problem, namely, no formation error information is incorporated in the performance index. Convergence to the formation is achieved by utilizing an exogenous system, thus expanding the steady-state formation space of the system. For the forward problem, we provide the existence condition for a nonzero steady state and characterize the steady-state space. For the inverse problem, we design both the input matrix and the exogenous system so that the desired formation can be achieved. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed results.

Optimal intrinsic formation using exogenous systems

TL;DR

The paper tackles intrinsic formation control for multi-agent systems by embedding an exogenous system into an infinite-horizon linear-quadratic framework, enabling convergence to formations without explicit formation-error terms in the cost. It derives conditions for nonzero steady states, characterizes the steady-state and maximal steady-state spaces, and then solves the inverse problem by designing the input matrix and exogenous system (with minimal dimension) so that a desired formation is achieved within the maximal space. It also provides a method to set the initial exogenous state to realize formation scaling, and validates the theory through illustrative 3D four-agent simulations that show secure, non-shortest-path convergence to line, square, and tetrahedral formations. Collectively, the approach extends intrinsic control to essentially arbitrary formations in any dimension while offering security benefits and design flexibility for large-scale multi-agent networks.

Abstract

This paper investigates the intrinsic formation problem of a multi-agent system using an exogenous system. The problem is formulated as an intrinsic infinite time-horizon linear quadratic optimal control problem, namely, no formation error information is incorporated in the performance index. Convergence to the formation is achieved by utilizing an exogenous system, thus expanding the steady-state formation space of the system. For the forward problem, we provide the existence condition for a nonzero steady state and characterize the steady-state space. For the inverse problem, we design both the input matrix and the exogenous system so that the desired formation can be achieved. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed results.

Paper Structure

This paper contains 12 sections, 9 theorems, 78 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Forward problem: steady state description
  4. Existence condition of a nonzero steady state
  5. Steady-state space
  6. Maximal steady-state space
  7. Inverse problem: system design
  8. Step 1: design of the input matrix $B$
  9. Step 2: design the exogenous system and the performance index
  10. Step 3: design the initial state of the exogenous system
  11. Illustrative examples
  12. Conclusion

Key Result

Lemma 1

(cf. kuvcera1973reviewTrentelman2001) Suppose $(\bar{A}, \bar{B})$ is stabilizable and $\bar{Q}$ is positive semi-definite, then there exists a smallest real symmetric positive semi-definite solution $P$ for the algebraic Riccati equation Rieq. Furthermore, is the optimal solution of the linear quadratic optimal control problem eq_optimal. The closed-loop matrix $\bar{A}_{cl}$ is stable if and on

Figures (8)

  • Figure 1: Evolution of states for line formation $x_{df}^1$
  • Figure 2: Trajectories of agents for line formation $x_{df}^1$
  • Figure 3: Evolution of states for square formation $x_{df}^2$
  • Figure 4: Trajectories of agents for square formation $x_{df}^2$
  • Figure 5: Evolution of states for tetrahedron formation $x_{df}^3$
  • ...and 3 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 3
  • ...and 7 more