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Performance bounds for multi-vehicle networks with local integrators

Jonas Hansson, Emma Tegling

TL;DR

This letter precisely characterize how a minimal condition number can be achieved in a collection of nth-order integrator systems, and generalizes these performance results to an arbitrary order.

Abstract

In this work, we consider the problem of coordinating a collection of $n$th-order integrator systems. The coordination is achieved through the novel serial-consensus design, which can be seen as a method for achieving a stable closed-loop while only using local relative measurements. Earlier work has shown that second-order serial consensus can stabilize a collection of double integrators with scalable performance conditions, independent of the number of agents and topology. In this paper, we generalize these performance results to an arbitrary order $n\geq 1$. The derived performance bound depends on the condition number, measured in the vector-induced maximum matrix norm, of a general diagonalizing matrix. We provide an exact characterization of how a minimal condition number can be achieved. Third-order serial consensus is illustrated through a case study of PI-controlled vehicular formation, where the added integrators are used to mitigate the effect of unmeasured load disturbances. The theoretical results are illustrated through examples.

Performance bounds for multi-vehicle networks with local integrators

TL;DR

This letter precisely characterize how a minimal condition number can be achieved in a collection of nth-order integrator systems, and generalizes these performance results to an arbitrary order.

Abstract

In this work, we consider the problem of coordinating a collection of th-order integrator systems. The coordination is achieved through the novel serial-consensus design, which can be seen as a method for achieving a stable closed-loop while only using local relative measurements. Earlier work has shown that second-order serial consensus can stabilize a collection of double integrators with scalable performance conditions, independent of the number of agents and topology. In this paper, we generalize these performance results to an arbitrary order . The derived performance bound depends on the condition number, measured in the vector-induced maximum matrix norm, of a general diagonalizing matrix. We provide an exact characterization of how a minimal condition number can be achieved. Third-order serial consensus is illustrated through a case study of PI-controlled vehicular formation, where the added integrators are used to mitigate the effect of unmeasured load disturbances. The theoretical results are illustrated through examples.

Paper Structure

This paper contains 13 sections, 3 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and network model
  4. Serial Consensus
  5. Scalable performance
  6. Main Results
  7. Case Study: PI-Control of Vehicle Formation
  8. Disturbance rejection
  9. Numerical Examples
  10. Minimizing norm
  11. PI control of vehicle formation
  12. Vehicle formation with load disturbance
  13. Discussion and Future Directions

Key Result

Theorem 1

The $n$th-order serial consensus system with $U(s)=0$, $p_k>0$ for all $k = 1,\ldots, n$, and $p_i\neq p_j$ for all $i\neq j$, with the states $\xi(t)=\left[(L^{n-1}x)^\top\!\!\!,\: (L^{n-2}\dot{x})^\top\!\!\!,\:(L^{n-3}\ddot{x})^\top\!\!\!,\:\cdots\!,\:(x^{(n-1)})^\top \right]^\top\!\!\!$ satisfy where $S$ is any invertible matrix that diagonlizes $A$ in eq:A.

Figures (3)

  • Figure 1: Illustration of vehicle formation control through serial consensus. In this example of a directed string formation, each vehicle measures the relative distance to its neighbor, and message passes its measurements to its follower.
  • Figure 2: Simulation of lead vehicle driving at velocity $v_\mathrm{ref}=10$ m/s and all other vehicles starting from a stand-still. Since the system has scalable performance, the transient error due to any initial condition will be bounded independent of the number of agents.
  • Figure 3: Simulation of $40$ agents driving uphill at an inclination ratio $\theta=0.1$, and with a leader velocity $10$ m/s. The agents that use a serial consensus-based PI controller experience a transient and then return to the desired spacing. The vehicles controlled with a proportional controller experience a transient before settling with stationary error.

Theorems & Definitions (11)

  • Definition 1: $n\textsuperscript{th}$-order serial consensus
  • Definition 2: Scalable Performance
  • Definition 3: Scalable $\mathcal{C}$-performance
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Example 1
  • Example 2
  • ...and 1 more