1 / 16

Undirected Formations

This document presents findings from the EECI Graduate School at Supelec on control undirected formations, particularly in the context of stability analysis for error systems. It explores the conditions for exponential stability in perturbed error systems starting from an equilibrium state. Through rigorous mathematical treatment, the study discusses the nature of equilibrium states, the behavior of associated norm constants, and the implications of sinusoidal dependencies among system variables. Results are derived considering infinitesimal rigidity and linearly independent configurations.

yael
Télécharger la présentation

Undirected Formations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Supelec EECI Graduate School in Control Undirected Formations A. S. Morse Yale University Gif – sur - Yvette May 24, 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAA

  2. 2 d1 1 d2 d3 3

  3. 1 3.989 4 6.5 6.509 2 3 3.003 3

  4. period = 270 seconds

  5. Let q be any positive number such that Let e(t) be the solution to the unperturbed error system starting at some state e(0) 2B. Then e(t) ! 0 as fast as e-¸ t! 0.

  6. There exists an open ball Babout e = 0 in R3 and a vector q(¹) 2 R3 depending continuously on ¹such that q(0) =0 and for every ¹ 2 B, q(¹) is an exponentially stable equilibrium of the perturbed error system Suppose error system is in an equilibrium state e = q(¹). Therefore the norm of each zi is constant. Therefore each z_i must be a constant or a linear combination of sinusoids. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

  7. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

  8. Suppose z is not in N. Then z1 andz2 are linearly independent. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

  9. Suppose error system is in an equilibrium state e = q(¹) and . . Then either

  10. Main Results for Triangles Pick ¹ so that ¹1 + ¹2 + ¹3 0 and so that ||¹|| is small enough so that {x,G} is infinitesimally rigid for all x in the set Suppose the error system is in equilibrium at e = q(¹). Then z is not in N and each ||zi||2= qi(¹) + di2 each zi is a linear combination of sinusoids each zi is nonconstant each zi2R2 not true in R3 So each xi(t) is also sinusoidal at frequency !

More Related