An optimal gains matrix for time-delay feedback control

James Biggs, Colin R. McInnes

Research output: Contribution to conferencePaper

Abstract

In this paper we propose an optimal time-delayed feedback control (TDFC) for tracking unstable periodic orbits (UPOs). It is shown that TDFC will drive a trajectory onto a periodic orbit while minimising an integral of a cost function of the error in periodicity and the control e®ort. This optimal TDFC relies on the linearisation about the delayed trajectory not the UPO itself and therefore can be implemented without a priori knowledge of a reference orbit. This optimal TDFC is applied to the problem of tracking an unstable periodic orbit in the nonlinear equations describing the circular restricted three-body problem. The results of this investigation demonstrate that TDFC could e±ciently drive a spacecraft onto a periodic orbit in the vicinity of a (UPO) halo orbit.

Conference

Conference2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09)
CityLondon, England
Period22/06/0924/06/09

Fingerprint

Delayed Feedback Control
Periodic Orbits
Feedback Control
Feedback control
Time Delay
Time delay
Orbits
Unstable
Trajectory
Restricted Three-body Problem
Trajectories
Spacecraft
Periodicity
Linearization
Cost Function
Nonlinear Equations
Nonlinear equations
Cost functions
Orbit
Demonstrate

Keywords

  • feedback control
  • chaos theory
  • control systems
  • orbits
  • periodic orbits

Cite this

Biggs, J., & McInnes, C. R. (2009). An optimal gains matrix for time-delay feedback control. Paper presented at 2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09), London, England, . https://doi.org/10.3182/20090622-3-UK-3004.00063
Biggs, James ; McInnes, Colin R. / An optimal gains matrix for time-delay feedback control. Paper presented at 2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09), London, England, .4 p.
@conference{8c899836dd5d410799f1040981fbee50,
title = "An optimal gains matrix for time-delay feedback control",
abstract = "In this paper we propose an optimal time-delayed feedback control (TDFC) for tracking unstable periodic orbits (UPOs). It is shown that TDFC will drive a trajectory onto a periodic orbit while minimising an integral of a cost function of the error in periodicity and the control e{\circledR}ort. This optimal TDFC relies on the linearisation about the delayed trajectory not the UPO itself and therefore can be implemented without a priori knowledge of a reference orbit. This optimal TDFC is applied to the problem of tracking an unstable periodic orbit in the nonlinear equations describing the circular restricted three-body problem. The results of this investigation demonstrate that TDFC could e±ciently drive a spacecraft onto a periodic orbit in the vicinity of a (UPO) halo orbit.",
keywords = "feedback control, chaos theory, control systems, orbits, periodic orbits",
author = "James Biggs and McInnes, {Colin R.}",
year = "2009",
month = "6",
day = "22",
doi = "10.3182/20090622-3-UK-3004.00063",
language = "English",
note = "2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09) ; Conference date: 22-06-2009 Through 24-06-2009",

}

Biggs, J & McInnes, CR 2009, 'An optimal gains matrix for time-delay feedback control' Paper presented at 2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09), London, England, 22/06/09 - 24/06/09, . https://doi.org/10.3182/20090622-3-UK-3004.00063

An optimal gains matrix for time-delay feedback control. / Biggs, James; McInnes, Colin R.

2009. Paper presented at 2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09), London, England, .

Research output: Contribution to conferencePaper

TY - CONF

T1 - An optimal gains matrix for time-delay feedback control

AU - Biggs, James

AU - McInnes, Colin R.

PY - 2009/6/22

Y1 - 2009/6/22

N2 - In this paper we propose an optimal time-delayed feedback control (TDFC) for tracking unstable periodic orbits (UPOs). It is shown that TDFC will drive a trajectory onto a periodic orbit while minimising an integral of a cost function of the error in periodicity and the control e®ort. This optimal TDFC relies on the linearisation about the delayed trajectory not the UPO itself and therefore can be implemented without a priori knowledge of a reference orbit. This optimal TDFC is applied to the problem of tracking an unstable periodic orbit in the nonlinear equations describing the circular restricted three-body problem. The results of this investigation demonstrate that TDFC could e±ciently drive a spacecraft onto a periodic orbit in the vicinity of a (UPO) halo orbit.

AB - In this paper we propose an optimal time-delayed feedback control (TDFC) for tracking unstable periodic orbits (UPOs). It is shown that TDFC will drive a trajectory onto a periodic orbit while minimising an integral of a cost function of the error in periodicity and the control e®ort. This optimal TDFC relies on the linearisation about the delayed trajectory not the UPO itself and therefore can be implemented without a priori knowledge of a reference orbit. This optimal TDFC is applied to the problem of tracking an unstable periodic orbit in the nonlinear equations describing the circular restricted three-body problem. The results of this investigation demonstrate that TDFC could e±ciently drive a spacecraft onto a periodic orbit in the vicinity of a (UPO) halo orbit.

KW - feedback control

KW - chaos theory

KW - control systems

KW - orbits

KW - periodic orbits

U2 - 10.3182/20090622-3-UK-3004.00063

DO - 10.3182/20090622-3-UK-3004.00063

M3 - Paper

ER -

Biggs J, McInnes CR. An optimal gains matrix for time-delay feedback control. 2009. Paper presented at 2nd IFAC Conference on the Analysis and Control of Chaotic Systems (CHAOS 09), London, England, . https://doi.org/10.3182/20090622-3-UK-3004.00063