Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft

James Biggs, William Holderbaum

Research output: Contribution to conferencePaper

Abstract

This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

Conference

Conference5th Wismar Symposium on Automatic Control, AUTSYM'08
CityWismar, Germany
Period18/09/0819/09/08

Fingerprint

Hamiltonians
Lie groups
attitude control
Attitude control
Spacecraft
control system
spacecraft
optimal control
algebra
Algebra
maximum principle
fold
Control systems
cost
Maximum principle
costs
Costs

Keywords

  • Hamiltonian systems
  • Lie groups
  • attitude control
  • spacecraft

Cite this

Biggs, J., & Holderbaum, W. (2008). Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft. Paper presented at 5th Wismar Symposium on Automatic Control, AUTSYM'08, Wismar, Germany, .
Biggs, James ; Holderbaum, William. / Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2) : an application to the attitude control of a spacecraft. Paper presented at 5th Wismar Symposium on Automatic Control, AUTSYM'08, Wismar, Germany, .
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abstract = "This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).",
keywords = "Hamiltonian systems, Lie groups, attitude control, spacecraft",
author = "James Biggs and William Holderbaum",
year = "2008",
language = "English",
note = "5th Wismar Symposium on Automatic Control, AUTSYM'08 ; Conference date: 18-09-2008 Through 19-09-2008",

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Biggs, J & Holderbaum, W 2008, 'Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft' Paper presented at 5th Wismar Symposium on Automatic Control, AUTSYM'08, Wismar, Germany, 18/09/08 - 19/09/08, .

Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2) : an application to the attitude control of a spacecraft. / Biggs, James; Holderbaum, William.

2008. Paper presented at 5th Wismar Symposium on Automatic Control, AUTSYM'08, Wismar, Germany, .

Research output: Contribution to conferencePaper

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T1 - Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2)

T2 - an application to the attitude control of a spacecraft

AU - Biggs, James

AU - Holderbaum, William

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N2 - This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

AB - This paper considers left-invariant control systems defined on the Lie groups SU(2) and SO(3). Such systems have a number of applications in both classical and quantum control problems. The purpose of this paper is two-fold. Firstly, the optimal control problem for a system varying on these Lie Groups, with cost that is quadratic in control is lifted to their Hamiltonian vector fields through the Maximum principle of optimal control and explicitly solved. Secondly, the control systems are integrated down to the level of the group to give the solutions for the optimal paths corresponding to the optimal controls. In addition it is shown here that integrating these equations on the Lie algebra su(2) gives simpler solutions than when these are integrated on the Lie algebra so(3).

KW - Hamiltonian systems

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KW - attitude control

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M3 - Paper

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Biggs J, Holderbaum W. Integrable Hamiltonian systems defined on the Lie groups SO(3) and SU(2): an application to the attitude control of a spacecraft. 2008. Paper presented at 5th Wismar Symposium on Automatic Control, AUTSYM'08, Wismar, Germany, .