Characterization of gradient control systems

Jorge Cortés; Arjan van der Schaft; Peter E. Crouch

doi:10.1137/S0363012903425568

Characterization of gradient control systems

Jorge Cortés, Arjan van der Schaft, Peter E. Crouch

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Abstract

Given a general nonlinear affine control system with outputs and a torsion-free affine connection defined on its state space, we investigate the gradient realization problem: we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo-Riemannian metric whose Levi-Civita connection is equal to the given affine connection. The results rely on a suitable notion of compatibility of the system with respect to the given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role throughout the discussion.

Original language	English
Pages (from-to)	1192-1214
Number of pages	23
Journal	SIAM journal on control and optimization
Volume	44
Issue number	4
DOIs	https://doi.org/10.1137/S0363012903425568
Publication status	Published - 2005

Keywords

Prolongation and gradient extension of a nonlinear system
Externally equivalent systems
Gradient control systems
Symmetric product

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Cortes2005characterizationFinal published version, 262 KB

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abstract = "Given a general nonlinear affine control system with outputs and a torsion-free affine connection defined on its state space, we investigate the gradient realization problem: we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo-Riemannian metric whose Levi-Civita connection is equal to the given affine connection. The results rely on a suitable notion of compatibility of the system with respect to the given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role throughout the discussion.",

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AU - Cortés, Jorge

AU - van der Schaft, Arjan

AU - Crouch, Peter E.

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AB - Given a general nonlinear affine control system with outputs and a torsion-free affine connection defined on its state space, we investigate the gradient realization problem: we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo-Riemannian metric whose Levi-Civita connection is equal to the given affine connection. The results rely on a suitable notion of compatibility of the system with respect to the given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role throughout the discussion.

KW - Prolongation and gradient extension of a nonlinear system

KW - Externally equivalent systems

KW - Gradient control systems

KW - Symmetric product

U2 - 10.1137/S0363012903425568

DO - 10.1137/S0363012903425568

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JO - SIAM journal on control and optimization

JF - SIAM journal on control and optimization

IS - 4

ER -

Characterization of gradient control systems

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