The equivalence of controlled Lagrangian and controlled Hamiltonian systems

Dong Eui Chang; Anthony M. Bloch; Naomi E. Leonard; Jerrold E. Marsden; Craig A. Woolsey

doi:10.1051/cocv:2002045

The equivalence of controlled Lagrangian and controlled Hamiltonian systems

Dong Eui Chang, Anthony M. Bloch, Naomi E. Leonard, Jerrold E. Marsden, Craig A. Woolsey

Research output: Contribution to journal › Article › peer-review

122 Scopus citations

Abstract

The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.

Original language	English (US)
Pages (from-to)	393-422
Number of pages	30
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	8
DOIs	https://doi.org/10.1051/cocv:2002045
State	Published - Jun 2002

All Science Journal Classification (ASJC) codes

Control and Systems Engineering
Control and Optimization
Computational Mathematics

Keywords

Controlled Hamiltonian
Controlled Lagrangian
Energy shaping
Equivalence
Lyapunov stability
Passivity

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10.1051/cocv:2002045

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abstract = "The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.",

keywords = "Controlled Hamiltonian, Controlled Lagrangian, Energy shaping, Equivalence, Lyapunov stability, Passivity",

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doi = "10.1051/cocv:2002045",

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T1 - The equivalence of controlled Lagrangian and controlled Hamiltonian systems

AU - Chang, Dong Eui

AU - Bloch, Anthony M.

AU - Leonard, Naomi E.

AU - Marsden, Jerrold E.

AU - Woolsey, Craig A.

PY - 2002/6

Y1 - 2002/6

N2 - The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.

AB - The purpose of this paper is to show that the method of controlled Lagrangians and its Hamiltonian counterpart (based on the notion of passivity) are equivalent under rather general hypotheses. We study the particular case of simple mechanical control systems (where the underlying Lagrangian is kinetic minus potential energy) subject to controls and external forces in some detail. The equivalence makes use of almost Poisson structures (Poisson brackets that may fail to satisfy the Jacobi identity) on the Hamiltonian side, which is the Hamiltonian counterpart of a class of gyroscopic forces on the Lagrangian side.

KW - Controlled Hamiltonian

KW - Controlled Lagrangian

KW - Energy shaping

KW - Equivalence

KW - Lyapunov stability

KW - Passivity

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SN - 1292-8119

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EP - 422

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

ER -

The equivalence of controlled Lagrangian and controlled Hamiltonian systems

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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