Optimal control of a beam with Kelvin–Voigt damping subject to forced vibrations using a piezoelectric patch actuator

A maximum principle is derived for the optimal control of a beam with Kelvin–Voigt damping subject to an external excitation with the control exercised by means of piezoelectric patch actuators. The objective functional is defined as a weighted quadratic functional of the displacement and velocity w...

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Bibliographic Details
Published in:Journal of vibration and control 2015-03, Vol.21 (4), p.701-713
Main Authors: Kucuk, Ismail, Yildirim, Kenan, Sadek, Ibrahim, Adali, Sarp
Format: Article
Language:English
Subjects:
Actuators
Adjoints
Damping
Ferroelectrics
Mathematical models
Maximum principle
Optimal control
Optimization
Partial differential equations
Piezoelectricity
Quantum physics
Terminals
Vibration
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Summary:A maximum principle is derived for the optimal control of a beam with Kelvin–Voigt damping subject to an external excitation with the control exercised by means of piezoelectric patch actuators. The objective functional is defined as a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time. A penalty term is also part of the objective functional defined as the control voltage used in the control process. The maximum principle makes use of a Hamiltonian defined in terms of an adjoint variable and the control function. The optimal control problem is expressed as a coupled system of partial differential equations in terms of state, adjoint and control variables subject to boundary, initial and terminal conditions. The solution is obtained by expanding the state and adjoint variables in terms of eigenfunctions and determining the optimal control using the maximum principle. Numerical examples are given to demonstrate the applicability and the efficiency of the proposed method.
ISSN:1077-5463
1741-2986
DOI:10.1177/1077546313488617