Finite-dimensional approximate modeling with error bounds of flexible vibrating systems based on partial eigenstructures

Jun Imai, Kiyoshi Wada

Research output: Contribution to journalArticle

Abstract

An approach to control-oriented uncertainty modeling is presented for a class of elastic vibrating systems such as flexible structures, beams and strings, described by partial differential equations. Uncertainty bounding techniques are developed using the upper and lower bounds of the unknown eigenparameters. The result forms a basis for a finite-dimensional controller design in which closed loop stability and performance are guaranteed. A feasible set of systems is defined of all systems governed by a class of differential equations with certain norm bounds of the unknown input and output operators and with partially known bounds of the eigenparameters. Then the perturbation magnitude covering the feasible set is evaluated in the frequency domain where a standard truncated modal model is chosen as the nominal one. An upper bound to the truncated error magnitude which is calculated by linear programming is proposed. It is demonstrated that all the parameters formulating a feasible set are derived by finite element analysis for a flexible beam example, and feasibility of the proposed scheme is also illustrated by numerical bounding resulte.

Original languageEnglish
Pages (from-to)36-44
Number of pages9
JournalElectrical Engineering in Japan (English translation of Denki Gakkai Ronbunshi)
Volume155
Issue number2
DOIs
Publication statusPublished - Apr 30 2006

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Flexible structures
Linear programming
Partial differential equations
Differential equations
Finite element method
Controllers
Uncertainty

Keywords

  • Controller design
  • Elastic systems
  • Finite-di-mensional approximation
  • Modal representation
  • Partial differential equations
  • Spectral systems

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

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title = "Finite-dimensional approximate modeling with error bounds of flexible vibrating systems based on partial eigenstructures",
abstract = "An approach to control-oriented uncertainty modeling is presented for a class of elastic vibrating systems such as flexible structures, beams and strings, described by partial differential equations. Uncertainty bounding techniques are developed using the upper and lower bounds of the unknown eigenparameters. The result forms a basis for a finite-dimensional controller design in which closed loop stability and performance are guaranteed. A feasible set of systems is defined of all systems governed by a class of differential equations with certain norm bounds of the unknown input and output operators and with partially known bounds of the eigenparameters. Then the perturbation magnitude covering the feasible set is evaluated in the frequency domain where a standard truncated modal model is chosen as the nominal one. An upper bound to the truncated error magnitude which is calculated by linear programming is proposed. It is demonstrated that all the parameters formulating a feasible set are derived by finite element analysis for a flexible beam example, and feasibility of the proposed scheme is also illustrated by numerical bounding resulte.",
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N2 - An approach to control-oriented uncertainty modeling is presented for a class of elastic vibrating systems such as flexible structures, beams and strings, described by partial differential equations. Uncertainty bounding techniques are developed using the upper and lower bounds of the unknown eigenparameters. The result forms a basis for a finite-dimensional controller design in which closed loop stability and performance are guaranteed. A feasible set of systems is defined of all systems governed by a class of differential equations with certain norm bounds of the unknown input and output operators and with partially known bounds of the eigenparameters. Then the perturbation magnitude covering the feasible set is evaluated in the frequency domain where a standard truncated modal model is chosen as the nominal one. An upper bound to the truncated error magnitude which is calculated by linear programming is proposed. It is demonstrated that all the parameters formulating a feasible set are derived by finite element analysis for a flexible beam example, and feasibility of the proposed scheme is also illustrated by numerical bounding resulte.

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KW - Spectral systems

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