A New Semi-Discretization of the Fully Clamped Euler-Bernoulli Beam Preserving Boundary Observability Uniformly
Year: 2024
Type: article
Abstract: This letter extends a Finite Difference model reduction method to the Euler-Bernoulli beam equation with fully clamped boundary conditions. The corresponding partial differential equation (PDE) is exactly observable in the energy space with a single boundary observer in arbitrarily short observation times. However, standard Finite Difference spatial discretization fails to achieve uniform exact observability as the mesh parameter approaches zero, with minimal observation time potentially depending on the filtering parameter. To address this, we propose a Finite Difference algorithm incorporating an averaging operator and discrete multipliers, leveraging Haraux’s theorem on the spectral gap to ensure uniform observability. This approach eliminates the need for artificial viscosity or Fourier filtering. Our method achieves uniform observability for arbitrarily small times with dual observers-the tip moment and average tip velocity-mirroring results from mixed Finite Elements applied to the wave equation with homogeneous Dirichlet boundary conditions, where dual controllers converge to the single controller of the PDE model [Castro, Micu-Numerische Mathematiik’06]. Our reduced model is applicable to more complex systems involving Euler-Bernoulli beam equations.
DOI: 10.13016/m2i6ur-yamr
Authors: Aydin, Ahmet Kaan, Haider, Md Zulfiqur, Özkan Özer, Ahmet
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6884863392
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