A New Semi-Discretization of the Fully Clamped Euler-Bernoulli Beam Preserving Boundary Observability Uniformly

• Published in: IEEE control systems letters 2024-12, p.1-1
• Main Authors: Aydin, Ahmet Kaan, Haider, Md Zulfiqur, Ozer, Ahmet Ozkan
• Format: Article
• Language: English
• Subjects: Average Operator; Boundary conditions; Discrete Multipliers; Eigenvalues and eigenfunctions; Euler-Bernoulli beam; Filtering; Finite difference methods; Finite Differences; Finite element analysis; Fourier series; Mathematical models; Observability; Uniform Observability; Viscosity
• ISSN: 2475-1456
• DOI: 10.1109/LCSYS.2024.3519379

This paper 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 Mathematik'06. Our reduced model is applicable to more complex systems involving Euler-Bernoulli beam equations.