Robust-Filtering of Sensor Data for the Finite Difference Model Reduction of a Piezoelectric Sandwich Beam

A space-discretized Finite Difference model reduction for the partial differential equations (PDE) of a piezoelectric sandwich beam with hinged boundary conditions is proposed. The PDE model is a Mead-Marcus type, and it describes transverse vibrations for a sandwich beam whose alternating outer (piezoelectric/elastic) layers constraining viscoelastic core layers which allow transverse the shear of the beam filaments. First, it is shown that the PDE model is exactly observable with a single boundary observer (sensor) design yet its Finite Difference model reduction is not able to retain the exact observability uniformly as the mesh parameter as h→0. This is mainly due to the spurious high-frequency eigenvalues prevent the sensor not being able to distinguish one vibrational frequency from another as h→0. For the robust and accurate design of the sensor, the low-frequency part of the solutions are controlled in order to eliminate the short wave-length (high-frequency) components of the solutions, the so-called direct Fourier filtering. After filtering, the uniform observability is recovered uniformly. The main hurdle in the proofs here is the coupling of the shear with bending dynamics different from a single-layer Euler-Bernoulli beam.