A Novel Finite Difference-Based Model Reduction and a Sensor Design for a Multilayer Smart Beam With Arbitrary Number of Layers

A Mead-Marcus type model describing the vibrations on a multilayer smart beam with arbitrary number of layers is considered with hinged boundary conditions. The model is known to be exactly observable in an appropriate Hilbert space with a single boundary sensor measurement. As a standard Finite Differences-based model reduction is considered, it is proved that the model reduction lacks exact observability uniformly as the mesh parameter goes to zero,
$h\to 0$
. This is a known phenomenon caused by spurious (artificial) high-frequency eigenvalues. First, it is proved that the exact observability can be retained by the implementation of the direct Fourier filtering technique. However, the optimality of the applied filtering demands further investigation. For this reason, an alternate model reduction is investigated by cleverly reducing the order of the model together with the consideration of equidistant grid points and averaging operators, as in Liu and Guo (2019, 2020, and 2021). This new model reduction successfully retains the exact observability uniformly as
$h\to 0$
. Moreover, it does not need a further numerical filtering. Our results are based on carefully analyzing the spectrum of the system matrix, and they are applicable to the standard Euler-Bernoulli and Rayleigh beam equations. The numerical simulations are provided to compare reduced models and to show the strength of introduced results.