2 research outputs found
Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model
A magnetizable piezoelectric beam model, free at both ends, is considered.
Piezoelectric materials have a strong interaction of electromagnetic and
acoustic waves, whose wave propagation speeds differ substantially. The
corresponding strongly-coupled PDE model describes the longitudinal vibrations
and the total charge accumulation at the electrodes of the beam. It is known
that the PDE model with appropriately chosen collocated state feedback
controllers is known to have exponentially stable solutions. However, the
collocated controller design is not always feasible since the performance of
controllers may not be good enough, and moreover, a small increment of feedback
controller gains can easily make the closed-loop system unstable. Therefore, a
non-collocated controller and observer design is considered for the first time
for this model. In particular, two state feedback controllers are designed at
the right end to recover the states so that the boundary output feedback
controllers can be designed as a replacement of the states with the estimate
from the observers on the left end. By a carefully-constructed Lyapunov
function, it is proved that the both the observer and the observer error
dynamics have uniformly exponential stable solutions. This framework offers a
substantial foundation for the model reductions by Finite Differences.Comment: 2 figure
The Exponential Stabilization of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers
This study investigates a strongly-coupled system of partial differential
equations (PDE) governing heat transfer in a copper rod, longitudinal
vibrations, and total charge accumulation at electrodes within a magnetizable
piezoelectric beam. Conducted within the transmission line framework, the
analysis reveals profound interactions between traveling electromagnetic and
mechanical waves in magnetizable piezoelectric beams, despite disparities in
their velocities. Findings suggest that in the open-loop scenario, the
interaction of heat and beam dynamics lacks exponential stability solely
considering thermal effects. To confront this challenge, two types of
boundary-type state feedback controllers are proposed: (i) employing static
feedback controllers entirely and (ii) adopting a hybrid approach wherein the
electrical controller dynamically enhances system dynamics. In both cases,
solutions of the PDE systems demonstrate exponential stability through
meticulously formulated Lyapunov functions with diverse multipliers. The
proposed proof technique establishes a robust foundation for demonstrating the
exponential stability of Finite-Difference-based model reductions as the
discretization parameter approaches zero.Comment: 1 figur