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Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model

Ahmet Ozkan Ozer, Uthman Rasaq, Ibrahim Khalilullah

Abstract

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.

Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model

Abstract

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.
Paper Structure (9 sections, 6 theorems, 47 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The PDE Model and Known Stabilization Results
  3. Closed-loop Model Under Consideration
  4. Non-collocated Observer Design and Error Tracking
  5. Our Contributions
  6. Exponential Stability by Lyapunov Approach
  7. Well-posedness
  8. Proposed Model Reduction
  9. Conclusion& Future Work

Key Result

Lemma 1

The energy $E(t)$ in naturalenergy is dissipative, i.e.

Figures (1)

  • Figure 1: Solutions are observed to decay to zero exponentially fast.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 1 more