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Observer Based Stabilization of Partial Differential Equations Governing Strain Gradient Micro Beams

Edalatzadeh, Mohammad Sajjad | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 46884 (08)
  4. University: Sharif University of Technology
  5. Department: Mechanical Engineering
  6. Advisor(s): Alasti, Aria; Fotouhi Firouzabad, Morteza
  7. Abstract:
  8. In recent decades, the study of Nano and Micro Electromechanical Systems (NEMS and MEMS) as in micro sensors and actuators or in Atomic Force Microscopes (AFM) attracted considerable attention. Since an indispensable part of such systems is a nano or micro flexible beam, the inevitable undesirable vibrations of the flexible beam necessitate the implementation of vibration control strategies. Recently, some cutting edge experiments on statics and dynamics behavior of micro beams have been conducted that reveal some deviations from the prediction of classical theories, and hence this leads to development of strain gradient elasticity theory. In addition, the often-adopted finite dimensional approximation of the Partial Differential Equations (PDEs) suffers from spillover instability in case of unmodeled dynamics take part in the control system. For this reason, the functional analysis and semi-group theory is employed to deal with the control of the system in an infinite dimensional space. In this research, the observability and observer design problem of a PDE governing the strain gradient micro beams are studied. In summary, the exact observability of the system with different observation operators is investigated. In this regard, the spectral theory of operator semigroups is utilized and then the results are verified by adopting a common so-called energy multiplier method. Based on the only non-collocated boundary measurement that is substantiated to give an exactly observable system, an infinite dimensional observer is designed. Moreover, the input-output stability of the system, the semigroup generation property of the observer state operator, the admissibility of the observer control operators, the exponential stability of error dynamic and of closed loop system are proved. The last pace toward practical implementation of the observer is taken by studying the restriction of the observer to a finite dimensional subspace. Some significant results are achieved and a novel theorem has been stated by comparing these results to those obtained from traditional observers, which are designed based on the truncated finite dimensional system
  9. Keywords:
  10. Vibration Control ; Observability ; Observer ; Strain Gradient Elasticity ; Microbeam ; Infinite Dimension System ; Modal Decomposition

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