Publication Date
8-2023
Advisor(s) - Committee Chair
Ahmet Ozkan Ozer, Mark Robinson, Mikhail Khenner, Dominic Lamphier
Degree Program
Department of Mathematics and Computer Science
Degree Type
Master of Science
Abstract
The one-dimensional Partial Differential Equation (PDE) model of the wave equation, which describes wave dynamics of a wide-range of controlled mechanical systems, with a state feedback controller at its boundary is known to have exponentially stable solutions. Moreover, it is also reported that the several model reductions of the wave equation by the standard Finite Differences and Finite Elements approximations suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of a uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. one common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, in which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough spectral estimation. The maximal decay rate is explicity provided as a function of the feedback gain and the filtering parameter. Unlike the existing literature, this approach allows finding the maximal decay rate in terms of the feedback gain and the filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results.
Piezoelectric materials exhibit electric responses to mechanical stress and mechanical responses to electric stress. A recently proposed PDE model, strongly coupling two wave equations, describes the longitudinal oscillations on the beam together with the total charge profile accumulated on the electrodes of the beam. The exponential stability of the solutions with two boundary feedback controllers is recently shown by proving an observability result for the control-free system, analogous to the existing results for the single wave equation case. Alternatively, the Lyapunov theory is cleverly adopted and the decay rate of the exponential stability of solutions is explicitly constructed in terms of the two feedback gains. Next, the Finite-Difference semi-discretization of the model is proposed. However, it is expected that this model reduction lacks exponential stability uniformly as the discretization parameter tends to zero. Therefor, the direct Fourier filtering technique, together with a thorough spectral estimation, is cleverly applied to retain the exponential decay rate, mimicking the PDE counterpart, uniformly with the Fourier-filtered solutions. It is quite fair to say that applying the Lyapunov technique to a discretized strongly-coupled controlled model is a totally new concept. Therefore, all the proofs are laid out in such a way that the maximal decay rate in terms of the filtering parameter and the optimal feedback gain are explicitly provided. Our results mimic the PDE counterpart.
Several numerical tests and simulations are provided to support our results.
Disciplines
Acoustics, Dynamics, and Controls | Applied Mathematics | Control Theory | Electro-Mechanical Systems | Engineering | Mechanical Engineering | Numerical Analysis and Computation | Partial Differential Equations | Physical Sciences and Mathematics
Recommended Citation
Sadeq Ibn Emran, md Rafi As, "A Robust Model Reduction for the Optimal Boundary Feedback Stabilization of Magnetizable Piezoelectric Beams" (2023). Masters Theses & Specialist Projects. Paper 3676.
https://digitalcommons.wku.edu/theses/3676
Included in
Acoustics, Dynamics, and Controls Commons, Control Theory Commons, Electro-Mechanical Systems Commons, Numerical Analysis and Computation Commons, Partial Differential Equations Commons