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Control of moving boundary parabolic partial differential equations (PDEs) Open Access


Other title
Boundary control
PDE Order-reduction
Moving boundary PDEs
PDE observer design
PDE backstepping
Type of item
Degree grantor
University of Alberta
Author or creator
Izadi, Mojtaba
Supervisor and department
Dubljevic, Stevan (Department of Chemical and Materials Engineering)
Examining committee member and department
Koch, Bob (Department of Mechanical Engineering)
Prasad, Vinay (Department of Chemical and Materials Engineering)
Daoutidis, Prodromos (Department of Chemical Engineering and Materials Science, University of Minnesota)
Forbes, Fraser (Department of Chemical and Materials Engineering)
Department of Chemical and Materials Engineering
Process Control
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Mathematical models of many transport processes are in the forms given by parabolic partial differential equations (PDEs). There are phenomena which may cause changes in shape and material properties of the process domain resulting in a moving boundary parabolic PDE model of the process. The focus of this thesis is to develop two control methods for parabolic PDE systems with time-dependent spatial domain. The first approach uses the PDE backstepping tool for stabilization of a class of one-dimensional unstable parabolic PDEs. In this method, an integral transformation maps the PDE system to a suitably selected exponentially stable target system. The kernel of transformation is defined by the solution of the kernel PDE that is of higher-order in space. It is shown that the kernel PDE is well-posed and a numerical solution is provided with the error analysis to establish the accuracy. The stabilizing control law is shown in the form of state-feedback with the gain in terms of kernel function. In addition, the backstepping-based observer design for state estimation of parabolic PDEs with time-dependent spatial domain is provided for a collocated boundary measurement and actuation. Specifically, the PDE system that describes the observation error dynamics is also transformed to the exponentially stable target system. The exponential stability of the closed-loop system with observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system. Modal decomposition techniques have been extensively used for the order-reduction of dissipative systems. The second approach is the use of Karhunen-Loeve (KL) decomposition to find the empirical eigenfunctions of the solution of moving boundary PDE systems. A mapping functional is obtained, which relates the evolution of the solution of the parabolic PDE with time-varying domain to a fixed reference configuration, while preserving space invariant properties of the initial solution ensemble. Subsequently, a low dimensional set of empirical eigenfunctions on the fixed domain is found and is mapped on the original time-varying domain resulting in the basis for the construction of the reduced-order model of the parabolic PDE system with time-varying domain. These modes are used as the basis set of functions in the Galerkin's method to find a reduced-order model for the optimal control design and state observation.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
M. Izadi, S. Dubljevic, ``Order-reduction of parabolic PDEs with time-varying domain using empirical eigenfunctions,'' AIChE Journal, vol. 59, issue 11, 4142-4150, 2013

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