A new method for constructing low-dimensional reduced models of dissipative partial differential equations is proposed. The original PDE, u(t) = F(u), is projected onto a linear subspace spanned by the so-called Snaps...
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A new method for constructing low-dimensional reduced models of dissipative partial differential equations is proposed. The original PDE, u(t) = F(u), is projected onto a linear subspace spanned by the so-called Snapshot Archetypes, that are selected spatial profiles of u(x, t). The selection rule of the Snapshot Archetypes characterizes the method. Two different selection methods are proposed. We provide an "energetic" criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics. This approach is tested for several PDE systems such as the Kuramoto-Sivashinsky equation, the Arneodo-Elezgaray reaction-diffusion model, and the self-ignition dynamics of a coal stockpile. The latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the Snapshot Archetype reduced models with respect to parameter variations. (C) 2003 Elsevier Science B.V. All rights reserved.
Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two-dimensional (2-D) parabolic partial differential equation with time-varying dom...
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Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two-dimensional (2-D) parabolic partial differential equation with time-varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time-varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite-dimensional plant in terms of a linear time-varying system. This reduced-order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2-D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. (c) 2014 American Institute of Chemical Engineers AIChE J, 61: 494-502, 2015
Accurate solutions of the distributed parameter system (DPS) may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such....
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Accurate solutions of the distributed parameter system (DPS) may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such. developing low-order models of high fidelity is important if the objective is accurate control of the DPS. When an exact model (system of partial differential equations (PDEs)) of the system is known, this work presents a method to develop a low-order model that assures convergent and consistent projection to a finite space. The resulting low-order model can then be used to design finite dimensional controllers. When there is no available first-principle model of the system, this work introduces a novel system identification method, that combines the characteristics of singular value decomposition (SVD) and the Karhunen-Loeve (KL) expansion for DPS to arrive at a low-order model that captures the dominant characteristics of the system. Here as well, the final model form allows for the synthesis of finite order controllers. Two non-linear reactor systems that can be described by systems of PDEs are provided to demonstrate the model identification methods. Feedback controllers are then synthesized based on these models to demonstrate their performance for disturbance rejection. (C) 2002 Elsevier Science Ltd. All rights reserved.
The problem of ranking, in which the goal is to learn a real-valued ranking function that induces a ranking over an instance space, has recently gained increasing attention in machine learning. We study a learning alg...
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The problem of ranking, in which the goal is to learn a real-valued ranking function that induces a ranking over an instance space, has recently gained increasing attention in machine learning. We study a learning algorithm for ranking generated by a regularized scheme with an l(1) regularizer. The algorithm is formulated in a data dependent hypothesis space. Such a space is spanned by empirical eigenfunctions which are constructed by a Mercer kernel and the learning data. We establish the computations of empirical eigenfunctions and the representer theorem for the algorithm. Particularly, we provide an analysis of the sparsity and convergence rates for the algorithm. The results show that our algorithm produces both satisfactory convergence rates and sparse representations under a mild condition, especially without assuming sparsity in terms of any basis. (C) 2015 Elsevier Inc. All rights reserved.
The proper orthogonal decomposition (POD) (also called Karhunen-Loeve expansion) has been recently used in turbulence to derive optimally fast converging bases of spatial functions, leading to efficient finite truncat...
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The proper orthogonal decomposition (POD) (also called Karhunen-Loeve expansion) has been recently used in turbulence to derive optimally fast converging bases of spatial functions, leading to efficient finite truncations. Whether a finite number of these modes can be used in numerical simulations to derive an ''accurate'' finite set of ordinary differential equations, over a certain range of bifurcation parameter values, still remains an open question. It is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite-dimensional system. In most cases, this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric POD basis. The Kuramoto-Sivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view. However, the conclusion is general and can be applied to other equations, such as the Navier-Stokes equations, the complex Ginzburg-Landau equation, and others.
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