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56
TrustRegion Proper Orthogonal Decomposition for Flow Control
 Institute for Computer
, 2000
"... . The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial di#erential equations, e.g. fluid flows. It can also be used to develop reduced order control models. The essential is the computation of POD basis functions that r ..."
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Cited by 42 (2 self)
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. The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial di#erential equations, e.g. fluid flows. It can also be used to develop reduced order control models. The essential is the computation of POD basis functions that represent the influence of the control action on the system in order to get a suitable control model. We present an approach where the suitable reduced order model is derived successively and give global convergence results. Keywords: proper orthogonal decomposition, flow control, reduced order modeling, trust region methods, global convergence 1. Introduction. We present a robust reduced order method for the control of complex timedependent physical processes governed by partial di#erential equations (PDE). Such a control problem often is hard to solve because of the high order system that describes the state (a large number of (finite element) basis elements for every point in the time d...
Reduced Order Controllers for Spatially Distributed Systems via Proper Orthogonal Decomposition
, 1999
"... A method for reducing controllers for systems described by partial differential equations (PDEs) is presented. This approach differs from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale finite eleme ..."
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Cited by 29 (4 self)
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A method for reducing controllers for systems described by partial differential equations (PDEs) is presented. This approach differs from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale finite element approximation of the PDE controller onto low order bases that are computed using the proper orthogonal decomposition (POD). Two methods for constructing input collections for POD, and hence low order bases, are discussed and computational results are included. The first uses the method of snapshots found in POD literature. The second is a new idea that uses an integral representation of the feedback control law. Specifically, the kernels, or functional gains, are used as data for POD. A low order controller derived by applying the POD process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably.
Reduced Order Model Compensator Control of Species Transport in a CVD Reactor
 in a CVD reactor. Optimal Control Application & Methods, 21:143 – 160
, 1999
"... We propose the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high pressure chemical vapor deposition (HPCVD) reactor. In this paper, we present a proofofconcept computational implementation of this metho ..."
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Cited by 20 (3 self)
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We propose the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high pressure chemical vapor deposition (HPCVD) reactor. In this paper, we present a proofofconcept computational implementation of this method with a simplified growth example for IIIV layers in which we implement Dirichlet boundary control of a dilute Group III reactant transported by convection and diffusion to an absorbing substrate with no reactions. We implement the modelbased feedback control using a reduced order state estimator based on observations of the flux of reactant at the substrate center. This is precisely the type of measurements available with current sensing technology. We demonstrate that the reduced order state estimator or compensator system is capable of substantial control authority when applied to the full system. In principle, these ideas can be extended to more general HPCVD control situations by inc...
Reduced order controllers for Burgers’ equation with a nonlinear observer
 Appl. Math. Comput. Sci
"... A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers ’ equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the pas ..."
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Cited by 18 (2 self)
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A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers ’ equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal decomposition (POD). However, the test problem was the twodimensional heat equation, a problem in which the physics dominates the system in such a way that controller efficacy is difficult to generalize. Here, we additionally incorporate a nonlinear observer by including the nonlinear terms of the state equation in the differential equation for the compensator.
Proper orthogonal decomposition for reduced order modeling: 2D heat flow
 Proc. of IEEE Int. Conf. on Control Applications
, 2003
"... Abstract — Modeling issues of infinite dimensional systems is studied in this paper. Although the modeling problem has been solved to some extent, use of decomposition techniques still pose several difficulties. A prime one of this is the amount of data to be processed. Method of snapshots integrate ..."
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Cited by 12 (6 self)
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Abstract — Modeling issues of infinite dimensional systems is studied in this paper. Although the modeling problem has been solved to some extent, use of decomposition techniques still pose several difficulties. A prime one of this is the amount of data to be processed. Method of snapshots integrated with POD is a remedy. The second difficulty is the fact that the decomposition followed by a projection yields an autonomous set of finite dimensional ODEs that is not useful for developing a concise understanding of the input operator of the system. A numerical approach to handle this issue is presented in this paper. As the example, we study 2D heat flow problem. The results obtained confirm the theoretical claims of the paper and emphasize that the technique presented here is not only applicable to infinite dimensional linear systems but also to nonlinear ones. I.
On model reduction for control design for distributed parameter systems, Volume dedicated to
, 2003
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Compensator Control For Chemical Vapor Deposition Film Growth Using Reduced Order Design Models
 North Carolina State University
, 1999
"... We present a summary of investigations on the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high pressure chemical vapor deposition (HPCVD) reactor that includes multiple species and controls, gas phase re ..."
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Cited by 8 (3 self)
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We present a summary of investigations on the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high pressure chemical vapor deposition (HPCVD) reactor that includes multiple species and controls, gas phase reactions, and time dependent tracking signals that are consistent with pulsed vapor reactant inputs. Numerical implementation of the modelbased feedback control uses a reduced order state estimator, based on partial state observations of the fluxes of reactants at the substrate center, which can be achieved with current sensing technology. We demonstrate that the reduced order state estimator or compensator system is capable of substantial control authority when applied to the full system. # Corresponding author: H.T. Banks, Center For Research in Scientific Computation, Box 8205, NCSU, Raleigh NC 27695 Telephone, 9195153968; Fax, 9195151636; Email, htbanks@eos.ncsu.edu 1 1 Introducti...
Interpolating local models of POD using fuzzy decision mechanisms
 9th Mechatronics Forum International Conference, Aug. 30
, 2004
"... In aerospace applications Proper Orthogonal Decomposition (POD) is used to obtain a dynamical model of aerodynamic flow control systems from simulation or experimental data. For different external test inputs (with disjoint frequency contents) the models obtained via POD are different, i.e. these mo ..."
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Cited by 5 (4 self)
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In aerospace applications Proper Orthogonal Decomposition (POD) is used to obtain a dynamical model of aerodynamic flow control systems from simulation or experimental data. For different external test inputs (with disjoint frequency contents) the models obtained via POD are different, i.e. these models are valid locally in a certain narrow frequency band. As the external input gets rich in the frequency content the model is unable to generate a good estimate of the states. Therefore, interpolation of these local models is necessary to capture a model that works in the whole frequency range of interest. We study a fuzzy system to perform a smooth transition from one model to another, and realize the transition scheme in the frequency domain. We illustrate the results on the one dimensional Burgers equation. 1
Output Feedback Control of Parabolic PDE Systems with Input Constraints 1
"... This paper proposes a methodology for output feedback control of parabolic PDE systems with input constraints. Initially, Galerkin’s method is used for the derivation of a finitedimensional ODE system that captures the dominant dynamics of the PDE system. This ODE system is then used as the basis f ..."
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This paper proposes a methodology for output feedback control of parabolic PDE systems with input constraints. Initially, Galerkin’s method is used for the derivation of a finitedimensional ODE system that captures the dominant dynamics of the PDE system. This ODE system is then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded output feedback control laws that use only measurements of the outputs and provide, at the same time, an explicit characterization of the set of admissible control actuator locations that can be used to guarantee closedloop stability for a given initial condition. Precise conditions that guarantee stability of the constrained closedloop parabolic PDE system are provided. The proposed output feedback design is shown to recover, asymptotically, the set of stabilizing actuator locations obtained under state feedback, as the separation between the fast and slow eigenvalues of the spatial differential operator increases.
NONLINEAR MODEL REDUCTION USING GROUP PROPER ORTHOGONAL DECOMPOSITION
"... Abstract. We propose a new method to reduce the cost of computing nonlinear terms in projection based reduced order models with global basis functions. We develop this method by extending ideas from the group finite element (GFE) method to proper orthogonal decomposition (POD) and call it the group ..."
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Cited by 4 (0 self)
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Abstract. We propose a new method to reduce the cost of computing nonlinear terms in projection based reduced order models with global basis functions. We develop this method by extending ideas from the group finite element (GFE) method to proper orthogonal decomposition (POD) and call it the group POD method. Here, a scalar twodimensional Burgers ’ equation is used as a model problem for the group POD method. Numerical results show that group POD models of Burgers ’ equation are as accurate and are computationally more efficient than standard POD models of Burgers ’ equation.