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286
Balanced model reduction via the proper orthogonal decomposition
 AIAA Journal
, 2002
"... A new method for performing a balanced reduction of a highorder linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshots is used to obtain lowrank, reducedrange approximationsto the system control ..."
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Cited by 132 (7 self)
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A new method for performing a balanced reduction of a highorder linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshots is used to obtain lowrank, reducedrange approximationsto the system controllability and observability grammiansin either the time or frequency domain.The approximationsare then used to obtain a balanced reducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized highorder system that models unsteady motion of a twodimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superior performance than those derived using a conventionalproper orthogonaldecomposition. Although further development is necessary, the concept also extends to nonlinear systems. W Nomenclature h = airfoil plunge displacement K = proper orthogonal decomposition (POD) kernel m = number of POD snapshots n = number of states in computational � uid dynamics (CFD) model nr = number of states in reducedordermodel R = correlation matrix T = matrix whose columns contain the balancing transformation vectors u; U = vector containing inputs for models, time and frequency domain Wc = controllabilitygrammian Wco = grammian product Wo = observability grammian x; X = aerodynamic state vector for CFD model, time and frequency domain xr = aerodynamic state vector for reducedordermodel y; Y = vector containing outputs of CFD model, time and frequency domain yr = vector containing outputs of reducedordermodel z = dual state vector for CFD model i = ith Hankel singular value = basis vector! = forcing frequency
A subspace approach to balanced truncation for model reduction of nonlinear control systems
 International Journal on Robust and Nonlinear Control
, 2002
"... of nonlinear control systems ..."
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DNSbased predictive control of turbulence: an optimal benchmark for feedback algorithms
, 1999
"... this paper describes and demonstrates one approach of determining such control strategies via optimal control theory and iterative direct numerical simulations. 2. Optimal and robust control in the predictive control framework 2.1. The seminal idea and an analogy to the game of chess The general i ..."
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Cited by 73 (7 self)
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this paper describes and demonstrates one approach of determining such control strategies via optimal control theory and iterative direct numerical simulations. 2. Optimal and robust control in the predictive control framework 2.1. The seminal idea and an analogy to the game of chess The general idea of the recedinghorizon predictive control setting (as formulated in continuous time) is shown in Figure 1. To put this approach into a more intuitive context, and to appreciate better the importance of the (somewhat mathematical) gradientbased optimization approach to the present problem, it is useful at the outset to compare and contrast the present approach to massivelyparallel bruteforce algorithms recently developed to play the game of chess. The parallels and the shortcomings of this analogy highlight well the problem at hand. t = 0 t = T ) Optimization of controls on horizon [0; T ]. . . . t = T a t = T a + T ) Optimization of controls on horizon [T a ; T a + T ]. . . . t = 2T a t = 2T a + T ) Optimization of controls on horizon [2T a ; 2T a + T ]. . . . t = 3T a : : : etc. Figure 1. The sequence of events in recedinghorizon predictive control. The heavy solid arrows indicate the flow advancement. The evolution of the "actual" flow response to several "test" distribution of controls is explored during the iterative flow prediction (dashed line) and adjoint computation (dotdashed line) stages, during which the control is optimized by a gradient algorithm. Once this iteration converges, the flow is "advanced" some portion Ta of the period T over which the control was optimized, and the optimization process is begun anew. The goal when playing chess is to capture the other player's king through an alternating series of discrete moves with the opponen...
Projectionbased approaches for model reduction of weakly nonlinear, timevarying systems
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
"... Abstract—The problem of automated macromodel generation is interesting from the viewpoint of systemlevel design because if small, accurate reducedorder models of system component blocks can be extracted, then much larger portions of a design, or more complicated systems, can be simulated or verifi ..."
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Cited by 59 (1 self)
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Abstract—The problem of automated macromodel generation is interesting from the viewpoint of systemlevel design because if small, accurate reducedorder models of system component blocks can be extracted, then much larger portions of a design, or more complicated systems, can be simulated or verified than if the analysis were to have to proceed at a detailed level. The prospect of generating the reduced model from a detailed analysis of component blocks is attractive because then the influence of secondorder device effects or parasitic components on the overall system performance can be assessed. In this way overly conservative design specifications can be avoided. This paper reports on experiences with extending model reduction techniques to nonlinear systems of differential–algebraic equations, specifically, systems representative of RF circuit components. The discussion proceeds from linear timevarying, to weakly nonlinear, to nonlinear timevarying analysis, relying generally on perturbational techniques to handle deviations from the linear timeinvariant case. The main intent is to explore which perturbational techniques work, which do not, and outline some problems that remain to be solved in developing robust, general nonlinear reduction methods. Index Terms—Circuit noise, circuit simulation, nonlinear systems, reducedorder systems, timevarying circuits. I.
Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations
 ICAM Report 990101, Virginia Tech; in
, 1999
"... Keywords: proper orthogonal decomposition, principal component analysis, KarhunenLo`eve expansion, heat equation, feedback control ..."
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Cited by 58 (6 self)
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Keywords: proper orthogonal decomposition, principal component analysis, KarhunenLo`eve expansion, heat equation, feedback control
Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor
, 1998
"... Proper orthogonal decomposition (which is also known as the Karhunen Lo`eve decomposition) is a reduction method that is used to obtain low dimensional dynamic models of distributed parameter systems. Roughly speaking, proper orthogonal decomposition (POD) is an optimal technique of finding a ba ..."
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Cited by 49 (5 self)
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Proper orthogonal decomposition (which is also known as the Karhunen Lo`eve decomposition) is a reduction method that is used to obtain low dimensional dynamic models of distributed parameter systems. Roughly speaking, proper orthogonal decomposition (POD) is an optimal technique of finding a basis which spans an ensemble of data, collected from an experiment or a numerical simulation of a dynamical system, in the sense that when these basis functions are used in a Galerkin procedure will yield a finite dimensional system with the smallest possible degrees of freedom. Thus the technique is well suited to treat optimal control and parameter estimation of distributed parameter systems. In this paper, the method is applied to analyze the complex flow phenomenon in a horizontal chemical vapor deposition (CVD) reactor. In particular, we show that POD can be used to efficiently approximate solutions to the compressible viscous flows coupled with the energy and the species equati...
Reduced order modeling of the Upper Tropical Pacific ocean model using proper orthogonal decomposition. Computers and Mathematics with Applications
, 2006
"... The proper orthogonal decomposition (POD) is shown an efficiently model reduction technique in simulating physical processes governed by partial differential equations. In this paper we make an initial effort to investigate problems related to POD reduced modeling of a largescale upper ocean circul ..."
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Cited by 43 (19 self)
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The proper orthogonal decomposition (POD) is shown an efficiently model reduction technique in simulating physical processes governed by partial differential equations. In this paper we make an initial effort to investigate problems related to POD reduced modeling of a largescale upper ocean circulation in the tropic Pacific domain. We constructed different POD models with different choices of snapshots and different number of POD basis functions. The results from these different POD models are compared with that of the original model. The main findings are: (1) the largescale seasonal variability of the tropic Pacific obtained by the original model can be captured well by a low dimensional system of order of 22, which is constructed by 20 snapshots and 7 leading POD basis functions. (2) RMS errors for the upper ocean layer thickness of the POD model of order of 22 is less than 1m that is less than 1from the POD model is around 0.99. (3) The modes that capture 0.99 energy are necessary to construct POD models in order to yield a high accuracy.
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...
Missing point estimation in models described by proper orthogonal decomposition
, 2007
"... This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used w ..."
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Cited by 42 (7 self)
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This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of datadependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25 % of the spatial grid points without compromising the accuracy of the reduced model.
Empirical model reduction of controlled nonlinear systems
 Proceedings of the IFAC World Congress
, 1999
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