Reduced order feedback synthesis for viscous incompressible flows

Ito, Kazufumi; Schroeter, Joel

doi:10.1016/s0895-7177(00)00237-5

Cited by 22 publications

(6 citation statements)

References 7 publications

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“…Furthermore, at least for systems with only quadratic nonlinearities -such as the Boussinesq equations -the reduced order model can be significantly less costly than classical discretization techniques such as the finite element method. However, none of the earlier examples of reduced order models for the unsteady incompressible NavierStokes and Boussinesq equations [7,10,13,23,24,[31][32][33][34] is endowed with practicable and rigorous error bounds.…”

Section: Introductionmentioning

confidence: 99%

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

Knezevic

Nguyen

Patera

2011

Math. Models Methods Appl. Sci.

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In this paper we present reduced basis approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold -to provide dimension reduction; an efficient POD-Greedy sampling method for identification of optimal and numerically stable approximationsto yield rapid convergence; accurate (Online) calculation of the solution-dependent stability factor by the Successive Constraint Method -to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the reduced basis approximation and associated outputs -to provide certainty in our predictions; and an Offline-Online computational decomposition strategy for our reduced basis approximation and associated error bound -to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional "complex" enclosure -a square with a small rectangle cut-out -parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the reduced basis approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

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Section: Introductionmentioning

confidence: 99%

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

Knezevic

Nguyen

Patera

2011

Math. Models Methods Appl. Sci.

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“…There are examples of rigorous reduced basis a posteriori error bounds for the steady Burgers' [37] and incompressible Navier-Stokes [28,36] equations; the new contribution of the current paper is treatment of the unsteady -paraboliccase. Although there are many examples of reduced order models for the unsteady incompressible Navier-Stokes equations [5,7,9,13,14,[17][18][19][20][21], none is endowed with rigorous a posteriori error bounds.…”

Section: Introductionmentioning

confidence: 99%

Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation

Nguyen

Rozza

Patera

2009

Calcolo

113

130

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“…We are able to construct an interpolation grid because the solutions to the SDRE continuously depend on the state. This technique is similar in spirit to gain scheduling and an optimal control two point boundary value problem described in [16]. We now proceed to develop the method mathematically.…”

Section: Interpolation Methodsmentioning

confidence: 99%

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

Banks¹,

Lewis²,

Tran³

2003

103

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State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

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Reduced order feedback synthesis for viscous incompressible flows

Cited by 22 publications

References 7 publications

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS

Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation

Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach

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