$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier–Stokes Equations

Dharmatti, Sheetal; Raymond, Jean‐Pierre; Thevenet, Laetitia

doi:10.1137/100782607

Cited by 30 publications

(12 citation statements)

References 11 publications

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“…However, in practice, the resulting system matrix A (∞) will be large scale and dense, which makes further computations barely feasible in terms of computation time and memory consumption. Also, a systematic error can easily be introduced by inaccurate computations with (20). Therefore, like in many similar applications, e.g., [2,25,28], we derive a suitable implicit implementation of the projection as it was proposed initially in [32].…”

Section: Projector-free Realization For Incompressible Flowsmentioning

confidence: 99%

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Robust output-feedback stabilization for incompressible flows using low-dimensional $$\mathcal {H}_{\infty }$$-controllers

2022

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Output-based controllers are known to be fragile with respect to model uncertainties. The standard $$\mathcal {H}_{\infty }$$ H ∞ -control theory provides a general approach to robust controller design based on the solution of the $$\mathcal {H}_{\infty }$$ H ∞ -Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples.

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Section: Projector-free Realization For Incompressible Flowsmentioning

confidence: 99%

“…The H ∞ -feedback control for the 2D incompressible Navier-Stokes equations has been treated in [20] from a theoretical perspective. Therein, unmodeled boundary inputs are considered and existence of optimal feedback solutions is shown with the help of Riccati equations.…”

Section: Introductionmentioning

confidence: 99%

Robust output-feedback stabilization for incompressible flows using low-dimensional $$\mathcal {H}_{\infty }$$-controllers

2022

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“…The H ∞ stabilization has been studied in [14], [12], [16], [33] by taking internal control for abstract parabolic systems and for Navier Stokes' equations in [9]. In [21] authors study the H ∞ boundary stabilization for Navier-Stokes equations. In the current work we extend results of [10] and [9] for H infinity stabilization of sabra shell model of turbulence.…”

Section: Introductionmentioning

confidence: 99%

Interior and H$^\infty$ feedback stabilization for sabra Shell model of turbulence

Biswas,

Dharmatti

2019

Preprint

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Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [18]. In this paper we have addressed stabilization problems related to sabra shell model of turbulence. We have studied internal stabilization via finite dimensional controller. Moreover we have also studied optimal robust control problem by solving an infinite time horizon max-min control problem. We first prove the H ∞ stabilization of the linearized system and charatarize it in terms of a feedback operator by solving an algebric ricatti equation. Finally we show that the control will asymptotically stabilize the nonlinear system.

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“…The Oseen linearization as a base for controller design for the incompressible Navier-Stokes equations has been thoroughly analyzed as, e.g., in [22,27,29] for Dirichlet boundary control in two and three dimensions and with mixed boundary conditions, in [1] in two and three dimensions including space discretizations and Riccati-based state feedback, in [16] in view of observer design, in [26] with extension methods for the Dirichlet control, and in the textbook [2] with fundamental results on spectral properties. As for robust control, the Oseen linearization has been investigated in [2,12] in view of the operator H ∞ -Riccati equations. Separately, the need and applicability for H ∞ -robust controllers to account for discretization or linearization errors in flow control setups have been discussed in [4,5].…”

mentioning

confidence: 99%

“…If v ∞ fulfills Assumption 3.1, then for γ > 1/ν, the Oseen operator A with Robin-type boundary conditions as defined in(11) with domain of definition D(A) as defined in(12) generates an analytic C 0 -semigroup on V 0 .…”

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confidence: 99%

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

Heiland

2022

MCRF

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<p style='text-indent:20px;'>Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system.</p>

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$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier–Stokes Equations

Cited by 30 publications

References 11 publications

Robust output-feedback stabilization for incompressible flows using low-dimensional $$\mathcal {H}_{\infty }$$-controllers

Robust output-feedback stabilization for incompressible flows using low-dimensional $$\mathcal {H}_{\infty }$$-controllers

Interior and H$^\infty$ feedback stabilization for sabra Shell model of turbulence

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

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