Backstepping boundary control of Navier-Stokes channel flow: a 3D extension

Cochran, Jennie; Vázquez, Rafael; Krstić, Miroslav

doi:10.1109/acc.2006.1655449

Cited by 26 publications

(33 citation statements)

References 13 publications

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“…CONCLUSIONS This paper presented the design of stabilizing boundary controllers for three different fluid systems. For more details about the system and its control law please refer to [4], [5], [6] for the vortex shedding, thermal convention loop and channel flow problems respectively.…”

Section: Control Lawmentioning

confidence: 99%

Control of Channel Flow Turbulence, Vortex Shedding, and Thermal Convection by Backstepping Boundary Control

Vázquez

Cochran

Aamo

et al. 2007

2007 American Control Conference

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Section: Control Lawmentioning

confidence: 99%

Control of Channel Flow Turbulence, Vortex Shedding, and Thermal Convection by Backstepping Boundary Control

Vázquez

Cochran

Aamo

et al. 2007

2007 American Control Conference

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“…Recently, the backstepping method for pde control, which has been used for several different applications [2], [13], [16], has been extended to turbulent flow control [17]. In a companion paper [8] we present a boundary control law that stabilizes a benchmark 3D linearized Navier-Stokes channel flow system. The derived velocity controller employs actuation in all three directions, but solely along one wall.…”

mentioning

confidence: 99%

“…Before presenting our new results, we start in Section II by reviewing the model of the channel flow, the linearization around the Poiseuille equilibrium profile, the transformation to Fourier space and lastly the control laws developed in [8] which stabilize the linearized system. We then continue to Section III where we derive explicit solutions to the control gain kernels for the case k x = 0.…”

mentioning

confidence: 99%

“…Instead, the gain kernels for these controllers are obtained by solving certain linear hyperbolic partial differential equations, which are precomputed offline. normal velocity −→ normal vorticity unstable stable In this paper we study the pdes derived in [8] that define the backstepping gain kernels. We focus first on the specific case of perterbations with no streamwise dependence.…”

mentioning

confidence: 99%

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Backstepping Boundary Control of Navier-Stokes Channel Flow: Explicit Gain Formulae in 3D

Cochran

Vázquez

Krstić

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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In a companion ACC '06 paper we present a boundary control law, designed using the backstepping method, that stabilizes the 3D Navier-Stokes system linearized around the much studied Poiseuille flow profile. This control law employs three controllers that actuate solely along one boundary. The controller that acts in the wall normal direction prepares the system for the backstepping method, while the other two controllers, which act in the streamwise and spanwise direction, are designed using backstepping to stabilize the system. They do so by decoupling the normal vorticity from the normal velocity and then stabilizing the normal velocity. Instead of solving Ricatti equations to find the kernels for the controller gain, the kernels are found by solving certain linear hyperbolic pdes offline. We study the pdes derived in the companion paper that define the gain kernels. We focus first on the specific case of perterbations with no streamwise dependence. This is equivalent to setting the streamwise wavenumber to zero. We derive explicit solutions to the gain kernels for this important case where transient growth is the largest. In addition, we solve a series of related pdes in order to find an approximate solution to the gain kernel pdes when the streamwise and spanwise wavenumbers are small.

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“…Then, only measurements of pressure and skin friction are required. A 3D channel flow, periodic in two directions, is also tractable, adding some refinements which include actuation of the spanwise velocity at the wall (see [9] for the new techniques and difficulties involved). All cited references consider the steady problem of stabilizing a given Poiseuille profile, therefore some modifications to account for the unsteady coefficients have to be done, in the same way the present paper extends the results of [42].…”

Section: 42mentioning

confidence: 99%

Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

Vázquez¹,

Trélat²,

Coron³

2008

Discrete &Amp; Continuous Dynamical Systems - B

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Abstract. We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control laws guaranteeing that the error between the state and the trajectory decays exponentially in L 2 , H 1 , and H 2 norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.

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Backstepping boundary control of Navier-Stokes channel flow: a 3D extension

Cited by 26 publications

References 13 publications

Control of Channel Flow Turbulence, Vortex Shedding, and Thermal Convection by Backstepping Boundary Control

Control of Channel Flow Turbulence, Vortex Shedding, and Thermal Convection by Backstepping Boundary Control

Backstepping Boundary Control of Navier-Stokes Channel Flow: Explicit Gain Formulae in 3D

Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

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