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Research Papers

Distributed Control of Two-Dimensional Navier–Stokes Equations in Fourier Spectral Simulations

[+] Author and Article Information
Behrooz Rahmani

Control Research Laboratory,
Department of Mechanical Engineering,
Yasouj University,
Yasouj 75914-353, Iran
e-mail: b_rahmani@yu.ac.ir

Amin Moosaie

Turbulence Research Laboratory,
Department of Mechanical Engineering,
Yasouj University,
Yasouj 75914-353, Iran

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 27, 2015; final manuscript received February 7, 2017; published online June 1, 2017. Assoc. Editor: Kevin Fite.

J. Dyn. Sys., Meas., Control 139(8), 081013 (Jun 01, 2017) (10 pages) Paper No: DS-15-1348; doi: 10.1115/1.4036070 History: Received July 27, 2015; Revised February 07, 2017

A method for distributed control of nonlinear flow equations is proposed. In this method, first, Takagi–Sugeno (T–S) fuzzy model is used to substitute the nonlinear partial differential equations (PDEs) governing the system by a set of linear PDEs, such that their fuzzy composition exactly recovers the original nonlinear equations. This is done to alleviate the mode-interaction phenomenon occurring in spectral treatment of nonlinear equations. Then, each of the so-obtained linear equations is converted to a set of ordinary differential equations (ODEs) using the fast Fourier transform (FFT) technique. Thus, the combination of T–S method and FFT technique leads to a number of ODEs for each grid point. For the stabilization of the dynamics of each grid point, the use is made of the parallel distributed compensation (PDC) method. The stability of the proposed control method is proved using the second Lyapunov theorem for fuzzy systems. In order to solve the nonlinear flow equation, a combination of FFT and Runge–Kutta methodologies is implemented. Simulation studies show the performance of the proposed method, for example, the smaller settling time and overshoot and also its relatively robustness with respect to the measurement noises.

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References

Figures

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Fig. 1

Response of uncontrolled Burgers equation to a sinusoidal initial state

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Fig. 2

Response of controlled Burgers equation to a random initial state for the Burgers equation

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Fig. 3

Control input designed for stabilization of Burgers equation

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Fig. 4

Presentation of the velocity change from function exp(sin(x)) to cos(x) for the Burgers equation

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Fig. 5

Control input designed for changing the velocity from function exp(sin(x)) to cos(x) for the Burgers equation

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Fig. 6

Presentation of the velocity change from function exp(sin(x)) to cos(x) for the Burgers perturbed equation

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Fig. 7

Presentation of the velocity change from function exp(sin(x)) to cos(x) for the Burgers equation with measurement white noise

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Fig. 8

Initial vorticity field for the control of 2D Navier–Stokes equations

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Fig. 9

Actual controlled vorticity field at t = 0.0001 s

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Fig. 10

Actual controlled vorticity field at t = 0.0002 s

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Fig. 11

Actual controlled vorticity field at t = 0.0005 s

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Fig. 12

Actual controlled vorticity field at t = 0.001 s

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Fig. 13

Actual controlled vorticity field at t = 0.01 s

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Fig. 14

The desired vorticity field

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Fig. 15

The error of vorticity field

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Fig. 16

Vorticity at point (x, y) = (π, π) versus time for stepwise time-dependent desired function

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Fig. 17

Vorticity at point (x, y) = (π, π) versus time for sine-type time-dependent desired function

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Fig. 18

Vorticity at point (x, y) = (π, π) versus time for uncertain Navier–Stokes equation and stepwise time-dependent desired function

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