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Reduced-order model based feedback control of the modified Hasegawa-Wakatani model
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10.1063/1.4796190
/content/aip/journal/pop/20/4/10.1063/1.4796190
http://aip.metastore.ingenta.com/content/aip/journal/pop/20/4/10.1063/1.4796190

Figures

Image of FIG. 1.
FIG. 1.

Schematic of the implementation of the full-state feedback control in the full linear (top) and full non linear (bottom) simulations. The entire state is first projected onto the unstable eigenvectors and the stable subspace of the balanced modes in order to compute the reduced-order state xr . The state is then multiplied by the gain K, computed based on the reduced-order model using LQR to obtain the control input .

Image of FIG. 2.
FIG. 2.

Schematic of the implementation of the observer-based feedback control in the linear (top) and nonlinear (bottom) simulations. The control input u and the sensor measurements y are used as inputs to the observer, which reconstruct the reduced-order state . This state is then multiplied by the gain Kr to obtain the control input u. Both the controller and the observer gain Kr and L are computed based on the reduced-order model.

Image of FIG. 3.
FIG. 3.

Actuator localized at the middle of the square plate and modeled as a distribution of the external potential that is added to the system. It is determined by the function , where and is a given parameter.

Image of FIG. 4.
FIG. 4.

(Left) The ion vorticity ζ and (right) density fluctuations n of the B-matrix defined in Eq. (14) . These two quantities are going to be the initial conditions of the nonlinear, full linear, and reduced model of the MHW equations.

Image of FIG. 5.
FIG. 5.

Representation of the two unstable eigenvectors of the linearized equations. The left part represents its real part, the right its imaginary part.

Image of FIG. 6.
FIG. 6.

Ion vorticity and density fluctuation (in color) of the full non linear MHW equations at three successive times with the B-matrix as the initial condition.

Image of FIG. 7.
FIG. 7.

The output correspond to the density fluctuation that occurs in the center of the square geometry with no control applied on the system.

Image of FIG. 8.
FIG. 8.

Error for balanced truncation ( ), balanced POD ( ), POD ( ), and upper and lower bound for the model reduction scheme.

Image of FIG. 9.
FIG. 9.

Full Linear model with 2 eigenvalues in the RHP.

Image of FIG. 10.
FIG. 10.

Full Linear model with 4 eigenvalues in the RHP.

Image of FIG. 11.
FIG. 11.

Full Linear model with 4 eigenvalues in the RHP: phase space plot.

Image of FIG. 12.
FIG. 12.

Full Linear model with 4 eigenvalues in the RHP: inside basin of attraction case.

Image of FIG. 13.
FIG. 13.

Sensors location.

Image of FIG. 14.
FIG. 14.

Output feedback: 4 RHP poles/Full density sensed.

Image of FIG. 15.
FIG. 15.

Output feedback: 4 RHP poles/4 density points sensed only.

Image of FIG. 16.
FIG. 16.

Nyquist diagram of the loop gain of the input sensitivity function for the unstable case with two right half plane eigenvalues.

Image of FIG. 17.
FIG. 17.

Nyquist diagram of the loop gain of the input sensitivity function for the unstable case with four right half plane eigenvalues.

Tables

Generic image for table
Table I.

Summary of the 3 systems that will be reduced then stabilized with only one actuator: for fixed α and μ, only κ is varied and obtain 3 different cases with 2, 4, or 8 right half plane (unstable) eigenvalues.

Generic image for table
Table II.

Summary of the 3 new reduced systems. r is the dimension of the stable reduced subsystem.

Generic image for table
Table III.

GM and PM deduced from the loop gain of the sensitivity function.

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/content/aip/journal/pop/20/4/10.1063/1.4796190
2013-04-02
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Reduced-order model based feedback control of the modified Hasegawa-Wakatani model
http://aip.metastore.ingenta.com/content/aip/journal/pop/20/4/10.1063/1.4796190
10.1063/1.4796190
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