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Limitations of linear control of thermal convection in a porous medium
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10.1063/1.2221354
Hui Zhao1 and Haim H. Bau1,a)
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Affiliations:
1 Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
a) Author to whom correspondence should be addressed. Electronic mail: bau@seas.upenn.edu
Phys. Fluids 18, 074109 (2006)
/content/aip/journal/pof2/18/7/10.1063/1.2221354
http://aip.metastore.ingenta.com/content/aip/journal/pof2/18/7/10.1063/1.2221354
View: Figures

## Figures

FIG. 1.

The critical Rayleigh number (solid line) for the transition from the motionless to the motion state and the corresponding imaginary part of the largest eigenvalue (dashed line) are depicted as functions of the ad hoc proportional controller gain .

FIG. 2.

The range of Rayleigh numbers for which the controlled system is stable as a function of the Rayleigh number for which the controller was designed . The solid line depicts the design Rayleigh number. The dashed and dotted lines correspond, respectively, to the linear, quadratic Gaussian controller and the suboptimal robust controller . The regions of stability and instability are indicated in the figure with the letters and , respectively.

FIG. 3.

Contours of the controller gain associated with an actuator located at are depicted as a function of location (a) and is depicted as a function of (b). The controller is designed to operate at .

FIG. 4.

The smallest bound of the transfer function for which a steady solution of the Riccati equation exists as a function of the Rayleigh number.

FIG. 5.

The pseudospectra of the linear operator A of the uncontrolled system when . The contour lines correspond to , , 10, , and . The disk has a radius of 100.

FIG. 6.

The pseudospectra of the linear operator of the system controlled with an ad hoc proportional controller when . The contour lines correspond to , , 10, , and . The disk has a radius of 100.

FIG. 7.

The transient growth of the system as a function of time. Ad hoc proportional controller, , and .

FIG. 8.

The maximum transient growth as a function of the Rayleigh number for the various control strategies. The solid, dashed, and dashed-dotted lines correspond, respectively, to the ad hoc proportional (with gain ), quadratic-Gaussian , and suboptimal robust controllers.

FIG. 9.

The maximum transient growth as a function of the relative weight . The solid and dashed lines correspond, respectively, to the controller with a state estimator and the controller without a state estimator. .

FIG. 10.

The transfer function norms: (a) 2-norm, (b) ∞-norm as functions of the Rayleigh number. The solid and dashed lines correspond, respectively, to quadratic-Gaussian and suboptimal robust controllers.

FIG. 11.

The temperature at is depicted as a function of time. Ad hoc proportional controller, , and . The disturbance amplitude is, respectively, 0.07 and 0.08 in (a) and (b).

FIG. 12.

The critical disturbance amplitude defining the basin of attraction of the controlled state as a function of the Rayleigh number (a) and as a function of (b). Ad hoc proportional controller. .

FIG. 13.

The critical amplitude of the optimal disturbance as a function of the Rayleigh number Ra for the optimal controller.

FIG. 14.

The maximum eigenvalue of the system as a function of the Rayleigh number (Ra).

/content/aip/journal/pof2/18/7/10.1063/1.2221354
2006-07-31
2014-04-19

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