Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T23:11:23.418Z Has data issue: false hasContentIssue false
  • English
  • Français

Empirical pseudo-balanced model reduction and feedback control of weakly nonlinear convection patterns

Published online by Cambridge University Press:  30 September 2010

A. C. OR  and
J. L. SPEYER
A. C. OR*
Affiliation:
Department of Aerospace and Mechanical Engineering, University of California, Los Angeles, CA 90095, USA
J. L. SPEYER
Affiliation:
Department of Aerospace and Mechanical Engineering, University of California, Los Angeles, CA 90095, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An empirical model reduction method is performed on nonlinear transient convection patterns near the threshold, using a pseudo-inverse-based projective method called the pseudo-balanced proper orthogonal decomposition (PBPOD). These transient patterns are large-scale amplitude/phase modulations in convection rolls, obtained by prescribing selected spatial input-shape functions. For the nonlinear convection patterns modelled, PBPOD appears to be very effective. Using the nonlinear front example, PBPOD is compared with other existing methods, such as POD and linearized BPOD. The limitations of the methods are discussed. Using a complex prescribed input, complex disturbances are generated and the outputs of the open-loop responses are compared between the original and the reduced-order models. The agreement is good. A feedback-control study is performed using the nonlinear front example. The controller is built by the pseudo-balanced reduced-order model, with a low-order nonlinear estimator. Closed-loop simulations show that the nonlinear travelling fronts can be effectively damped out by the feedback-control actions.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 662 , 10 November 2010 , pp. 36 - 65
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barbagallo, A., Sipp, D. & Schmid, P. 2009 Closed-loop control of an open cavity flow. Part 2. Reduced-order models based on balanced truncation. J. Fluid Mech. 641, 1–50.CrossRefGoogle Scholar
Chen, C. T. 1970 Introduction to Linear System Theory. Rinehart and Winston.Google Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1982 Hydrodynamic Stability. Cambridge Monographs on Mechanics and Applied Mechanics, Cambridge University Press.Google Scholar
Gottlieb, D. & Orszag, S. A. 1981 Numerical Analysis of Spectral Methods: Theory and Applications. SIAM.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1998 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Howle, L. E. 1997 Linear stability analysis of controlled Rayleigh–Bénard convection using shadowgraphic measurement. Phys. Fluids 9, 3111–3113.CrossRefGoogle Scholar
Lall, S., Marsden, J. E. & Glavaski, S. 2002 A subspace approach to balanced truncation for model reduction of nonlinear control systems. Intl J. Robust Nonlinear Control 12, 519–535.CrossRefGoogle Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2009 Reduced order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. (in press).CrossRefGoogle Scholar
Ma, Z., Rowley, C. W. & Tadmor, G. 2010 Snapshot-based balanced truncation for linear time-periodic systems. IEEE Trans. Autom. Control 55 (2), 469–473.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.CrossRefGoogle Scholar
Or, A. C., Cortelezzi, L. & Speyer, J. L. 2001 Robust feedback control of Rayleigh–Bénard convection. J. Fluid Mech. 437, 175–202.CrossRefGoogle Scholar
Or, A. C. & Speyer, J. L. 2003 Active suppression of finite-amplitude Rayleigh–Bénard convection. J. Fluid Mech. 483, 111–128.CrossRefGoogle Scholar
Or, A. C. & Speyer, J. L. 2005 Gain-scheduled controller for the suppression of convection at high Rayleigh number. Phys. Rev. E 71, 046302-1.CrossRefGoogle ScholarPubMed
Or, A. C. & Speyer, J. L. 2008 Model reduction of input-output dynamical systems by proper orthogonal decomposition. J. Guid. Control Dyn. 31, 322–328.CrossRefGoogle Scholar
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15, 997–1013.CrossRefGoogle Scholar
Scherpen, J. M. A. 1993 Balancing for nonlinear systems. Syst. Control Lett. 21, 143–153.CrossRefGoogle Scholar
Schlüter, A., Lortz, D. & Busse, F. H. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144.CrossRefGoogle Scholar
Tang, J. & Bau, H. H. 1998 Experiments on the stabilization of the no-motion state of a fluid layer heated from below and cooled above. J. Fluid Mech. 363, 153–171.CrossRefGoogle Scholar
Willcox, K. & Peraire, J. 2002 Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40, 2323–2330.CrossRefGoogle Scholar