LQG-Balanced Truncation Low-Order Controller for Stabilization of Laminar Flows

  • Peter BennerEmail author
  • Jan Heiland
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 127)


Recent theoretical and simulation results have shown that Riccati based feedback can stabilize flows at moderate Reynolds numbers. We extend this established control setup by the method of LQG-balanced truncation. In view of practical implementation, we introduce a controller that bases only on outputs rather than on the full state of the system. Also, we provide a very low dimensional observer so that the control actuation can be computed in an online fashion.


Navier-Stokes equation model order reduction stabilization output feedback 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005)Google Scholar
  2. 2.
    Bänsch, E., Benner, P.: Stabilization of incompressible flow problems by Riccati-based feedback. In: Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S. (eds.) Constrained Optimization and Optimal Control for Partial Differential Equations, Basel, Switzerland. International Series of Numerical Mathematics, vol. 160, pp. 5–20. Birkhäuser (2012)Google Scholar
  3. 3.
    Bänsch, E., Benner, P., Saak, J., Weichelt, H.K.: Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flow. Preprint SPP1253-154, DFG-SPP1253 (2013)Google Scholar
  4. 4.
    Benner, P., Li, J.-R., Penzl, T.: Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer. Lin. Alg. Appl. 15(9), 755–777 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benner, P., Saak, J., Stoll, M., Weichelt, H.: Efficient solution of large-scale saddle point systems arising in Riccati-based boundary feedback stabilization of incompressible Stokes flow. SIAM J. Sci. Comput. 170, S150–S170 (2013)Google Scholar
  6. 6.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)Google Scholar
  7. 7.
    Gugercin, S., Stykel, T., Wyatt, S.: Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput. 35(5), B1010–B1033 (2013)Google Scholar
  8. 8.
    Heiland, J.: lqgbt-oseen – Python module for LQG-BT of linearized flow equations, v1.0 (2014),
  9. 9.
    Heinkenschloss, M., Sorensen, D.C., Sun, K.: Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations. SIAM J. Sci. Comput. 30(2), 1038–1063 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hinze, M.: Control of weakly conductive fluids by near wall Lorentz forces. SFB609- Preprint 19-2004, Technische Universität Dresden (2004)Google Scholar
  11. 11.
    Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Cont. Optim. 40(3), 925–946 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hooshyar, N., Hamersma, P.J., Mudde, R.F., van Ommen, J.R.: Intensified operation of slurry bubble columns using structured gas injection. Canadian J. Chem. Engrg. 88(4), 533–542 (2010)Google Scholar
  13. 13.
    Johnson, C., Rannacher, R., Boman, M.: Numerics and hydrodynamic stability: Toward error control in computational fluid dynamics. SIAM J. Numer. Anal. 32(4), 1058–1079 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jonckheere, E.A., Silverman, L.M.: A new set of invariants for linear systems – application to reduced order compensator design. IEEE Trans. Automat. Control 28, 953–964 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    King, R. (ed.): Active flow control. Papers contributed to the conference ‘Active flow control 2006, September 27-29. Springer, Berlin (2006)Google Scholar
  16. 16.
    King, R. (ed.): Active Flow Control II: Papers Contributed to the Conference ’Active Flow Control II 2010’, May 26-28. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Springer, Berlin (2010)Google Scholar
  17. 17.
    Locatelli, A.: Optimal Control: An Introduction. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  18. 18.
    Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: FFC: the FEniCS form compiler. In: Automated Solution of Differential Equations by the Finite Element Method, pp. 227–238. Springer, Berlin (2012)CrossRefGoogle Scholar
  19. 19.
    Möckel, J., Reis, T., Stykel, T.: Linear-quadratic gaussian balancing for model reduction of differential-algebraic systems. Internat. J. Control 84(10), 1627–1643 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mustafa, D., Glover, K.: Controller design by \(\mathcal{H}_\infty\)-balanced truncation. IEEE Trans. Automat. Control 36(6), 668–682 (1991)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Raymond, J.-P.: Local boundary feedback stabilization of the Navier-Stokes equations. In: Control Systems: Theory, Numerics and Applications, Proceedings of Science. SISSA, Rome, March 30-April 1 (2005),
  22. 22.
    Storkaas, E., Skogestad, S., Alstad, V.: Stabilization of desired flow regimes in pipelines. In: AIChE Annual Meeting (2001)Google Scholar
  23. 23.
    Stykel, T.: Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl. 415(2-3), 262–289 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using the finite element technique. Internat. J. Comput. & Fluids 1(1), 73–100 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tombs, M., Postlethwaite, I.: Truncated balanced realization of a stable non-minimal state-space system. Internat. J. Control 46(4), 1319–1330 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

Personalised recommendations