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Parallel Implementation of LQG Balanced Truncation for Large-Scale Systems

  • Jose M. Badía
  • Peter Benner
  • Rafael Mayo
  • Enrique S. Quintana-Ortí
  • Gregorio Quintana-Ortí
  • Alfredo Remón
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

Model reduction of large-scale linear time-invariant systems is an ubiquitous task in control and simulation of complex dynamical processes. We discuss how LQG balanced truncation can be applied to reduce the order of large-scale control problems arising from the spatial discretization of time-dependent partial differential equations. Numerical examples on a parallel computer demonstrate the effectiveness of our approach.

Keywords

model reduction LQG balancing algebraic Riccati equation Newton’s method parallel algorithms 

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References

  1. 1.
    Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Badía, J.M., et al.: Parallel algorithms for balanced truncation model reduction of sparse systems. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds.) PARA 2004. LNCS, vol. 3732, pp. 267–275. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Badía, J.M., et al.: Parallel solution of large-scale and sparse generalized algebraic Riccati equations. In: Nagel, W.E., Walter, W.V., Lehner, W. (eds.) Euro-Par 2006. LNCS, vol. 4128, pp. 710–719. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Badía, J.M., et al.: Balanced truncation model reduction of large and sparse generalized linear systems. Technical Report CSC/06-04, Technical University of Chemnitz, Chemnitz, 09107 Chemnitz, Germany (2006)Google Scholar
  5. 5.
    Benner, P., Byers, R.: An exact line search method for solving generalized continuous-time algebraic Riccati equations. IEEE Trans. Automat. Control 43(1), 101–107 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benner, P., Mehrmann, V., Sorensen, D. (eds.): MFCS 1976. LNCS, vol. 45. Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Jonckheere, E., Silverman, L.: A new set of invariants for linear systems—application to reduced order compensator. IEEE Trans. Automat. Control AC-28, 953–964 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Automat. Control AC-13, 114–115 (1968)CrossRefGoogle Scholar
  9. 9.
    Lancaster, P., Rodman, L.: The Algebraic Riccati Equation. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  10. 10.
    Li, J.-R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Obinata, G., Anderson, B.: Model Reduction for Control System Design. Communications and Control Engineering Series. Springer, London (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Penzl, T.: A cyclic low rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401–1418 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Penzl, T.: Lyapack Users Guide. Technical Report SFB393/00-33, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, FRG (2000), available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html
  14. 14.
    Wachspress, E.: The ADI model problem, Available from the author (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jose M. Badía
    • 1
  • Peter Benner
    • 2
  • Rafael Mayo
    • 1
  • Enrique S. Quintana-Ortí
    • 1
  • Gregorio Quintana-Ortí
    • 1
  • Alfredo Remón
    • 1
  1. 1.Depto. de Ingeniería y Ciencia de ComputadoresUniversidad Jaume ICastellónSpain
  2. 2.Fakultät für MathematikTechnische Universität ChemnitzChemnitzGermany

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